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</button> <ul id="toc-Quantum_information_processing-sublist" class="vector-toc-list"> <li id="toc-Quantum_information" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_information"> <div class="vector-toc-text"> 2.1 Quantum information </div> </a> <ul id="toc-Quantum_information-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unitary_operators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unitary_operators"> <div class="vector-toc-text"> 2.2 Unitary operators </div> </a> <ul id="toc-Unitary_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_parallelism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_parallelism"> <div class="vector-toc-text"> 2.3 Quantum parallelism </div> </a> <ul id="toc-Quantum_parallelism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_programming" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_programming"> <div class="vector-toc-text"> 2.4 Quantum programming </div> </a> <ul id="toc-Quantum_programming-sublist" class="vector-toc-list"> <li id="toc-Gate_array" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Gate_array"> <div class="vector-toc-text"> 2.4.1 Gate array </div> </a> <ul id="toc-Gate_array-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement-based_quantum_computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Measurement-based_quantum_computing"> <div class="vector-toc-text"> 2.4.2 Measurement-based quantum computing </div> </a> <ul id="toc-Measurement-based_quantum_computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adiabatic_quantum_computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Adiabatic_quantum_computing"> <div class="vector-toc-text"> 2.4.3 Adiabatic quantum computing </div> </a> <ul id="toc-Adiabatic_quantum_computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Neuromorphic_quantum_computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Neuromorphic_quantum_computing"> <div class="vector-toc-text"> 2.4.4 Neuromorphic quantum computing </div> </a> <ul id="toc-Neuromorphic_quantum_computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_quantum_computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Topological_quantum_computing"> <div class="vector-toc-text"> 2.4.5 Topological quantum computing </div> </a> <ul id="toc-Topological_quantum_computing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_Turing_machine" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantum_Turing_machine"> <div class="vector-toc-text"> 2.4.6 Quantum Turing machine </div> </a> <ul id="toc-Quantum_Turing_machine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noisy_intermediate-scale_quantum_computing" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Noisy_intermediate-scale_quantum_computing"> <div class="vector-toc-text"> 2.4.7 Noisy intermediate-scale quantum computing </div> </a> <ul id="toc-Noisy_intermediate-scale_quantum_computing-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_cryptography_and_cybersecurity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_cryptography_and_cybersecurity"> <div class="vector-toc-text"> 2.5 Quantum cryptography and cybersecurity </div> </a> <ul id="toc-Quantum_cryptography_and_cybersecurity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Communication" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Communication"> <div class="vector-toc-text"> 3 Communication </div> </a> <ul id="toc-Communication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algorithms"> <div class="vector-toc-text"> 4 Algorithms </div> </a> <button aria-controls="toc-Algorithms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Algorithms subsection </button> <ul id="toc-Algorithms-sublist" class="vector-toc-list"> <li id="toc-Simulation_of_quantum_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simulation_of_quantum_systems"> <div class="vector-toc-text"> 4.1 Simulation of quantum systems </div> </a> <ul id="toc-Simulation_of_quantum_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Post-quantum_cryptography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Post-quantum_cryptography"> <div class="vector-toc-text"> 4.2 Post-quantum cryptography </div> </a> <ul id="toc-Post-quantum_cryptography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Search_problems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Search_problems"> <div class="vector-toc-text"> 4.3 Search problems </div> </a> <ul id="toc-Search_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_annealing" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_annealing"> <div class="vector-toc-text"> 4.4 Quantum annealing </div> </a> <ul id="toc-Quantum_annealing-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Machine_learning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Machine_learning"> <div class="vector-toc-text"> 4.5 Machine learning </div> </a> <ul id="toc-Machine_learning-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Engineering" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Engineering"> <div class="vector-toc-text"> 5 Engineering </div> </a> <button aria-controls="toc-Engineering-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Engineering subsection </button> <ul id="toc-Engineering-sublist" class="vector-toc-list"> <li id="toc-Challenges" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Challenges"> <div class="vector-toc-text"> 5.1 Challenges </div> </a> <ul id="toc-Challenges-sublist" class="vector-toc-list"> <li id="toc-Decoherence" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Decoherence"> <div class="vector-toc-text"> 5.1.1 Decoherence </div> </a> <ul id="toc-Decoherence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_supremacy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_supremacy"> <div class="vector-toc-text"> 5.2 Quantum supremacy </div> </a> <ul id="toc-Quantum_supremacy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Skepticism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Skepticism"> <div class="vector-toc-text"> 5.3 Skepticism </div> </a> <ul id="toc-Skepticism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physical_realizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physical_realizations"> <div class="vector-toc-text"> 5.4 Physical realizations </div> </a> <ul id="toc-Physical_realizations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Potential_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Potential_applications"> <div class="vector-toc-text"> 6 Potential applications </div> </a> <ul id="toc-Potential_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> 7 Theory </div> </a> <button aria-controls="toc-Theory-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Theory subsection </button> <ul id="toc-Theory-sublist" class="vector-toc-list"> <li id="toc-Computability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computability"> <div class="vector-toc-text"> 7.1 Computability </div> </a> <ul id="toc-Computability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complexity"> <div class="vector-toc-text"> 7.2 Complexity </div> </a> <ul id="toc-Complexity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 8 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 9 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 10 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 11 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 12 Further reading </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further reading subsection </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Textbooks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Textbooks"> <div class="vector-toc-text"> 12.1 Textbooks </div> </a> <ul id="toc-Textbooks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Academic_papers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Academic_papers"> <div class="vector-toc-text"> 12.2 Academic papers </div> </a> <ul id="toc-Academic_papers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Quantum computing</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 36 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" > 36 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Kwantumberekening" title="Kwantumberekening – Afrikaans" lang="af" hreflang="af" data-title="Kwantumberekening" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%88%D8%B3%D8%A8%D8%A9_%D9%83%D9%85%D9%88%D9%85%D9%8A%D8%A9" title="حوسبة كمومية – Arabic" lang="ar" hreflang="ar" data-title="حوسبة كمومية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Computaci%C3%B3n_cu%C3%A1ntica" title="Computación cuántica – Asturian" lang="ast" hreflang="ast" data-title="Computación cuántica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8B%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%9F%E0%A6%BE%E0%A6%AE_%E0%A6%95%E0%A6%AE%E0%A7%8D%E0%A6%AA%E0%A6%BF%E0%A6%89%E0%A6%9F%E0%A6%BF%E0%A6%82" title="কোয়ান্টাম কম্পিউটিং – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম কম্পিউটিং" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Kvantno_ra%C4%8Dunarstvo" title="Kvantno računarstvo – Bosnian" lang="bs" hreflang="bs" data-title="Kvantno računarstvo" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca badge-Q70894304 mw-list-item" title=""><a href="https://ca.wikipedia.org/wiki/Computaci%C3%B3_qu%C3%A0ntica" title="Computació quàntica – Catalan" lang="ca" hreflang="ca" data-title="Computació quàntica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvantedatabehandling" title="Kvantedatabehandling – Danish" lang="da" hreflang="da" data-title="Kvantedatabehandling" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Quantenrechner" title="Quantenrechner – German" lang="de" hreflang="de" data-title="Quantenrechner" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el badge-Q70894304 mw-list-item" title=""><a href="https://el.wikipedia.org/wiki/%CE%9A%CE%B2%CE%B1%CE%BD%CF%84%CE%B9%CE%BA%CE%AE_%CF%80%CE%BB%CE%B7%CF%81%CE%BF%CF%86%CE%BF%CF%81%CE%B9%CE%BA%CE%AE" title="Κβαντική πληροφορική – Greek" lang="el" hreflang="el" data-title="Κβαντική πληροφορική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Computaci%C3%B3n_cu%C3%A1ntica" title="Computación cuántica – Spanish" lang="es" hreflang="es" data-title="Computación cuántica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo badge-Q70894304 mw-list-item" title=""><a href="https://eo.wikipedia.org/wiki/Kvantumkomputiko" title="Kvantumkomputiko – Esperanto" lang="eo" hreflang="eo" data-title="Kvantumkomputiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Konputazio_kuantiko" title="Konputazio kuantiko – Basque" lang="eu" hreflang="eu" data-title="Konputazio kuantiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B1%D8%A7%DB%8C%D8%A7%D9%86%D8%B4_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="رایانش کوانتومی – Persian" lang="fa" hreflang="fa" data-title="رایانش کوانتومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Informatique_quantique" title="Informatique quantique – French" lang="fr" hreflang="fr" data-title="Informatique quantique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%ADomhaireacht_chandamach" title="Ríomhaireacht chandamach – Irish" lang="ga" hreflang="ga" data-title="Ríomhaireacht chandamach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-ko badge-Q70894304 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90_%EC%BB%B4%ED%93%A8%ED%8C%85" title="양자 컴퓨팅 – Korean" lang="ko" hreflang="ko" data-title="양자 컴퓨팅" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Komputasi_kuantum" title="Komputasi kuantum – Indonesian" lang="id" hreflang="id" data-title="Komputasi kuantum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Ukucikiza_ngohoyana" title="Ukucikiza ngohoyana – Zulu" lang="zu" hreflang="zu" data-title="Ukucikiza ngohoyana" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target">IsiZulu</a></li><li class="interlanguage-link interwiki-it badge-Q70894304 mw-list-item" title=""><a href="https://it.wikipedia.org/wiki/Computazione_quantistica" title="Computazione quantistica – Italian" lang="it" hreflang="it" data-title="Computazione quantistica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%97%D7%A9%D7%95%D7%91_%D7%A7%D7%95%D7%95%D7%A0%D7%98%D7%99" title="מחשוב קוונטי – Hebrew" lang="he" hreflang="he" data-title="מחשוב קוונטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%BD%D1%82%D1%8B%D0%BD_%D0%BA%D0%BE%D0%BC%D0%BF%D1%8C%D1%8E%D1%82%D0%B5%D1%80" title="Квантын компьютер – Mongolian" lang="mn" hreflang="mn" data-title="Квантын компьютер" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Kwantum_rekenen" title="Kwantum rekenen – Dutch" lang="nl" hreflang="nl" data-title="Kwantum rekenen" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja badge-Q70894304 mw-list-item" title=""><a href="https://ja.wikipedia.org/wiki/%E9%87%8F%E5%AD%90%E3%82%B3%E3%83%B3%E3%83%94%E3%83%A5%E3%83%BC%E3%83%86%E3%82%A3%E3%83%B3%E3%82%B0" title="量子コンピューティング – Japanese" lang="ja" hreflang="ja" data-title="量子コンピューティング" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a 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id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Quantum_computers&redirect=no" class="mw-redirect" title="Quantum computers">Quantum computers</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Computer hardware technology that uses quantum mechanics</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bloch_sphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Bloch_sphere.svg/220px-Bloch_sphere.svg.png" decoding="async" width="220" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Bloch_sphere.svg/330px-Bloch_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Bloch_sphere.svg/440px-Bloch_sphere.svg.png 2x" data-file-width="238" data-file-height="252" /></a><figcaption><a href="/wiki/Bloch_sphere" title="Bloch sphere">Bloch sphere</a> representation of a qubit. The state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/259aaac5393082d769c30b0b58fd0fafe4032d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.251ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle }"> is a point on the surface of the sphere, partway between the poles, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }">.</figcaption></figure> A quantum computer is a <a href="/wiki/Computer" title="Computer">computer</a> that exploits <a href="/wiki/Quantum_mechanical" class="mw-redirect" title="Quantum mechanical">quantum mechanical</a> phenomena. On small scales, physical matter exhibits properties of <a href="/wiki/Wave-particle_duality" class="mw-redirect" title="Wave-particle duality">both particles and waves</a>, and quantum computing leverages this behavior using specialized hardware. <a href="/wiki/Classical_physics" title="Classical physics">Classical physics</a> cannot explain the operation of these quantum devices, and a scalable quantum computer could perform some calculations <a href="/wiki/Exponential_growth" title="Exponential growth">exponentially</a> faster<a href="#cite_note-1">[a]</a> than any modern "classical" computer. Theoretically a large-scale quantum computer could <a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">break some widely used encryption schemes</a> and aid physicists in performing <a href="/wiki/Quantum_simulator" title="Quantum simulator">physical simulations</a>; however, the current state of the art is largely experimental and impractical, with several obstacles to useful applications. The basic <a href="/wiki/Unit_of_information" class="mw-redirect" title="Unit of information">unit of information</a> in quantum computing, the <a href="/wiki/Qubit" title="Qubit">qubit</a> (or "quantum bit"), serves the same function as the <a href="/wiki/Bit" title="Bit">bit</a> in classical computing. However, unlike a classical bit, which can be in one of two states (a <a href="/wiki/Binary_number" title="Binary number">binary</a>), a qubit can exist in a <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a> of its two "basis" states, which loosely means that it is in both states simultaneously. When <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measuring</a> a qubit, the result is a <a href="/wiki/Born_rule" title="Born rule">probabilistic output</a> of a classical bit. If a quantum computer manipulates the qubit in a particular way, <a href="/wiki/Wave_interference" title="Wave interference">wave interference</a> effects can amplify the desired measurement results. The design of <a href="/wiki/Quantum_algorithms" class="mw-redirect" title="Quantum algorithms">quantum algorithms</a> involves creating procedures that allow a quantum computer to perform calculations efficiently and quickly. Quantum computers are not yet practical for real work. Physically engineering high-quality qubits has proven challenging. If a physical qubit is not sufficiently <a href="/wiki/Isolated_system" title="Isolated system">isolated</a> from its environment, it suffers from <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">quantum decoherence</a>, introducing <a href="/wiki/Noise_(signal_processing)" title="Noise (signal processing)">noise</a> into calculations. National governments have invested heavily in experimental research that aims to develop scalable qubits with longer coherence times and lower error rates. Example implementations include <a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">superconductors</a> (which isolate an <a href="/wiki/Electrical_current" class="mw-redirect" title="Electrical current">electrical current</a> by eliminating <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a>) and <a href="/wiki/Trapped_ion_quantum_computer" class="mw-redirect" title="Trapped ion quantum computer">ion traps</a> (which confine a single <a href="/wiki/Atom" title="Atom">atomic particle</a> using <a href="/wiki/Electromagnetic_fields" class="mw-redirect" title="Electromagnetic fields">electromagnetic fields</a>). In principle, a classical computer can solve the same computational problems as a quantum computer, given enough time. Quantum advantage comes in the form of <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a> rather than <a href="/wiki/Computability" title="Computability">computability</a>, and <a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">quantum complexity theory</a> shows that some quantum algorithms are exponentially more efficient than the best-known classical algorithms. A large-scale quantum computer could in theory solve computational problems unsolvable by a classical computer in any reasonable amount of time. This concept of extra ability has been called "<a href="/wiki/Quantum_supremacy" title="Quantum supremacy">quantum supremacy</a>". While such claims have drawn significant attention to the discipline, near-term practical use cases remain limited. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=1" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For a chronological guide, see <a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline of quantum computing and communication</a>.</div> For many years, the fields of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a> formed distinct academic communities.<a href="#cite_note-FOOTNOTEAaronson2013132-2">[1]</a> <a href="/wiki/Modern_quantum_theory" class="mw-redirect" title="Modern quantum theory">Modern quantum theory</a> developed in the 1920s to explain the <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a> observed at atomic scales,<a href="#cite_note-3">[2]</a> and <a href="/wiki/Digital_computers" class="mw-redirect" title="Digital computers">digital computers</a> emerged in the following decades to replace <a href="/wiki/Human_computers" class="mw-redirect" title="Human computers">human computers</a> for tedious calculations.<a href="#cite_note-4">[3]</a> Both disciplines had practical applications during <a href="/wiki/World_War_II" title="World War II">World War II</a>; computers played a major role in <a href="/wiki/World_War_II_cryptography" title="World War II cryptography">wartime cryptography</a>,<a href="#cite_note-5">[4]</a> and quantum physics was essential for <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a> used in the <a href="/wiki/Manhattan_Project" title="Manhattan Project">Manhattan Project</a>.<a href="#cite_note-6">[5]</a> As <a href="/wiki/Physicists" class="mw-redirect" title="Physicists">physicists</a> applied quantum mechanical models to computational problems and swapped digital <a href="/wiki/Bit" title="Bit">bits</a> for <a href="/wiki/Qubits" class="mw-redirect" title="Qubits">qubits</a>, the fields of quantum mechanics and computer science began to converge. In 1980, <a href="/wiki/Paul_Benioff" title="Paul Benioff">Paul Benioff</a> introduced the <a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">quantum Turing machine</a>, which uses quantum theory to describe a simplified computer.<a href="#cite_note-The_computer_as_a_physical_system-7">[6]</a> When digital computers became faster, physicists faced an <a href="/wiki/Exponential_time" class="mw-redirect" title="Exponential time">exponential</a> increase in overhead when <a href="/wiki/Simulating_quantum_dynamics" class="mw-redirect" title="Simulating quantum dynamics">simulating quantum dynamics</a>,<a href="#cite_note-8">[7]</a> prompting <a href="/wiki/Yuri_Manin" title="Yuri Manin">Yuri Manin</a> and <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> to independently suggest that hardware based on quantum phenomena might be more efficient for computer simulation.<a href="#cite_note-manin1980vychislimoe-9">[8]</a><a href="#cite_note-10">[9]</a><a href="#cite_note-FOOTNOTENielsenChuang2010214-11">[10]</a> In a 1984 paper, <a href="/wiki/Charles_H._Bennett_(physicist)" title="Charles H. Bennett (physicist)">Charles Bennett</a> and <a href="/wiki/Gilles_Brassard" title="Gilles Brassard">Gilles Brassard</a> applied quantum theory to <a href="/wiki/Cryptography" title="Cryptography">cryptography</a> protocols and demonstrated that quantum key distribution could enhance <a href="/wiki/Information_security" title="Information security">information security</a>.<a href="#cite_note-bb84-12">[11]</a><a href="#cite_note-personal-13">[12]</a> <a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a> then emerged for solving <a href="/wiki/Oracle_machine" title="Oracle machine">oracle problems</a>, such as <a href="/wiki/Deutsch%27s_algorithm" class="mw-redirect" title="Deutsch's algorithm">Deutsch's algorithm</a> in 1985,<a href="#cite_note-14">[13]</a> the <a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani algorithm</a> in 1993,<a href="#cite_note-15">[14]</a> and <a href="/wiki/Simon%27s_algorithm" class="mw-redirect" title="Simon's algorithm">Simon's algorithm</a> in 1994.<a href="#cite_note-16">[15]</a> These algorithms did not solve practical problems, but demonstrated mathematically that one could gain more information by querying a <a href="/wiki/Black_box" title="Black box">black box</a> with a quantum state in <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a>, sometimes referred to as quantum parallelism.<a href="#cite_note-FOOTNOTENielsenChuang201030-32-17">[16]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png/200px-Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png" decoding="async" width="200" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png/300px-Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png/400px-Peter_Shor_2017_Dirac_Medal_Award_Ceremony.png 2x" data-file-width="759" data-file-height="977" /></a><figcaption><a href="/wiki/Peter_Shor" title="Peter Shor">Peter Shor</a> (pictured here in 2017) showed in 1994 that a scalable quantum computer would be able to break <a href="/wiki/RSA_encryption" class="mw-redirect" title="RSA encryption">RSA encryption</a>.</figcaption></figure> <a href="/wiki/Peter_Shor" title="Peter Shor">Peter Shor</a> built on these results with <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">his 1994 algorithm</a> for breaking the widely used <a href="/wiki/RSA_(cryptosystem)" title="RSA (cryptosystem)">RSA</a> and <a href="/wiki/Diffie%E2%80%93Hellman" class="mw-redirect" title="Diffie–Hellman">Diffie–Hellman</a> encryption protocols,<a href="#cite_note-FOOTNOTEShor1994-18">[17]</a> which drew significant attention to the field of quantum computing. In 1996, <a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's algorithm</a> established a quantum speedup for the widely applicable <a href="/wiki/Unstructured_data" title="Unstructured data">unstructured</a> search problem.<a href="#cite_note-19">[18]</a><a href="#cite_note-FOOTNOTENielsenChuang20107-20">[19]</a> The same year, <a href="/wiki/Seth_Lloyd" title="Seth Lloyd">Seth Lloyd</a> proved that quantum computers could simulate quantum systems without the exponential overhead present in classical simulations,<a href="#cite_note-273.5278.1073-21">[20]</a> validating Feynman's 1982 conjecture.<a href="#cite_note-22">[21]</a> Over the years, <a href="/wiki/Experimental_physics" title="Experimental physics">experimentalists</a> have constructed small-scale quantum computers using <a href="/wiki/Trapped_ion_quantum_computer" class="mw-redirect" title="Trapped ion quantum computer">trapped ions</a> and superconductors.<a href="#cite_note-FOOTNOTEGrumblingHorowitz2019164–169-23">[22]</a> In 1998, a two-qubit quantum computer demonstrated the feasibility of the technology,<a href="#cite_note-24">[23]</a><a href="#cite_note-25">[24]</a> and subsequent experiments have increased the number of qubits and reduced error rates.<a href="#cite_note-FOOTNOTEGrumblingHorowitz2019164–169-23">[22]</a> In 2019, <a href="/wiki/Google_AI" title="Google AI">Google AI</a> and <a href="/wiki/NASA" title="NASA">NASA</a> announced that they had achieved <a href="/wiki/Quantum_supremacy" title="Quantum supremacy">quantum supremacy</a> with a 54-qubit machine, performing a computation that is impossible for any classical computer.<a href="#cite_note-26">[25]</a><a href="#cite_note-1910.11333-27">[26]</a><a href="#cite_note-28">[27]</a> However, the validity of this claim is still being actively researched.<a href="#cite_note-29">[28]</a><a href="#cite_note-30">[29]</a> In December 2023, physicists, for the first time, reported the entanglement of individual molecules, which may have significant applications in quantum computing.<a href="#cite_note-PHYS-20231207-31">[30]</a> <div class="mw-heading mw-heading2"><h2 id="Quantum_information_processing">Quantum information processing</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=2" title="Edit section: Quantum information processing">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></div> <a href="/wiki/Computer_engineer" class="mw-redirect" title="Computer engineer">Computer engineers</a> typically describe a <a href="/wiki/Modern_computer" class="mw-redirect" title="Modern computer">modern computer</a>'s operation in terms of <a href="/wiki/Classical_electrodynamics" class="mw-redirect" title="Classical electrodynamics">classical electrodynamics</a>. Within these "classical" computers, some components (such as <a href="/wiki/Semiconductors" class="mw-redirect" title="Semiconductors">semiconductors</a> and <a href="/wiki/Random_number_generators" class="mw-redirect" title="Random number generators">random number generators</a>) may rely on quantum behavior, but these components are not <a href="/wiki/Isolated_system" title="Isolated system">isolated</a> from their environment, so any <a href="/wiki/Quantum_information" title="Quantum information">quantum information</a> quickly <a href="/wiki/Quantum_decoherence" title="Quantum decoherence">decoheres</a>. While <a href="/wiki/Programmers" class="mw-redirect" title="Programmers">programmers</a> may depend on <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a> when designing a <a href="/wiki/Randomized_algorithm" title="Randomized algorithm">randomized algorithm</a>, quantum mechanical notions like superposition and <a href="/wiki/Quantum_interference" class="mw-redirect" title="Quantum interference">interference</a> are largely irrelevant for <a href="/wiki/Program_analysis" title="Program analysis">program analysis</a>. <a href="/wiki/Quantum_program" class="mw-redirect" title="Quantum program">Quantum programs</a>, in contrast, rely on precise control of <a href="/wiki/Quantum_coherence" class="mw-redirect" title="Quantum coherence">coherent</a> quantum systems. Physicists <a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">describe these systems mathematically</a> using <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>. <a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> model <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitudes</a>, <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> model <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a>, and <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> model the operations that can be performed on these states. Programming a quantum computer is then a matter of <a href="/wiki/Function_composition_(computer_science)" title="Function composition (computer science)">composing</a> operations in such a way that the resulting program computes a useful result in theory and is implementable in practice. As physicist <a href="/wiki/Charles_H._Bennett_(physicist)" title="Charles H. Bennett (physicist)">Charlie Bennett</a> describes the relationship between quantum and classical computers,<a href="#cite_note-32">[31]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">A classical computer is a quantum computer ... so we shouldn't be asking about "where do quantum speedups come from?" We should say, "well, all computers are quantum. ... Where do classical slowdowns come from?"</blockquote> <div class="mw-heading mw-heading3"><h3 id="Quantum_information">Quantum information</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=3" title="Edit section: Quantum information">edit</a>]</div> Just as the bit is the basic concept of classical information theory, the <a href="/wiki/Qubit" title="Qubit">qubit</a> is the fundamental unit of <a href="/wiki/Quantum_information" title="Quantum information">quantum information</a>. The same term qubit is used to refer to an abstract mathematical model and to any physical system that is represented by that model. A classical bit, by definition, exists in either of two physical states, which can be denoted 0 and 1. A qubit is also described by a state, and two states often written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> serve as the quantum counterparts of the classical states 0 and 1. However, the quantum states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> belong to a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, meaning that they can be multiplied by constants and added together, and the result is again a valid quantum state. Such a combination is known as a superposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }">.<a href="#cite_note-FOOTNOTENielsenChuang201013-33">[32]</a><a href="#cite_note-FOOTNOTEMermin200717-34">[33]</a> A two-dimensional <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> mathematically represents a qubit state. Physicists typically use <a href="/wiki/Dirac_notation" class="mw-redirect" title="Dirac notation">Dirac notation</a> for quantum mechanical <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"> 'ket <a href="/wiki/Psi_(Greek)" title="Psi (Greek)">psi</a>' for a vector labeled <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> . Because a qubit is a two-state system, any qubit state takes the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha |0\rangle +\beta |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha |0\rangle +\beta |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10ab241f89535b2255509d41f39bfbbed35830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.088ex; height:2.843ex;" alt="{\displaystyle \alpha |0\rangle +\beta |1\rangle }"> , where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> are the standard basis states,<a href="#cite_note-36">[b]</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> are the <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitudes</a>, which are in general <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>.<a href="#cite_note-FOOTNOTEMermin200717-34">[33]</a> If either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> is zero, the qubit is effectively a classical bit; when both are nonzero, the qubit is in superposition. Such a <a href="/wiki/Quantum_state_vector" class="mw-redirect" title="Quantum state vector">quantum state vector</a> acts similarly to a (classical) <a href="/wiki/Probability_vector" title="Probability vector">probability vector</a>, with one key difference: unlike probabilities, probability amplitudes are not necessarily positive numbers.<a href="#cite_note-FOOTNOTEAaronson2013110-37">[35]</a> Negative amplitudes allow for destructive wave interference. When a qubit is <a href="/wiki/Quantum_measurement" class="mw-redirect" title="Quantum measurement">measured</a> in the <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a>, the result is a classical bit. The <a href="/wiki/Born_rule" title="Born rule">Born rule</a> describes the <a href="/wiki/Norm-squared" class="mw-redirect" title="Norm-squared">norm-squared</a> correspondence between amplitudes and probabilities—when measuring a qubit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha |0\rangle +\beta |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha |0\rangle +\beta |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10ab241f89535b2255509d41f39bfbbed35830b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.088ex; height:2.843ex;" alt="{\displaystyle \alpha |0\rangle +\beta |1\rangle }">, the state <a href="/wiki/Wave_function_collapse" title="Wave function collapse">collapses</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa34618537661f2d4d710cc26e8afe891f50f7b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.836ex; height:3.343ex;" alt="{\displaystyle |\alpha |^{2}}">, or to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\beta |^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\beta |^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4caecb560883af3b9c0c3a1d0e13aae75f121d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.68ex; height:3.343ex;" alt="{\displaystyle |\beta |^{2}}">. Any valid qubit state has coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18cd7473cdb894839d10852890517b1fb687c73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.617ex; height:3.343ex;" alt="{\displaystyle |\alpha |^{2}+|\beta |^{2}=1}">. As an example, measuring the qubit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d1003ac2c368cb0f7535b4902062b29af36df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.115ex; height:3.176ex;" alt="{\displaystyle 1/{\sqrt {2}}|0\rangle +1/{\sqrt {2}}|1\rangle }"> would produce either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed066a3ad158da0ad6d6a421a606b1c8a35eb95b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |0\rangle }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f53021ca18e77477ee5bd3c1523e5830189ec5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |1\rangle }"> with equal probability. Each additional qubit doubles the <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension</a> of the <a href="/wiki/State_space_(physics)" class="mw-redirect" title="State space (physics)">state space</a>.<a href="#cite_note-FOOTNOTEMermin200718-35">[34]</a> As an example, the vector <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/√2⁠|00⟩ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/√2⁠|01⟩ represents a two-qubit state, a <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of the qubit |0⟩ with the qubit <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/√2⁠|0⟩ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/√2⁠|1⟩. This vector inhabits a four-dimensional <a href="/wiki/Vector_space" title="Vector space">vector space</a> spanned by the basis vectors |00⟩, |01⟩, |10⟩, and |11⟩. The <a href="/wiki/Bell_state" title="Bell state">Bell state</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/√2⁠|00⟩ + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠1/√2⁠|11⟩ is impossible to decompose into the tensor product of two individual qubits—the two qubits are <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entangled</a> because their probability amplitudes are <a href="/wiki/Correlated" class="mw-redirect" title="Correlated">correlated</a>. In general, the vector space for an n-qubit system is 2n-dimensional, and this makes it challenging for a classical computer to simulate a quantum one: representing a 100-qubit system requires storing 2100 classical values. <div class="mw-heading mw-heading3"><h3 id="Unitary_operators">Unitary operators</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=4" title="Edit section: Unitary operators">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Unitarity_(physics)" title="Unitarity (physics)">Unitarity (physics)</a></div> The state of this one-qubit <a href="/wiki/Quantum_memory" title="Quantum memory">quantum memory</a> can be manipulated by applying <a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gates</a>, analogous to how classical memory can be manipulated with <a href="/wiki/Logic_gate" title="Logic gate">classical logic gates</a>. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d92d08efa8ccef15f7593ccfdb140edef506d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.192ex; height:6.176ex;" alt="{\displaystyle X:={\begin{pmatrix}0&1\\1&0\end{pmatrix}}.}"> Mathematically, the application of such a logic gate to a quantum state vector is modelled with <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X|0\rangle =|1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X|0\rangle =|1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab196442d13e4f433213e12d3fe0f14dd94088e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.506ex; height:2.843ex;" alt="{\displaystyle X|0\rangle =|1\rangle }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X|1\rangle =|0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X|1\rangle =|0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753cd0153c35b8d10551bf7fd9da25fb274edb6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.506ex; height:2.843ex;" alt="{\displaystyle X|1\rangle =|0\rangle }">.</dd></dl> The mathematics of single qubit gates can be extended to operate on multi-qubit quantum memories in two important ways. One way is simply to select a qubit and apply that gate to the target qubit while leaving the remainder of the memory unaffected. Another way is to apply the gate to its target only if another part of the memory is in a desired state. These two choices can be illustrated using another example. The possible states of a two-qubit quantum memory are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>01</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>10</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>11</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55c7ee07f3a61799148cc67ebc11b82df3e3ca2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:65.126ex; height:12.509ex;" alt="{\displaystyle |00\rangle :={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}};\quad |01\rangle :={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}};\quad |10\rangle :={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}};\quad |11\rangle :={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}.}"> The <a href="/wiki/Controlled_NOT_gate" title="Controlled NOT gate">controlled NOT (CNOT)</a> gate can then be represented using the following matrix: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {CNOT} :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>CNOT</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CNOT} :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17fcf6a57b6d3b886d9d5073032b1e7695518aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:27.735ex; height:12.509ex;" alt="{\displaystyle \operatorname {CNOT} :={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}.}"> As a mathematical consequence of this definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {CNOT} |00\rangle =|00\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>CNOT</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>00</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {CNOT} |00\rangle =|00\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0a5336f1e26a22df63063316d4b0dbaf266328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.146ex; height:2.843ex;" alt="{\textstyle \operatorname {CNOT} |00\rangle =|00\rangle }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {CNOT} |01\rangle =|01\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>CNOT</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>01</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>01</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {CNOT} |01\rangle =|01\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b89c86631eaaf54086afc0199de442765d8aa72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.146ex; height:2.843ex;" alt="{\textstyle \operatorname {CNOT} |01\rangle =|01\rangle }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {CNOT} |10\rangle =|11\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>CNOT</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>10</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>11</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {CNOT} |10\rangle =|11\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41135fc98eb537e37245efa719378d9f726f5754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.146ex; height:2.843ex;" alt="{\textstyle \operatorname {CNOT} |10\rangle =|11\rangle }">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {CNOT} |11\rangle =|10\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>CNOT</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>11</mn> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>10</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \operatorname {CNOT} |11\rangle =|10\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5dd1790f487ed75a103c6d09e4aa61d6135bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.146ex; height:2.843ex;" alt="{\textstyle \operatorname {CNOT} |11\rangle =|10\rangle }">. In other words, the CNOT applies a NOT gate (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d80c41192705e1a6c6de1d65e16d7f70fbac391" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\textstyle X}"> from before) to the second qubit if and only if the first qubit is in the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e02220355a0a3bf0dfe8884b3023a41b86f6cc14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\textstyle |1\rangle }">. If the first qubit is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |0\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55f385865bf20d44afef7add01ed2679901e9c4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\textstyle |0\rangle }">, nothing is done to either qubit. In summary, quantum computation can be described as a network of quantum logic gates and measurements. However, any <a href="/wiki/Deferred_measurement_principle" title="Deferred measurement principle">measurement can be deferred</a> to the end of quantum computation, though this deferment may come at a computational cost, so most <a href="/wiki/Quantum_circuit" title="Quantum circuit">quantum circuits</a> depict a network consisting only of quantum logic gates and no measurements. <div class="mw-heading mw-heading3"><h3 id="Quantum_parallelism">Quantum parallelism</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=5" title="Edit section: Quantum parallelism">edit</a>]</div> Quantum parallelism is the heuristic that quantum computers can be thought of as evaluating a function for multiple input values simultaneously. This can be achieved by preparing a quantum system in a superposition of input states and applying a unitary transformation that encodes the function to be evaluated. The resulting state encodes the function's output values for all input values in the superposition, allowing for the computation of multiple outputs simultaneously. This property is key to the speedup of many quantum algorithms. However, "parallelism" in this sense is insufficient to speed up a computation, because the measurement at the end of the computation gives only one value. To be useful, a quantum algorithm must also incorporate some other conceptual ingredient.<a href="#cite_note-FOOTNOTENielsenChuang201030–32-38">[36]</a><a href="#cite_note-FOOTNOTEMermin200738–39-39">[37]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_programming">Quantum programming</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=6" title="Edit section: Quantum programming">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></div> There are a number of <a href="/wiki/Model_of_computation" title="Model of computation">models of computation</a> for quantum computing, distinguished by the basic elements in which the computation is decomposed. <div class="mw-heading mw-heading4"><h4 id="Gate_array">Gate array </h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=7" title="Edit section: Gate array">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Quantum_Toffoli_Gate_Implementation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Quantum_Toffoli_Gate_Implementation.svg/250px-Quantum_Toffoli_Gate_Implementation.svg.png" decoding="async" width="250" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Quantum_Toffoli_Gate_Implementation.svg/375px-Quantum_Toffoli_Gate_Implementation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Quantum_Toffoli_Gate_Implementation.svg/500px-Quantum_Toffoli_Gate_Implementation.svg.png 2x" data-file-width="311" data-file-height="82" /></a><figcaption>A quantum circuit diagram implementing a <a href="/wiki/Toffoli_gate" title="Toffoli gate">Toffoli gate</a> from <a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">more primitive gates</a></figcaption></figure> A <a href="/wiki/Quantum_circuit" title="Quantum circuit">quantum gate array</a> decomposes computation into a sequence of few-qubit <a href="/wiki/Quantum_gate" class="mw-redirect" title="Quantum gate">quantum gates</a>. A quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements. Any quantum computation (which is, in the above formalism, any <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{n}\times 2^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{n}\times 2^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224a4a2c00116d57f7d93bd1116d1518837f1c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.602ex; height:2.343ex;" alt="{\displaystyle 2^{n}\times 2^{n}}"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a <a href="/wiki/Quantum_logic_gate#Universal_quantum_gates" title="Quantum logic gate">universal gate set</a>, since a computer that can run such circuits is a <a href="/wiki/Universal_quantum_computer" class="mw-redirect" title="Universal quantum computer">universal quantum computer</a>. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the <a href="/wiki/Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay-Kitaev theorem</a>. Implementation of Boolean functions using the few-qubit quantum gates is presented here.<a href="#cite_note-40">[38]</a> <div class="mw-heading mw-heading4"><h4 id="Measurement-based_quantum_computing">Measurement-based quantum computing</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=8" title="Edit section: Measurement-based quantum computing">edit</a>]</div> A <a href="/wiki/Measurement-based_quantum_computer" class="mw-redirect" title="Measurement-based quantum computer">measurement-based quantum computer</a> decomposes computation into a sequence of <a href="/wiki/Bell_state#Bell_state_measurement" title="Bell state">Bell state measurements</a> and single-qubit <a href="/wiki/Quantum_gate" class="mw-redirect" title="Quantum gate">quantum gates</a> applied to a highly entangled initial state (a <a href="/wiki/Cluster_state" title="Cluster state">cluster state</a>), using a technique called <a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a>. <div class="mw-heading mw-heading4"><h4 id="Adiabatic_quantum_computing">Adiabatic quantum computing</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=9" title="Edit section: Adiabatic quantum computing">edit</a>]</div> An <a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">adiabatic quantum computer</a>, based on <a href="/wiki/Quantum_annealing" title="Quantum annealing">quantum annealing</a>, decomposes computation into a slow continuous transformation of an initial <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> into a final Hamiltonian, whose ground states contain the solution.<a href="#cite_note-Das_2008_1061–1081-41">[39]</a> <div class="mw-heading mw-heading4"><h4 id="Neuromorphic_quantum_computing">Neuromorphic quantum computing</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=10" title="Edit section: Neuromorphic quantum computing">edit</a>]</div> Neuromorphic quantum computing (abbreviated as ‘n.quantum computing’) is an unconventional computing type of computing that uses <a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">neuromorphic computing</a> to perform quantum operations. It was suggested that quantum algorithms, which are algorithms that run on a realistic model of quantum computation, can be computed equally efficiently with neuromorphic quantum computing. Both, traditional quantum computing and neuromorphic quantum computing are physics-based unconventional computing approaches to computations and do not follow the <a href="/wiki/Von_Neumann_architecture" title="Von Neumann architecture">von Neumann architecture</a>. They both construct a system (a circuit) that represents the physical problem at hand and then leverage their respective physics properties of the system to seek the “minimum”. Neuromorphic quantum computing and quantum computing share similar physical properties during computation. <div class="mw-heading mw-heading4"><h4 id="Topological_quantum_computing">Topological quantum computing</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=11" title="Edit section: Topological quantum computing">edit</a>]</div> A <a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">topological quantum computer</a> decomposes computation into the braiding of <a href="/wiki/Anyon" title="Anyon">anyons</a> in a 2D lattice.<a href="#cite_note-Nayaketal2008-42">[40]</a> <div class="mw-heading mw-heading4"><h4 id="Quantum_Turing_machine">Quantum Turing machine</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=12" title="Edit section: Quantum Turing machine">edit</a>]</div> A <a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">quantum Turing machine</a> is the quantum analog of a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>.<a href="#cite_note-The_computer_as_a_physical_system-7">[6]</a> All of these models of computation—quantum circuits,<a href="#cite_note-43">[41]</a> <a href="/wiki/One-way_quantum_computer" title="One-way quantum computer">one-way quantum computation</a>,<a href="#cite_note-44">[42]</a> adiabatic quantum computation,<a href="#cite_note-45">[43]</a> and topological quantum computation<a href="#cite_note-FLW02-46">[44]</a>—have been shown to be equivalent to the quantum Turing machine; given a perfect implementation of one such quantum computer, it can simulate all the others with no more than polynomial overhead. This equivalence need not hold for practical quantum computers, since the overhead of simulation may be too large to be practical. <div class="mw-heading mw-heading4"><h4 id="Noisy_intermediate-scale_quantum_computing">Noisy intermediate-scale quantum computing</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=13" title="Edit section: Noisy intermediate-scale quantum computing">edit</a>]</div> The <a href="/wiki/Threshold_theorem" title="Threshold theorem">threshold theorem</a> shows how increasing the number of qubits can mitigate errors,<a href="#cite_note-FOOTNOTENielsenChuang2010481-47">[45]</a> yet fully fault-tolerant quantum computing remains "a rather distant dream".<a href="#cite_note-preskill2018-48">[46]</a> According to some researchers, noisy intermediate-scale quantum (<a href="/wiki/NISQ" class="mw-redirect" title="NISQ">NISQ</a>) machines may have specialized uses in the near future, but <a href="/wiki/Noise_(signal_processing)" title="Noise (signal processing)">noise</a> in quantum gates limits their reliability.<a href="#cite_note-preskill2018-48">[46]</a> Scientists at <a href="/wiki/Harvard_University" title="Harvard University">Harvard</a> University successfully created "quantum circuits" that correct errors more efficiently than alternative methods, which may potentially remove a major obstacle to practical quantum computers.<a href="#cite_note-49">[47]</a><a href="#cite_note-50">[48]</a> The Harvard research team was supported by <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</a>, <a href="/wiki/QuEra_Computing" class="mw-redirect" title="QuEra Computing">QuEra Computing</a>, <a href="/wiki/California_Institute_of_Technology" title="California Institute of Technology">Caltech</a>, and <a href="/wiki/Princeton_University" title="Princeton University">Princeton</a> University and funded by <a href="/wiki/DARPA" title="DARPA">DARPA</a>'s Optimization with Noisy Intermediate-Scale Quantum devices (ONISQ) program.<a href="#cite_note-51">[49]</a><a href="#cite_note-52">[50]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_cryptography_and_cybersecurity">Quantum cryptography and cybersecurity</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=14" title="Edit section: Quantum cryptography and cybersecurity">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Laser_optique.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Laser_optique.jpg/220px-Laser_optique.jpg" decoding="async" width="220" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Laser_optique.jpg/330px-Laser_optique.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Laser_optique.jpg/440px-Laser_optique.jpg 2x" data-file-width="510" data-file-height="450" /></a><figcaption>Example of a quantum cryptosystem layout</figcaption></figure> Quantum computing has significant potential applications in the fields of cryptography and cybersecurity. Quantum cryptography, which relies on the principles of quantum mechanics, offers the possibility of secure communication channels that are resistant to eavesdropping. Quantum key distribution (QKD) protocols, such as BB84, enable the secure exchange of cryptographic keys between parties, ensuring the confidentiality and integrity of communication. Moreover, quantum random number generators (QRNGs) can produce high-quality random numbers, which are essential for secure encryption. However, quantum computing also poses challenges to traditional cryptographic systems. Shor's algorithm, a quantum algorithm for integer factorization, could potentially break widely used public-key cryptography schemes like RSA, which rely on the difficulty of factoring large numbers. Post-quantum cryptography, which involves the development of cryptographic algorithms that are resistant to attacks by both classical and quantum computers, is an active area of research aimed at addressing this concern. Ongoing research in quantum cryptography and post-quantum cryptography is crucial for ensuring the security of communication and data in the face of evolving quantum computing capabilities. Advances in these fields, such as the development of new QKD protocols, the improvement of QRNGs, and the standardization of post-quantum cryptographic algorithms, will play a key role in maintaining the integrity and confidentiality of information in the quantum era.<a href="#cite_note-53">[51]</a> <div class="mw-heading mw-heading2"><h2 id="Communication">Communication</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=15" title="Edit section: Communication">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></div> <a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a> enables new ways to transmit data securely; for example, <a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">quantum key distribution</a> uses entangled quantum states to establish secure <a href="/wiki/Cryptographic_keys" class="mw-redirect" title="Cryptographic keys">cryptographic keys</a>.<a href="#cite_note-54">[52]</a> When a sender and receiver exchange quantum states, they can guarantee that an <a href="/wiki/Adversary_(cryptography)" title="Adversary (cryptography)">adversary</a> does not intercept the message, as any unauthorized eavesdropper would disturb the delicate quantum system and introduce a detectable change.<a href="#cite_note-55">[53]</a> With appropriate <a href="/wiki/Cryptographic_protocols" class="mw-redirect" title="Cryptographic protocols">cryptographic protocols</a>, the sender and receiver can thus establish shared private information resistant to eavesdropping.<a href="#cite_note-bb84-12">[11]</a><a href="#cite_note-56">[54]</a> Modern <a href="/wiki/Fiber-optic_cables" class="mw-redirect" title="Fiber-optic cables">fiber-optic cables</a> can transmit quantum information over relatively short distances. Ongoing experimental research aims to develop more reliable hardware (such as quantum repeaters), hoping to scale this technology to long-distance <a href="/wiki/Quantum_networks" class="mw-redirect" title="Quantum networks">quantum networks</a> with end-to-end entanglement. Theoretically, this could enable novel technological applications, such as distributed quantum computing and enhanced <a href="/wiki/Quantum_sensing" class="mw-redirect" title="Quantum sensing">quantum sensing</a>.<a href="#cite_note-57">[55]</a><a href="#cite_note-58">[56]</a> <div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=16" title="Edit section: Algorithms">edit</a>]</div> Progress in finding <a href="/wiki/Quantum_algorithms" class="mw-redirect" title="Quantum algorithms">quantum algorithms</a> typically focuses on this quantum circuit model, though exceptions like the <a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">quantum adiabatic algorithm</a> exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.<a href="#cite_note-zoo-59">[57]</a> Quantum algorithms that offer more than a polynomial speedup over the best-known classical algorithm include Shor's algorithm for factoring and the related quantum algorithms for computing <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithms</a>, solving <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a>, and more generally solving the <a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">hidden subgroup problem</a> for <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> finite groups.<a href="#cite_note-zoo-59">[57]</a> These algorithms depend on the primitive of the <a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">quantum Fourier transform</a>. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, but evidence suggests that this is unlikely.<a href="#cite_note-60">[58]</a> Certain oracle problems like <a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's problem</a> and the <a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani problem</a> do give provable speedups, though this is in the <a href="/wiki/Quantum_complexity_theory#Quantum_query_complexity" title="Quantum complexity theory">quantum query model</a>, which is a restricted model where lower bounds are much easier to prove and doesn't necessarily translate to speedups for practical problems. Other problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomials</a>, and the <a href="/wiki/Quantum_algorithm_for_linear_systems_of_equations" class="mw-redirect" title="Quantum algorithm for linear systems of equations">quantum algorithm for linear systems of equations</a> have quantum algorithms appearing to give super-polynomial speedups and are <a href="/wiki/BQP" title="BQP">BQP</a>-complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that no quantum algorithm gives a super-polynomial speedup, which is believed to be unlikely.<a href="#cite_note-FOOTNOTENielsenChuang201042-61">[59]</a> Some quantum algorithms, like <a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's algorithm</a> and <a href="/wiki/Amplitude_amplification" title="Amplitude amplification">amplitude amplification</a>, give polynomial speedups over corresponding classical algorithms.<a href="#cite_note-zoo-59">[57]</a> Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems.<a href="#cite_note-FOOTNOTENielsenChuang20107-20">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Simulation_of_quantum_systems">Simulation of quantum systems</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=17" title="Edit section: Simulation of quantum systems">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_simulation" class="mw-redirect" title="Quantum simulation">Quantum simulation</a></div> Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, <a href="/wiki/Quantum_simulator" title="Quantum simulator">quantum simulation</a> may be an important application of quantum computing.<a href="#cite_note-62">[60]</a> Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a <a href="/wiki/Collider" title="Collider">collider</a>.<a href="#cite_note-63">[61]</a> In June 2023, IBM computer scientists reported that a quantum computer produced better results for a physics problem than a conventional supercomputer.<a href="#cite_note-NYT-20230614-64">[62]</a><a href="#cite_note-NAT-20230614-65">[63]</a> About 2% of the annual global energy output is used for <a href="/wiki/Nitrogen_fixation" title="Nitrogen fixation">nitrogen fixation</a> to produce <a href="/wiki/Ammonia" title="Ammonia">ammonia</a> for the <a href="/wiki/Haber_process" title="Haber process">Haber process</a> in the agricultural fertilizer industry (even though naturally occurring organisms also produce ammonia). Quantum simulations might be used to understand this process and increase the energy efficiency of production.<a href="#cite_note-66">[64]</a> It is expected that an early use of quantum computing will be modeling that improves the efficiency of the Haber–Bosch process<a href="#cite_note-67">[65]</a> by the mid-2020s<a href="#cite_note-68">[66]</a> although some have predicted it will take longer.<a href="#cite_note-69">[67]</a> <div class="mw-heading mw-heading3"><h3 id="Post-quantum_cryptography">Post-quantum cryptography</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=18" title="Edit section: Post-quantum cryptography">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></div> A notable application of quantum computation is for <a href="/wiki/Cryptanalysis" title="Cryptanalysis">attacks</a> on cryptographic systems that are currently in use. <a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a>, which underpins the security of <a href="/wiki/Public_key_cryptography" class="mw-redirect" title="Public key cryptography">public key cryptographic</a> systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> (e.g., products of two 300-digit primes).<a href="#cite_note-70">[68]</a> By comparison, a quantum computer could solve this problem exponentially faster using Shor's algorithm to find its factors.<a href="#cite_note-FOOTNOTENielsenChuang2010216-71">[69]</a> This ability would allow a quantum computer to break many of the <a href="/wiki/Cryptography" title="Cryptography">cryptographic</a> systems in use today, in the sense that there would be a <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular <a href="/wiki/Asymmetric_Algorithms" class="mw-redirect" title="Asymmetric Algorithms">public key ciphers</a> are based on the difficulty of factoring integers or the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> problem, both of which can be solved by Shor's algorithm. In particular, the <a href="/wiki/RSA_(algorithm)" class="mw-redirect" title="RSA (algorithm)">RSA</a>, <a href="/wiki/Diffie%E2%80%93Hellman" class="mw-redirect" title="Diffie–Hellman">Diffie–Hellman</a>, and <a href="/wiki/Elliptic_curve_Diffie%E2%80%93Hellman" class="mw-redirect" title="Elliptic curve Diffie–Hellman">elliptic curve Diffie–Hellman</a> algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security. Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of post-quantum cryptography.<a href="#cite_note-pqcrypto_survey-72">[70]</a><a href="#cite_note-73">[71]</a> Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the <a href="/wiki/McEliece_cryptosystem" title="McEliece cryptosystem">McEliece cryptosystem</a> based on a problem in <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>.<a href="#cite_note-pqcrypto_survey-72">[70]</a><a href="#cite_note-74">[72]</a> <a href="/wiki/Lattice-based_cryptography" title="Lattice-based cryptography">Lattice-based cryptosystems</a> are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a> <a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">hidden subgroup problem</a>, which would break many lattice based cryptosystems, is a well-studied open problem.<a href="#cite_note-75">[73]</a> It has been proven that applying Grover's algorithm to break a <a href="/wiki/Symmetric-key_algorithm" title="Symmetric-key algorithm">symmetric (secret key) algorithm</a> by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case,<a href="#cite_note-bennett_1997-76">[74]</a> meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see <a href="/wiki/Key_size#Effect_of_quantum_computing_attacks_on_key_strength" title="Key size">Key size</a>). <div class="mw-heading mw-heading3"><h3 id="Search_problems">Search problems</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=19" title="Edit section: Search problems">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's algorithm</a></div> The most well-known example of a problem that allows for a polynomial quantum speedup is unstructured search, which involves finding a marked item out of a list of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> items in a database. This can be solved by Grover's algorithm using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({\sqrt {n}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\sqrt {n}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5526ab1252c0f682bbe07c0ad67c0f29de5522b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.913ex; height:3.009ex;" alt="{\displaystyle O({\sqrt {n}})}"> queries to the database, quadratically fewer than the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6becc31c61ad3420a1e4ee9e39c28baf73bda24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \Omega (n)}"> queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Many examples of provable quantum speedups for query problems are based on Grover's algorithm, including <a href="/wiki/BHT_algorithm" title="BHT algorithm">Brassard, Høyer, and Tapp's algorithm</a> for finding collisions in two-to-one functions,<a href="#cite_note-77">[75]</a> and Farhi, Goldstone, and Gutmann's algorithm for evaluating NAND trees.<a href="#cite_note-78">[76]</a> Problems that can be efficiently addressed with Grover's algorithm have the following properties:<a href="#cite_note-79">[77]</a><a href="#cite_note-80">[78]</a> <ol><li>There is no searchable structure in the collection of possible answers,</li> <li>The number of possible answers to check is the same as the number of inputs to the algorithm, and</li> <li>There exists a Boolean function that evaluates each input and determines whether it is the correct answer.</li></ol> For problems with all these properties, the running time of Grover's algorithm on a quantum computer scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied<a href="#cite_note-81">[79]</a> is a <a href="/wiki/Boolean_satisfiability_problem" title="Boolean satisfiability problem">Boolean satisfiability problem</a>, where the database through which the algorithm iterates is that of all possible answers. An example and possible application of this is a <a href="/wiki/Password_cracking" title="Password cracking">password cracker</a> that attempts to guess a password. Breaking <a href="/wiki/Symmetric-key_algorithm" title="Symmetric-key algorithm">symmetric ciphers</a> with this algorithm is of interest to government agencies.<a href="#cite_note-82">[80]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_annealing">Quantum annealing</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=20" title="Edit section: Quantum annealing">edit</a>]</div> <a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a> relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which slowly evolves to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process. Adiabatic optimization may be helpful for solving <a href="/wiki/Computational_biology" title="Computational biology">computational biology</a> problems.<a href="#cite_note-83">[81]</a> <div class="mw-heading mw-heading3"><h3 id="Machine_learning">Machine learning</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=21" title="Edit section: Machine learning">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></div> Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> tasks.<a href="#cite_note-preskill2018-48">[46]</a><a href="#cite_note-84">[82]</a> For example, the <a href="/wiki/HHL_Algorithm" class="mw-redirect" title="HHL Algorithm">HHL Algorithm</a>, named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts.<a href="#cite_note-preskill2018-48">[46]</a><a href="#cite_note-Quantum_algorithm_for_solving_linear_systems_of_equations_by_Harrow_et_al.-85">[83]</a> Some research groups have recently explored the use of quantum annealing hardware for training <a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machines</a> and <a href="/wiki/Deep_neural_networks" class="mw-redirect" title="Deep neural networks">deep neural networks</a>.<a href="#cite_note-86">[84]</a><a href="#cite_note-87">[85]</a><a href="#cite_note-88">[86]</a> Deep generative chemistry models emerge as powerful tools to expedite <a href="/wiki/Drug_discovery" title="Drug discovery">drug discovery</a>. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems<a href="#cite_note-273.5278.1073-21">[20]</a> and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models<a href="#cite_note-89">[87]</a> including quantum GANs<a href="#cite_note-90">[88]</a> may eventually be developed into ultimate generative chemistry algorithms. <div class="mw-heading mw-heading2"><h2 id="Engineering">Engineering</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=22" title="Edit section: Engineering">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_(39188583425).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_%2839188583425%29.jpg/220px-A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_%2839188583425%29.jpg" decoding="async" width="220" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_%2839188583425%29.jpg/330px-A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_%2839188583425%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_%2839188583425%29.jpg/440px-A_Wafer_of_the_Latest_D-Wave_Quantum_Computers_%2839188583425%29.jpg 2x" data-file-width="5158" data-file-height="3522" /></a><figcaption>A <a href="/wiki/Wafer_(electronics)" title="Wafer (electronics)">wafer</a> of <a href="/wiki/Adiabatic_quantum_computer" class="mw-redirect" title="Adiabatic quantum computer">adiabatic quantum computers</a></figcaption></figure> As of 2023,<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Quantum_computing&action=edit">[update]</a> classical computers outperform quantum computers for all real-world applications. While current quantum computers may speed up solutions to particular mathematical problems, they give no computational advantage for practical tasks. Scientists and engineers are exploring multiple technologies for quantum computing hardware and hope to develop scalable quantum architectures, but serious obstacles remain.<a href="#cite_note-good-for-nothing-91">[89]</a><a href="#cite_note-CACM-92">[90]</a> <div class="mw-heading mw-heading3"><h3 id="Challenges">Challenges</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=23" title="Edit section: Challenges">edit</a>]</div> There are a number of technical challenges in building a large-scale quantum computer.<a href="#cite_note-93">[91]</a> Physicist <a href="/wiki/David_P._DiVincenzo" class="mw-redirect" title="David P. DiVincenzo">David DiVincenzo</a> has listed <a href="/wiki/DiVincenzo%27s_criteria" title="DiVincenzo's criteria">these requirements</a> for a practical quantum computer:<a href="#cite_note-94">[92]</a> <ul><li>Physically scalable to increase the number of qubits</li> <li>Qubits that can be initialized to arbitrary values</li> <li>Quantum gates that are faster than <a href="/wiki/Decoherence" class="mw-redirect" title="Decoherence">decoherence</a> time</li> <li>Universal gate set</li> <li>Qubits that can be read easily.</li></ul> Sourcing parts for quantum computers is also very difficult. <a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting quantum computers</a>, like those constructed by <a href="/wiki/Google" title="Google">Google</a> and <a href="/wiki/IBM" title="IBM">IBM</a>, need <a href="/wiki/Helium-3" title="Helium-3">helium-3</a>, a <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear</a> research byproduct, and special <a href="/wiki/Superconducting" class="mw-redirect" title="Superconducting">superconducting</a> cables made only by the Japanese company Coax Co.<a href="#cite_note-95">[93]</a> The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of <a href="/w/index.php?title=Quantum_controllers&action=edit&redlink=1" class="new" title="Quantum controllers (page does not exist)">quantum controllers</a> that enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.<a href="#cite_note-96">[94]</a> <div class="mw-heading mw-heading4"><h4 id="Decoherence">Decoherence</h4>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=24" title="Edit section: Decoherence">edit</a>]</div> One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for <a href="/wiki/Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">NMR</a> and <a href="/wiki/MRI" class="mw-redirect" title="MRI">MRI</a> technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.<a href="#cite_note-DiVincenzo_1995-97">[95]</a> Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a <a href="/wiki/Dilution_refrigerator" title="Dilution refrigerator">dilution refrigerator</a><a href="#cite_note-98">[96]</a>) in order to prevent significant decoherence.<a href="#cite_note-99">[97]</a> A 2020 study argues that <a href="/wiki/Ionizing_radiation" title="Ionizing radiation">ionizing radiation</a> such as <a href="/wiki/Cosmic_rays" class="mw-redirect" title="Cosmic rays">cosmic rays</a> can nevertheless cause certain systems to decohere within milliseconds.<a href="#cite_note-100">[98]</a> As a result, time-consuming tasks may render some quantum algorithms inoperable, as attempting to maintain the state of qubits for a long enough duration will eventually corrupt the superpositions.<a href="#cite_note-101">[99]</a> These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical <a href="/wiki/Pulse_shaping" title="Pulse shaping">pulse shaping</a>. Error rates are typically proportional to the ratio of operating time to decoherence time; hence any operation must be completed much more quickly than the decoherence time. As described by the <a href="/wiki/Threshold_theorem" title="Threshold theorem">threshold theorem</a>, if the error rate is small enough, it is thought to be possible to use <a href="/wiki/Quantum_error_correction" title="Quantum error correction">quantum error correction</a> to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often-cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing. Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction.<a href="#cite_note-102">[100]</a> With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds. However, the encoding and error-correction overheads increase the size of a real fault-tolerant quantum computer by several orders of magnitude. Careful estimates<a href="#cite_note-:1-103">[101]</a><a href="#cite_note-:2-104">[102]</a> show that at least 3 million physical qubits would factor 2,048-bit integer in 5 months on a fully error-corrected trapped-ion quantum computer. In terms of the number of physical qubits, to date, this remains the lowest estimate<a href="#cite_note-105">[103]</a> for practically useful integer factorization problem sizing 1,024-bit or larger. Another approach to the stability-decoherence problem is to create a <a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">topological quantum computer</a> with <a href="/wiki/Anyon" title="Anyon">anyons</a>, <a href="/wiki/Quasi-particle" class="mw-redirect" title="Quasi-particle">quasi-particles</a> used as threads, and relying on <a href="/wiki/Braid_theory" class="mw-redirect" title="Braid theory">braid theory</a> to form stable logic gates.<a href="#cite_note-106">[104]</a><a href="#cite_note-107">[105]</a> <div class="mw-heading mw-heading3"><h3 id="Quantum_supremacy">Quantum supremacy</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=25" title="Edit section: Quantum supremacy">edit</a>]</div> Physicist <a href="/wiki/John_Preskill" title="John Preskill">John Preskill</a> coined the term quantum supremacy to describe the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers.<a href="#cite_note-108">[106]</a><a href="#cite_note-109">[107]</a><a href="#cite_note-110">[108]</a> The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.<a href="#cite_note-111">[109]</a> In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the <a href="/wiki/Sycamore_processor" title="Sycamore processor">Sycamore quantum computer</a> more than 3,000,000 times faster than they could be done on <a href="/wiki/Summit_(supercomputer)" title="Summit (supercomputer)">Summit</a>, generally considered the world's fastest computer.<a href="#cite_note-1910.11333-27">[26]</a><a href="#cite_note-112">[110]</a><a href="#cite_note-113">[111]</a> This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed,<a href="#cite_note-114">[112]</a><a href="#cite_note-115">[113]</a> and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to the gap between Sycamore and classical supercomputers<a href="#cite_note-116">[114]</a><a href="#cite_note-117">[115]</a><a href="#cite_note-118">[116]</a> and even beating it.<a href="#cite_note-119">[117]</a><a href="#cite_note-120">[118]</a><a href="#cite_note-121">[119]</a> In December 2020, a group at <a href="/wiki/University_of_Science_and_Technology_of_China" title="University of Science and Technology of China">USTC</a> implemented a type of <a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a> on 76 photons with a <a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">photonic quantum computer</a>, <a href="/wiki/Jiuzhang_(quantum_computer)" title="Jiuzhang (quantum computer)">Jiuzhang</a>, to demonstrate quantum supremacy.<a href="#cite_note-122">[120]</a><a href="#cite_note-123">[121]</a><a href="#cite_note-124">[122]</a> The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds.<a href="#cite_note-:6-125">[123]</a> Claims of quantum supremacy have generated hype around quantum computing,<a href="#cite_note-126">[124]</a> but they are based on contrived benchmark tasks that do not directly imply useful real-world applications.<a href="#cite_note-good-for-nothing-91">[89]</a><a href="#cite_note-127">[125]</a> In January 2024, a study published in Physical Review Letters provided direct verification of quantum supremacy experiments by computing exact amplitudes for experimentally generated bitstrings using a new-generation Sunway supercomputer, demonstrating a significant leap in simulation capability built on a multiple-amplitude tensor network contraction algorithm. This development underscores the evolving landscape of quantum computing, highlighting both the progress and the complexities involved in validating quantum supremacy claims.<a href="#cite_note-128">[126]</a> <div class="mw-heading mw-heading3"><h3 id="Skepticism">Skepticism</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=26" title="Edit section: Skepticism">edit</a>]</div> Despite high hopes for quantum computing, significant progress in hardware, and optimism about future applications, a 2023 <a href="/wiki/Nature_(journal)" title="Nature (journal)">Nature</a> spotlight article summarized current quantum computers as being "For now, [good for] absolutely nothing".<a href="#cite_note-good-for-nothing-91">[89]</a> The article elaborated that quantum computers are yet to be more useful or efficient than conventional computers in any case, though it also argued that in the long term such computers are likely to be useful. A 2023 <a href="/wiki/Communications_of_the_ACM" title="Communications of the ACM">Communications of the ACM</a> article<a href="#cite_note-CACM-92">[90]</a> found that current quantum computing algorithms are "insufficient for practical quantum advantage without significant improvements across the software/hardware stack". It argues that the most promising candidates for achieving speedup with quantum computers are "small-data problems", for example in chemistry and materials science. However, the article also concludes that a large range of the potential applications it considered, such as machine learning, "will not achieve quantum advantage with current quantum algorithms in the foreseeable future", and it identified I/O constraints that make speedup unlikely for "big data problems, unstructured linear systems, and database search based on Grover's algorithm". This state of affairs can be traced to several current and long-term considerations. <ul><li>Conventional computer hardware and algorithms are not only optimized for practical tasks, but are still improving rapidly, particularly <a href="/wiki/GPU" class="mw-redirect" title="GPU">GPU</a> accelerators.</li> <li>Current quantum computing hardware generates only a limited amount of <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">entanglement</a> before getting overwhelmed by noise.</li> <li>Quantum algorithms provide speedup over conventional algorithms only for some tasks, and matching these tasks with practical applications proved challenging. Some promising tasks and applications require resources far beyond those available today.<a href="#cite_note-129">[127]</a><a href="#cite_note-130">[128]</a> In particular, processing large amounts of non-quantum data is a challenge for quantum computers.<a href="#cite_note-CACM-92">[90]</a></li> <li>Some promising algorithms have been "dequantized", i.e., their non-quantum analogues with similar complexity have been found.</li> <li>If <a href="/wiki/Quantum_error_correction" title="Quantum error correction">quantum error correction</a> is used to scale quantum computers to practical applications, its overhead may undermine speedup offered by many quantum algorithms.<a href="#cite_note-CACM-92">[90]</a></li> <li>Complexity analysis of algorithms sometimes makes abstract assumptions that do not hold in applications. For example, input data may not already be available encoded in quantum states, and "oracle functions" used in Grover's algorithm often have internal structure that can be exploited for faster algorithms.</li></ul> In particular, building computers with large numbers of qubits may be futile if those qubits are not connected well enough and cannot maintain sufficiently high degree of entanglement for a long time. When trying to outperform conventional computers, quantum computing researchers often look for new tasks that can be solved on quantum computers, but this leaves the possibility that efficient non-quantum techniques will be developed in response, as seen for Quantum supremacy demonstrations. Therefore, it is desirable to prove lower bounds on the complexity of best possible non-quantum algorithms (which may be unknown) and show that some quantum algorithms asymptomatically improve upon those bounds. Some researchers have expressed skepticism that scalable quantum computers could ever be built, typically because of the issue of maintaining coherence at large scales, but also for other reasons. <a href="/wiki/Bill_Unruh" class="mw-redirect" title="Bill Unruh">Bill Unruh</a> doubted the practicality of quantum computers in a paper published in 1994.<a href="#cite_note-131">[129]</a> <a href="/wiki/Paul_Davies" title="Paul Davies">Paul Davies</a> argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the <a href="/wiki/Holographic_principle" title="Holographic principle">holographic principle</a>.<a href="#cite_note-132">[130]</a> Skeptics like <a href="/wiki/Gil_Kalai" title="Gil Kalai">Gil Kalai</a> doubt that quantum supremacy will ever be achieved.<a href="#cite_note-133">[131]</a><a href="#cite_note-134">[132]</a><a href="#cite_note-135">[133]</a> Physicist <a href="/wiki/Mikhail_Dyakonov" title="Mikhail Dyakonov">Mikhail Dyakonov</a> has expressed skepticism of quantum computing as follows: <dl><dd>"So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10300... Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never."<a href="#cite_note-136">[134]</a><a href="#cite_note-137">[135]</a></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Physical_realizations">Physical realizations</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=27" title="Edit section: Physical realizations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/List_of_proposed_quantum_registers" title="List of proposed quantum registers">List of proposed quantum registers</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:IBM_Q_system_(Fraunhofer_2).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/IBM_Q_system_%28Fraunhofer_2%29.jpg/260px-IBM_Q_system_%28Fraunhofer_2%29.jpg" decoding="async" width="260" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/IBM_Q_system_%28Fraunhofer_2%29.jpg/390px-IBM_Q_system_%28Fraunhofer_2%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/IBM_Q_system_%28Fraunhofer_2%29.jpg/520px-IBM_Q_system_%28Fraunhofer_2%29.jpg 2x" data-file-width="5166" data-file-height="3584" /></a><figcaption><a href="/wiki/IBM_Q_System_One" title="IBM Q System One">Quantum System One</a>, a quantum computer by <a href="/wiki/IBM" title="IBM">IBM</a> from 2019 with 20 superconducting qubits<a href="#cite_note-138">[136]</a></figcaption></figure> A practical quantum computer must use a physical system as a programmable quantum register.<a href="#cite_note-139">[137]</a> Researchers are exploring several technologies as candidates for reliable qubit implementations.<a href="#cite_note-FOOTNOTEGrumblingHorowitz2019127-140">[138]</a> <a href="/wiki/Superconductors" class="mw-redirect" title="Superconductors">Superconductors</a> and <a href="/wiki/Trapped_ion" class="mw-redirect" title="Trapped ion">trapped ions</a> are some of the most developed proposals, but experimentalists are considering other hardware possibilities as well.<a href="#cite_note-FOOTNOTEGrumblingHorowitz2019114-141">[139]</a> The first quantum logic gates were implemented with <a href="/wiki/Trapped_ion" class="mw-redirect" title="Trapped ion">trapped ions</a> and prototype general purpose machines with up to 20 qubits have been realized. However, the technology behind these devices combines complex vacuum equipment, lasers, microwave and radio frequency equipment making full scale processors difficult to integrate with standard computing equipment. Moreover, the trapped ion system itself has engineering challenges to overcome.<a href="#cite_note-FOOTNOTEGrumblingHorowitz2019119-142">[140]</a> The largest commercial systems are based on <a href="/wiki/Superconductor" class="mw-redirect" title="Superconductor">superconductor</a> devices and have scaled to 2000 qubits. However, the error rates for larger machines have been on the order of 5%. Technologically these devices are all cryogenic and scaling to large numbers of qubits requires wafer-scale integration, a serious engineering challenge by itself.<a href="#cite_note-FOOTNOTEGrumblingHorowitz2019126-143">[141]</a> Research efforts to create stabler qubits for quantum computing include topological quantum computer approaches. For example, <a href="/wiki/Microsoft_Azure_Quantum" title="Microsoft Azure Quantum">Microsoft</a> is working on a computer based on the quantum properties of two-dimensional quasiparticles called <a href="/wiki/Anyons" class="mw-redirect" title="Anyons">anyons</a>.<a href="#cite_note-144">[142]</a><a href="#cite_note-145">[143]</a><a href="#cite_note-146">[144]</a> <div class="mw-heading mw-heading2"><h2 id="Potential_applications">Potential applications</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=28" title="Edit section: Potential applications">edit</a>]</div> With focus on business management's point of view, the potential applications of quantum computing into four major categories are cybersecurity, data analytics and artificial intelligence, optimization and simulation, and data management and searching.<a href="#cite_note-147">[145]</a> Investment in quantum computing research has increased in the public and private sectors.<a href="#cite_note-148">[146]</a><a href="#cite_note-149">[147]</a> As one consulting firm summarized,<a href="#cite_note-150">[148]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">... investment dollars are pouring in, and quantum-computing start-ups are proliferating. ... While quantum computing promises to help businesses solve problems that are beyond the reach and speed of conventional <a href="/wiki/High-performance_computers" class="mw-redirect" title="High-performance computers">high-performance computers</a>, use cases are largely experimental and hypothetical at this early stage.</blockquote> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=29" title="Edit section: Theory">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Computability">Computability</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=30" title="Edit section: Computability">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Computability_theory" title="Computability theory">Computability theory</a></div> Any <a href="/wiki/Computational_problem" title="Computational problem">computational problem</a> solvable by a classical computer is also solvable by a quantum computer.<a href="#cite_note-FOOTNOTENielsenChuang201029-151">[149]</a> Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, which underlies the operation of quantum computers. Conversely, any problem solvable by a quantum computer is also solvable by a classical computer. It is possible to simulate both quantum and classical computers manually with just some paper and a pen, if given enough time. More formally, any quantum computer can be simulated by a <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a>. In other words, quantum computers provide no additional power over classical computers in terms of <a href="/wiki/Computability" title="Computability">computability</a>. This means that quantum computers cannot solve <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable problems</a> like the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a>, and the existence of quantum computers does not disprove the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a>.<a href="#cite_note-FOOTNOTENielsenChuang2010126-152">[150]</a> <div class="mw-heading mw-heading3"><h3 id="Complexity">Complexity</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=31" title="Edit section: Complexity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></div> While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve certain problems faster than classical computers. For instance, it is known that quantum computers can efficiently <a href="/wiki/Integer_factorization" title="Integer factorization">factor integers</a>, while this is not believed to be the case for classical computers. The class of <a href="/wiki/Computational_problem" title="Computational problem">problems</a> that can be efficiently solved by a quantum computer with bounded error is called <a href="/wiki/BQP" title="BQP">BQP</a>, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with an error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to <a href="/wiki/Bounded-error_probabilistic_polynomial" class="mw-redirect" title="Bounded-error probabilistic polynomial">BPP</a> ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time <a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">probabilistic Turing machines</a> with bounded error.<a href="#cite_note-FOOTNOTENielsenChuang201041-153">[151]</a> It is known that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {BPP\subseteq BQP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">P</mi> <mo>⊆</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">Q</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {BPP\subseteq BQP}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59af787cd2b2478c3c326d722fbffbc7839c1abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.366ex; height:2.343ex;" alt="{\displaystyle {\mathsf {BPP\subseteq BQP}}}"> and is widely suspected that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {BQP\subsetneq BPP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">Q</mi> <mi mathvariant="sans-serif">P</mi> <mo>⊊</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {BQP\subsetneq BPP}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/106869205d6945f52c45f002a693dfb27d54574c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.366ex; height:2.676ex;" alt="{\displaystyle {\mathsf {BQP\subsetneq BPP}}}">, which intuitively would mean that quantum computers are more powerful than classical computers in terms of <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a>.<a href="#cite_note-FOOTNOTENielsenChuang2010201-154">[152]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:BQP_complexity_class_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/BQP_complexity_class_diagram.svg/220px-BQP_complexity_class_diagram.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/BQP_complexity_class_diagram.svg/330px-BQP_complexity_class_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/BQP_complexity_class_diagram.svg/440px-BQP_complexity_class_diagram.svg.png 2x" data-file-width="518" data-file-height="414" /></a><figcaption>The suspected relationship of BQP to several classical complexity classes<a href="#cite_note-FOOTNOTENielsenChuang201042-61">[59]</a></figcaption></figure> The exact relationship of BQP to <a href="/wiki/P_(complexity)" title="P (complexity)">P</a>, <a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a>, and <a href="/wiki/PSPACE_(complexity)" class="mw-redirect" title="PSPACE (complexity)">PSPACE</a> is not known. However, it is known that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">P</mi> <mo>⊆</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">Q</mi> <mi mathvariant="sans-serif">P</mi> <mo>⊆</mo> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">S</mi> <mi mathvariant="sans-serif">P</mi> <mi mathvariant="sans-serif">A</mi> <mi mathvariant="sans-serif">C</mi> <mi mathvariant="sans-serif">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30ec0b4e850418f5478e8a8e91fd2ccb20d0ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.115ex; height:2.343ex;" alt="{\displaystyle {\mathsf {P\subseteq BQP\subseteq PSPACE}}}">; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the <a href="/wiki/Discrete_logarithm_problem" class="mw-redirect" title="Discrete logarithm problem">discrete logarithm problem</a> are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {NP\nsubseteq BQP}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> <mi mathvariant="sans-serif">P</mi> <mo>⊈</mo> <mi mathvariant="sans-serif">B</mi> <mi mathvariant="sans-serif">Q</mi> <mi mathvariant="sans-serif">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {NP\nsubseteq BQP}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b2cc3f547f6bad0a6b40df6a38349b52541c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.976ex; height:3.176ex;" alt="{\displaystyle {\mathsf {NP\nsubseteq BQP}}}">; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> problems (if an NP-complete problem were in BQP, then it would follow from <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hardness</a> that all problems in NP are in BQP).<a href="#cite_note-BernVazi-155">[153]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=32" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Quantum_computing" class="extiw" title="commons:Category:Quantum computing">Quantum computing</a>.</div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 21em;"> <ul><li><a href="/wiki/D-Wave_Systems" title="D-Wave Systems">D-Wave Systems</a> – Canadian quantum computing company</li> <li><a href="/wiki/Electronic_quantum_holography" title="Electronic quantum holography">Electronic quantum holography</a> – Information storage technology</li> <li><a href="/wiki/Glossary_of_quantum_computing" title="Glossary of quantum computing">Glossary of quantum computing</a></li> <li><a href="/wiki/IARPA" class="mw-redirect" title="IARPA">IARPA</a> – American government agencyPages displaying short descriptions of redirect targets</li> <li><a href="/wiki/IonQ" title="IonQ">IonQ</a> – US information technology company</li> <li><a href="/wiki/List_of_emerging_technologies" title="List of emerging technologies">List of emerging technologies</a> – New technologies actively in development</li> <li><a href="/wiki/List_of_quantum_processors" title="List of quantum processors">List of quantum processors</a> – List of quantum computer components</li> <li><a href="/wiki/Magic_state_distillation" title="Magic state distillation">Magic state distillation</a> – Quantum computing algorithm</li> <li><a href="/wiki/Metacomputing" title="Metacomputing">Metacomputing</a> – Computing for the purpose of computing</li> <li><a href="/wiki/Natural_computing" title="Natural computing">Natural computing</a> – terminology introduced to encompass three classes of methodsPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Optical_computing" title="Optical computing">Optical computing</a> – Computer that uses photons or light waves</li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a> – device which can be used to store or transfer information between independent qubits in a quantum computerPages displaying wikidata descriptions as a fallback</li> <li><a href="/wiki/Quantum_cognition" title="Quantum cognition">Quantum cognition</a> – Application of quantum theory mathematics to cognitive phenomena</li> <li><a href="/wiki/Quantum_volume" title="Quantum volume">Quantum volume</a> – Metric for a quantum computer's capabilities</li> <li><a href="/wiki/Quantum_weirdness" title="Quantum weirdness">Quantum weirdness</a> – Unintuitive aspects of quantum mechanics</li> <li><a href="/wiki/Rigetti_Computing" title="Rigetti Computing">Rigetti Computing</a> – American quantum computing company</li> <li><a href="/wiki/Supercomputer" title="Supercomputer">Supercomputer</a> – Type of extremely powerful computer</li> <li><a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">Theoretical computer science</a> – Subfield of computer science and mathematics</li> <li><a href="/wiki/Unconventional_computing" title="Unconventional computing">Unconventional computing</a> – Computing by new or unusual methods</li> <li><a href="/wiki/Valleytronics" title="Valleytronics">Valleytronics</a> – Experimental area in semiconductors</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=33" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> As used in this article, "exponentially faster" has a precise <a href="/wiki/Computational_complexity" title="Computational complexity">complexity theoretical</a> meaning. Usually, it means that as a function of input size in bits, the best known classical algorithm for a problem requires an <a href="/wiki/Exponential_growth" title="Exponential growth">exponentially growing</a> number of steps, while a quantum algorithm uses only a polynomial number of steps. </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> The <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> is also the computational basis.<a href="#cite_note-FOOTNOTEMermin200718-35">[34]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=34" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEAaronson2013132-2"><a href="#cite_ref-FOOTNOTEAaronson2013132_2-0">^</a> <a href="#CITEREFAaronson2013">Aaronson 2013</a>, p. 132. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBhatta2020" class="citation journal cs1">Bhatta, Varun S. (10 May 2020). <a rel="nofollow" class="external text" href="https://www.currentscience.ac.in/Volumes/118/09/1365.pdf">"Plurality of Wave–Particle Duality"</a> (PDF). Current Science. 118 (9): 1365. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.18520%2Fcs%2Fv118%2Fi9%2F1365-1374">10.18520/cs/v118/i9/1365-1374</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0011-3891">0011-3891</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216143449">216143449</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Current+Science&rft.atitle=Plurality+of+Wave%E2%80%93Particle+Duality&rft.volume=118&rft.issue=9&rft.pages=1365&rft.date=2020-05-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216143449%23id-name%3DS2CID&rft.issn=0011-3891&rft_id=info%3Adoi%2F10.18520%2Fcs%2Fv118%2Fi9%2F1365-1374&rft.aulast=Bhatta&rft.aufirst=Varun+S.&rft_id=https%3A%2F%2Fwww.currentscience.ac.in%2FVolumes%2F118%2F09%2F1365.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCeruzzi2012" class="citation book cs1">Ceruzzi, Paul E. 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American Chemical Society. Retrieved 12 April 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=cen.acs.org&rft.atitle=Chemistry+is+quantum+computing%27s+killer+app&rft.date=2017-10-30&rft.aulast=Bourzac&rft.aufirst=Katherine&rft_id=https%3A%2F%2Fcen.acs.org%2Farticles%2F95%2Fi43%2FChemistry-quantum-computings-killer-app.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"> </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLenstra2000" class="citation journal cs1">Lenstra, Arjen K. (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150410234239/http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf">"Integer Factoring"</a> (PDF). Designs, Codes and Cryptography. 19 (2/3): 101–128. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1008397921377">10.1023/A:1008397921377</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9816153">9816153</a>. Archived from <a rel="nofollow" class="external text" href="http://sage.math.washington.edu/edu/124/misc/arjen_lenstra_factoring.pdf">the original</a> (PDF) on 10 April 2015.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Designs%2C+Codes+and+Cryptography&rft.atitle=Integer+Factoring&rft.volume=19&rft.issue=2%2F3&rft.pages=101-128&rft.date=2000&rft_id=info%3Adoi%2F10.1023%2FA%3A1008397921377&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9816153%23id-name%3DS2CID&rft.aulast=Lenstra&rft.aufirst=Arjen+K.&rft_id=http%3A%2F%2Fsage.math.washington.edu%2Fedu%2F124%2Fmisc%2Farjen_lenstra_factoring.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"> </li> <li id="cite_note-FOOTNOTENielsenChuang2010216-71"><a href="#cite_ref-FOOTNOTENielsenChuang2010216_71-0">^</a> <a href="#CITEREFNielsenChuang2010">Nielsen & Chuang 2010</a>, p. 216. </li> <li id="cite_note-pqcrypto_survey-72">^ <a href="#cite_ref-pqcrypto_survey_72-0">a</a> <a href="#cite_ref-pqcrypto_survey_72-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernstein2009" class="citation book cs1">Bernstein, Daniel J. 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SIAM Journal on Computing. 26 (5): 1411–1473. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.144.7852">10.1.1.144.7852</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0097539796300921">10.1137/S0097539796300921</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Computing&rft.atitle=Quantum+Complexity+Theory&rft.volume=26&rft.issue=5&rft.pages=1411-1473&rft.date=1997&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.144.7852%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1137%2FS0097539796300921&rft.aulast=Bernstein&rft.aufirst=Ethan&rft.au=Vazirani%2C+Umesh&rft_id=http%3A%2F%2Fwww.cs.berkeley.edu%2F~vazirani%2Fbv.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=35" title="Edit section: Sources">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAaronson2013" class="citation book cs1"><a href="/wiki/Scott_Aaronson" title="Scott Aaronson">Aaronson, Scott</a> (2013). Quantum Computing Since Democritus. Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511979309">10.1017/CBO9780511979309</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-19956-8" title="Special:BookSources/978-0-521-19956-8"><bdi>978-0-521-19956-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/829706638">829706638</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing+Since+Democritus&rft.pub=Cambridge+University+Press&rft.date=2013&rft_id=info%3Aoclcnum%2F829706638&rft_id=info%3Adoi%2F10.1017%2FCBO9780511979309&rft.isbn=978-0-521-19956-8&rft.aulast=Aaronson&rft.aufirst=Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrumblingHorowitz2019" class="citation book cs1">Grumbling, Emily; Horowitz, Mark, eds. (2019). Quantum Computing: Progress and Prospects. Washington, DC: The National Academies Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.17226%2F25196">10.17226/25196</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-309-47970-7" title="Special:BookSources/978-0-309-47970-7"><bdi>978-0-309-47970-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1091904777">1091904777</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125635007">125635007</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing%3A+Progress+and+Prospects&rft.place=Washington%2C+DC&rft.pub=The+National+Academies+Press&rft.date=2019&rft_id=info%3Aoclcnum%2F1091904777&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125635007%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.17226%2F25196&rft.isbn=978-0-309-47970-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMermin2007" class="citation book cs1"><a href="/wiki/N._David_Mermin" title="N. David Mermin">Mermin, N. David</a> (2007). Quantum Computer Science: An Introduction. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511813870">10.1017/CBO9780511813870</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-511-34258-5" title="Special:BookSources/978-0-511-34258-5"><bdi>978-0-511-34258-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/422727925">422727925</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computer+Science%3A+An+Introduction&rft.date=2007&rft_id=info%3Aoclcnum%2F422727925&rft_id=info%3Adoi%2F10.1017%2FCBO9780511813870&rft.isbn=978-0-511-34258-5&rft.aulast=Mermin&rft.aufirst=N.+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNielsenChuang2010" class="citation book cs1"><a href="/wiki/Michael_Nielsen" title="Michael Nielsen">Nielsen, Michael</a>; <a href="/wiki/Isaac_L._Chuang" class="mw-redirect" title="Isaac L. Chuang">Chuang, Isaac</a> (2010). <a href="/wiki/Quantum_Computation_and_Quantum_Information" title="Quantum Computation and Quantum Information">Quantum Computation and Quantum Information</a> (10th anniversary ed.). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511976667">10.1017/CBO9780511976667</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-511-99277-3" title="Special:BookSources/978-0-511-99277-3"><bdi>978-0-511-99277-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/700706156">700706156</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:59717455">59717455</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computation+and+Quantum+Information&rft.edition=10th+anniversary&rft.date=2010&rft_id=info%3Aoclcnum%2F700706156&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59717455%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FCBO9780511976667&rft.isbn=978-0-511-99277-3&rft.aulast=Nielsen&rft.aufirst=Michael&rft.au=Chuang%2C+Isaac&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShor1994" class="citation conference cs1"><a href="/wiki/Peter_Shor" title="Peter Shor">Shor, Peter W.</a> (1994). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. <a href="/wiki/Symposium_on_Foundations_of_Computer_Science" title="Symposium on Foundations of Computer Science">Symposium on Foundations of Computer Science</a>. <a href="/wiki/Santa_Fe,_New_Mexico" title="Santa Fe, New Mexico">Santa Fe, New Mexico</a>: <a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a>. pp. 124–134. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FSFCS.1994.365700">10.1109/SFCS.1994.365700</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8186-6580-6" title="Special:BookSources/978-0-8186-6580-6"><bdi>978-0-8186-6580-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Algorithms+for+Quantum+Computation%3A+Discrete+Logarithms+and+Factoring&rft.place=Santa+Fe%2C+New+Mexico&rft.pages=124-134&rft.pub=IEEE&rft.date=1994&rft_id=info%3Adoi%2F10.1109%2FSFCS.1994.365700&rft.isbn=978-0-8186-6580-6&rft.aulast=Shor&rft.aufirst=Peter+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=36" title="Edit section: Further reading">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <div class="mw-heading mw-heading3"><h3 id="Textbooks">Textbooks</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=37" title="Edit section: Textbooks">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAkama2014" class="citation book cs1">Akama, Seiki (2014). Elements of Quantum Computing: History, Theories and Engineering Applications. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-08284-4">10.1007/978-3-319-08284-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-08284-4" title="Special:BookSources/978-3-319-08284-4"><bdi>978-3-319-08284-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/884786739">884786739</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Quantum+Computing%3A+History%2C+Theories+and+Engineering+Applications&rft.pub=Springer&rft.date=2014&rft_id=info%3Aoclcnum%2F884786739&rft_id=info%3Adoi%2F10.1007%2F978-3-319-08284-4&rft.isbn=978-3-319-08284-4&rft.aulast=Akama&rft.aufirst=Seiki&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenentiCasatiRossiniStrini2019" class="citation book cs1">Benenti, Giuliano; Casati, Giulio; Rossini, Davide; Strini, Giuliano (2019). Principles of Quantum Computation and Information: A Comprehensive Textbook (2nd ed.). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F10909">10.1142/10909</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-3237-23-0" title="Special:BookSources/978-981-3237-23-0"><bdi>978-981-3237-23-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1084428655">1084428655</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62280636">62280636</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Computation+and+Information%3A+A+Comprehensive+Textbook&rft.edition=2nd&rft.date=2019&rft_id=info%3Aoclcnum%2F1084428655&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62280636%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2F10909&rft.isbn=978-981-3237-23-0&rft.aulast=Benenti&rft.aufirst=Giuliano&rft.au=Casati%2C+Giulio&rft.au=Rossini%2C+Davide&rft.au=Strini%2C+Giuliano&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBernhardt2019" class="citation book cs1">Bernhardt, Chris (2019). Quantum Computing for Everyone. MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-35091-4" title="Special:BookSources/978-0-262-35091-4"><bdi>978-0-262-35091-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1082867954">1082867954</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing+for+Everyone&rft.pub=MIT+Press&rft.date=2019&rft_id=info%3Aoclcnum%2F1082867954&rft.isbn=978-0-262-35091-4&rft.aulast=Bernhardt&rft.aufirst=Chris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHidary2021" class="citation book cs1">Hidary, Jack D. (2021). Quantum Computing: An Applied Approach (2nd ed.). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-83274-2">10.1007/978-3-030-83274-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03-083274-2" title="Special:BookSources/978-3-03-083274-2"><bdi>978-3-03-083274-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1272953643">1272953643</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:238223274">238223274</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing%3A+An+Applied+Approach&rft.edition=2nd&rft.date=2021&rft_id=info%3Aoclcnum%2F1272953643&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A238223274%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-030-83274-2&rft.isbn=978-3-03-083274-2&rft.aulast=Hidary&rft.aufirst=Jack+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHiroshiMasahito2006" class="citation book cs1">Hiroshi, Imai; Masahito, Hayashi, eds. (2006). Quantum Computation and Information: From Theory to Experiment. Topics in Applied Physics. Vol. 102. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-33133-6">10.1007/3-540-33133-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33133-9" title="Special:BookSources/978-3-540-33133-9"><bdi>978-3-540-33133-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computation+and+Information%3A+From+Theory+to+Experiment&rft.series=Topics+in+Applied+Physics&rft.date=2006&rft_id=info%3Adoi%2F10.1007%2F3-540-33133-6&rft.isbn=978-3-540-33133-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughesIsaacsonPerrySun2021" class="citation book cs1">Hughes, Ciaran; Isaacson, Joshua; Perry, Anastasia; Sun, Ranbel F.; Turner, Jessica (2021). <a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1007/978-3-030-61601-4.pdf">Quantum Computing for the Quantum Curious</a> (PDF). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-61601-4">10.1007/978-3-030-61601-4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03-061601-4" title="Special:BookSources/978-3-03-061601-4"><bdi>978-3-03-061601-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1244536372">1244536372</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:242566636">242566636</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing+for+the+Quantum+Curious&rft.date=2021&rft_id=info%3Aoclcnum%2F1244536372&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A242566636%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-030-61601-4&rft.isbn=978-3-03-061601-4&rft.aulast=Hughes&rft.aufirst=Ciaran&rft.au=Isaacson%2C+Joshua&rft.au=Perry%2C+Anastasia&rft.au=Sun%2C+Ranbel+F.&rft.au=Turner%2C+Jessica&rft_id=https%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2F10.1007%2F978-3-030-61601-4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJaeger2007" class="citation book cs1">Jaeger, Gregg (2007). Quantum Information: An Overview. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-36944-0">10.1007/978-0-387-36944-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-36944-0" title="Special:BookSources/978-0-387-36944-0"><bdi>978-0-387-36944-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/186509710">186509710</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information%3A+An+Overview&rft.date=2007&rft_id=info%3Aoclcnum%2F186509710&rft_id=info%3Adoi%2F10.1007%2F978-0-387-36944-0&rft.isbn=978-0-387-36944-0&rft.aulast=Jaeger&rft.aufirst=Gregg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnstonHarriganGimeno-Segovia2019" class="citation book cs1">Johnston, Eric R.; Harrigan, Nic; Gimeno-Segovia, Mercedes (2019). Programming Quantum Computers: Essential Algorithms and Code Samples. O'Reilly Media, Incorporated. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4920-3968-6" title="Special:BookSources/978-1-4920-3968-6"><bdi>978-1-4920-3968-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1111634190">1111634190</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Programming+Quantum+Computers%3A+Essential+Algorithms+and+Code+Samples&rft.pub=O%27Reilly+Media%2C+Incorporated&rft.date=2019&rft_id=info%3Aoclcnum%2F1111634190&rft.isbn=978-1-4920-3968-6&rft.aulast=Johnston&rft.aufirst=Eric+R.&rft.au=Harrigan%2C+Nic&rft.au=Gimeno-Segovia%2C+Mercedes&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKayeLaflammeMosca2007" class="citation book cs1">Kaye, Phillip; <a href="/wiki/Raymond_Laflamme" title="Raymond Laflamme">Laflamme, Raymond</a>; <a href="/wiki/Michele_Mosca" title="Michele Mosca">Mosca, Michele</a> (2007). An Introduction to Quantum Computing. OUP Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-857000-4" title="Special:BookSources/978-0-19-857000-4"><bdi>978-0-19-857000-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/85896383">85896383</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Quantum+Computing&rft.pub=OUP+Oxford&rft.date=2007&rft_id=info%3Aoclcnum%2F85896383&rft.isbn=978-0-19-857000-4&rft.aulast=Kaye&rft.aufirst=Phillip&rft.au=Laflamme%2C+Raymond&rft.au=Mosca%2C+Michele&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKitaevShenVyalyi2002" class="citation book cs1"><a href="/wiki/Alexei_Kitaev" title="Alexei Kitaev">Kitaev, Alexei Yu.</a>; Shen, Alexander H.; Vyalyi, Mikhail N. (2002). Classical and Quantum Computation. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3229-5" title="Special:BookSources/978-0-8218-3229-5"><bdi>978-0-8218-3229-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/907358694">907358694</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+and+Quantum+Computation&rft.pub=American+Mathematical+Soc.&rft.date=2002&rft_id=info%3Aoclcnum%2F907358694&rft.isbn=978-0-8218-3229-5&rft.aulast=Kitaev&rft.aufirst=Alexei+Yu.&rft.au=Shen%2C+Alexander+H.&rft.au=Vyalyi%2C+Mikhail+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurgalinBorzunov2021" class="citation book cs1">Kurgalin, Sergei; Borzunov, Sergei (2021). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/978-3-030-65052-0">Concise Guide to Quantum Computing: Algorithms, Exercises, and Implementations</a>. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-030-65052-0">10.1007/978-3-030-65052-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-030-65052-0" title="Special:BookSources/978-3-030-65052-0"><bdi>978-3-030-65052-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Concise+Guide+to+Quantum+Computing%3A+Algorithms%2C+Exercises%2C+and+Implementations&rft.pub=Springer&rft.date=2021&rft_id=info%3Adoi%2F10.1007%2F978-3-030-65052-0&rft.isbn=978-3-030-65052-0&rft.aulast=Kurgalin&rft.aufirst=Sergei&rft.au=Borzunov%2C+Sergei&rft_id=https%3A%2F%2Fdx.doi.org%2F10.1007%2F978-3-030-65052-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStolzeSuter2004" class="citation book cs1">Stolze, Joachim; Suter, Dieter (2004). Quantum Computing: A Short Course from Theory to Experiment. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9783527617760">10.1002/9783527617760</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-61776-0" title="Special:BookSources/978-3-527-61776-0"><bdi>978-3-527-61776-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/212140089">212140089</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Computing%3A+A+Short+Course+from+Theory+to+Experiment&rft.date=2004&rft_id=info%3Aoclcnum%2F212140089&rft_id=info%3Adoi%2F10.1002%2F9783527617760&rft.isbn=978-3-527-61776-0&rft.aulast=Stolze&rft.aufirst=Joachim&rft.au=Suter%2C+Dieter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSusskindFriedman2014" class="citation book cs1"><a href="/wiki/Leonard_Susskind" title="Leonard Susskind">Susskind, Leonard</a>; Friedman, Art (2014). Quantum Mechanics: The Theoretical Minimum. <a href="/wiki/New_York_City" title="New York City">New York</a>: <a href="/wiki/Basic_Books" title="Basic Books">Basic Books</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-465-08061-8" title="Special:BookSources/978-0-465-08061-8"><bdi>978-0-465-08061-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics%3A+The+Theoretical+Minimum&rft.place=New+York&rft.pub=Basic+Books&rft.date=2014&rft.isbn=978-0-465-08061-8&rft.aulast=Susskind&rft.aufirst=Leonard&rft.au=Friedman%2C+Art&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWichert2020" class="citation book cs1">Wichert, Andreas (2020). Principles of Quantum Artificial Intelligence: Quantum Problem Solving and Machine Learning (2nd ed.). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F11938">10.1142/11938</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-2431-7" title="Special:BookSources/978-981-12-2431-7"><bdi>978-981-12-2431-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1178715016">1178715016</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:225498497">225498497</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Quantum+Artificial+Intelligence%3A+Quantum+Problem+Solving+and+Machine+Learning&rft.edition=2nd&rft.date=2020&rft_id=info%3Aoclcnum%2F1178715016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A225498497%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2F11938&rft.isbn=978-981-12-2431-7&rft.aulast=Wichert&rft.aufirst=Andreas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWong2022" class="citation book cs1">Wong, Thomas (2022). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220129214631/http://www.thomaswong.net/introduction-to-classical-and-quantum-computing-1e.pdf">Introduction to Classical and Quantum Computing</a> (PDF). Rooted Grove. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/979-8-9855931-0-5" title="Special:BookSources/979-8-9855931-0-5"><bdi>979-8-9855931-0-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1308951401">1308951401</a>. Archived from <a rel="nofollow" class="external text" href="http://www.thomaswong.net/introduction-to-classical-and-quantum-computing-1e.pdf">the original</a> (PDF) on 29 January 2022. Retrieved 6 February 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Classical+and+Quantum+Computing&rft.pub=Rooted+Grove&rft.date=2022&rft_id=info%3Aoclcnum%2F1308951401&rft.isbn=979-8-9855931-0-5&rft.aulast=Wong&rft.aufirst=Thomas&rft_id=http%3A%2F%2Fwww.thomaswong.net%2Fintroduction-to-classical-and-quantum-computing-1e.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZengChenZhouWen2019" class="citation book cs1">Zeng, Bei; Chen, Xie; Zhou, Duan-Lu; Wen, Xiao-Gang (2019). Quantum Information Meets Quantum Matter. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1508.02595">1508.02595</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4939-9084-9">10.1007/978-1-4939-9084-9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-9084-9" title="Special:BookSources/978-1-4939-9084-9"><bdi>978-1-4939-9084-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1091358969">1091358969</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118528258">118528258</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Information+Meets+Quantum+Matter&rft.date=2019&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118528258%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-1-4939-9084-9&rft_id=info%3Aoclcnum%2F1091358969&rft_id=info%3Aarxiv%2F1508.02595&rft.isbn=978-1-4939-9084-9&rft.aulast=Zeng&rft.aufirst=Bei&rft.au=Chen%2C+Xie&rft.au=Zhou%2C+Duan-Lu&rft.au=Wen%2C+Xiao-Gang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Academic_papers">Academic papers</h3>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=38" title="Edit section: Academic papers">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbbotDoeringCavesLidar2003" class="citation journal cs1"><a href="/wiki/Derek_Abbott" title="Derek Abbott">Abbot, Derek</a>; <a href="/wiki/Charles_R._Doering" title="Charles R. Doering">Doering, Charles R.</a>; <a href="/wiki/Carlton_M._Caves" title="Carlton M. Caves">Caves, Carlton M.</a>; <a href="/wiki/Daniel_Lidar" title="Daniel Lidar">Lidar, Daniel M.</a>; <a href="/wiki/Howard_Brandt" title="Howard Brandt">Brandt, Howard E.</a>; et al. (2003). "Dreams versus Reality: Plenary Debate Session on Quantum Computing". Quantum Information Processing. 2 (6): 449–472. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0310130">quant-ph/0310130</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003QuIP....2..449A">2003QuIP....2..449A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AQINP.0000042203.24782.9a">10.1023/B:QINP.0000042203.24782.9a</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F45526">2027.42/45526</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:34885835">34885835</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum+Information+Processing&rft.atitle=Dreams+versus+Reality%3A+Plenary+Debate+Session+on+Quantum+Computing&rft.volume=2&rft.issue=6&rft.pages=449-472&rft.date=2003&rft_id=info%3Ahdl%2F2027.42%2F45526&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A34885835%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2003QuIP....2..449A&rft_id=info%3Aarxiv%2Fquant-ph%2F0310130&rft_id=info%3Adoi%2F10.1023%2FB%3AQINP.0000042203.24782.9a&rft.aulast=Abbot&rft.aufirst=Derek&rft.au=Doering%2C+Charles+R.&rft.au=Caves%2C+Carlton+M.&rft.au=Lidar%2C+Daniel+M.&rft.au=Brandt%2C+Howard+E.&rft.au=Hamilton%2C+Alexander+R.&rft.au=Ferry%2C+David+K.&rft.au=Gea-Banacloche%2C+Julio&rft.au=Bezrukov%2C+Sergey+M.&rft.au=Kish%2C+Laszlo+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerthiaume1998" class="citation book cs1">Berthiaume, Andre (1 December 1998). "Quantum Computation". Solution Manual for Quantum Mechanics. pp. 233–234. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789814541893_0016">10.1142/9789814541893_0016</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4541-88-6" title="Special:BookSources/978-981-4541-88-6"><bdi>978-981-4541-88-6</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:128255429">128255429</a> – via Semantic Scholar.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantum+Computation&rft.btitle=Solution+Manual+for+Quantum+Mechanics&rft.pages=233-234&rft.date=1998-12-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A128255429%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2F9789814541893_0016&rft.isbn=978-981-4541-88-6&rft.aulast=Berthiaume&rft.aufirst=Andre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiVincenzo2000" class="citation journal cs1"><a href="/wiki/David_DiVincenzo" title="David DiVincenzo">DiVincenzo, David P.</a> (2000). "The Physical Implementation of Quantum Computation". Fortschritte der Physik. 48 (9–11): 771–783. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0002077">quant-ph/0002077</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000ForPh..48..771D">2000ForPh..48..771D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F1521-3978%28200009%2948%3A9%2F11%3C771%3A%3AAID-PROP771%3E3.0.CO%3B2-E">10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15439711">15439711</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fortschritte+der+Physik&rft.atitle=The+Physical+Implementation+of+Quantum+Computation&rft.volume=48&rft.issue=9%E2%80%9311&rft.pages=771-783&rft.date=2000&rft_id=info%3Aarxiv%2Fquant-ph%2F0002077&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15439711%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2F1521-3978%28200009%2948%3A9%2F11%3C771%3A%3AAID-PROP771%3E3.0.CO%3B2-E&rft_id=info%3Abibcode%2F2000ForPh..48..771D&rft.aulast=DiVincenzo&rft.aufirst=David+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiVincenzo1995" class="citation journal cs1">DiVincenzo, David P. (1995). "Quantum Computation". Science. 270 (5234): 255–261. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995Sci...270..255D">1995Sci...270..255D</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.242.2165">10.1.1.242.2165</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.270.5234.255">10.1126/science.270.5234.255</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220110562">220110562</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Quantum+Computation&rft.volume=270&rft.issue=5234&rft.pages=255-261&rft.date=1995&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.242.2165%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A220110562%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1126%2Fscience.270.5234.255&rft_id=info%3Abibcode%2F1995Sci...270..255D&rft.aulast=DiVincenzo&rft.aufirst=David+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"> Table 1 lists switching and dephasing times for various systems.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynman1982" class="citation journal cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Feynman, Richard</a> (1982). "Simulating physics with computers". International Journal of Theoretical Physics. 21 (6–7): 467–488. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982IJTP...21..467F">1982IJTP...21..467F</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.9310">10.1.1.45.9310</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02650179">10.1007/BF02650179</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124545445">124545445</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Theoretical+Physics&rft.atitle=Simulating+physics+with+computers&rft.volume=21&rft.issue=6%E2%80%937&rft.pages=467-488&rft.date=1982&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.45.9310%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124545445%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF02650179&rft_id=info%3Abibcode%2F1982IJTP...21..467F&rft.aulast=Feynman&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJeutner2021" class="citation journal cs1">Jeutner, Valentin (2021). <a rel="nofollow" class="external text" href="https://lup.lub.lu.se/record/e034e7b7-d17c-4863-9cee-7e654f97225b">"The Quantum Imperative: Addressing the Legal Dimension of Quantum Computers"</a>. Morals & Machines. 1 (1): 52–59. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5771%2F2747-5174-2021-1-52">10.5771/2747-5174-2021-1-52</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:236664155">236664155</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Morals+%26+Machines&rft.atitle=The+Quantum+Imperative%3A+Addressing+the+Legal+Dimension+of+Quantum+Computers&rft.volume=1&rft.issue=1&rft.pages=52-59&rft.date=2021&rft_id=info%3Adoi%2F10.5771%2F2747-5174-2021-1-52&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A236664155%23id-name%3DS2CID&rft.aulast=Jeutner&rft.aufirst=Valentin&rft_id=https%3A%2F%2Flup.lub.lu.se%2Frecord%2Fe034e7b7-d17c-4863-9cee-7e654f97225b&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrantzKjaergaardYanOrlando2019" class="citation journal cs1">Krantz, P.; Kjaergaard, M.; Yan, F.; Orlando, T. P.; Gustavsson, S.; Oliver, W. D. (17 June 2019). "A Quantum Engineer's Guide to Superconducting Qubits". <a href="/wiki/Applied_Physics_Reviews" title="Applied Physics Reviews">Applied Physics Reviews</a>. 6 (2): 021318. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1904.06560">1904.06560</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019ApPRv...6b1318K">2019ApPRv...6b1318K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.5089550">10.1063/1.5089550</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1931-9401">1931-9401</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119104251">119104251</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Physics+Reviews&rft.atitle=A+Quantum+Engineer%27s+Guide+to+Superconducting+Qubits&rft.volume=6&rft.issue=2&rft.pages=021318&rft.date=2019-06-17&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119104251%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2019ApPRv...6b1318K&rft_id=info%3Aarxiv%2F1904.06560&rft.issn=1931-9401&rft_id=info%3Adoi%2F10.1063%2F1.5089550&rft.aulast=Krantz&rft.aufirst=P.&rft.au=Kjaergaard%2C+M.&rft.au=Yan%2C+F.&rft.au=Orlando%2C+T.+P.&rft.au=Gustavsson%2C+S.&rft.au=Oliver%2C+W.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMitchell1998" class="citation web cs1">Mitchell, Ian (1998). <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/mitchell98computing.html">"Computing Power into the 21st Century: Moore's Law and Beyond"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Computing+Power+into+the+21st+Century%3A+Moore%27s+Law+and+Beyond&rft.date=1998&rft.aulast=Mitchell&rft.aufirst=Ian&rft_id=http%3A%2F%2Fciteseer.ist.psu.edu%2Fmitchell98computing.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSimon1994" class="citation web cs1">Simon, Daniel R. (1994). <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/simon94power.html">"On the Power of Quantum Computation"</a>. Institute of Electrical and Electronics Engineers Computer Society Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=On+the+Power+of+Quantum+Computation&rft.pub=Institute+of+Electrical+and+Electronics+Engineers+Computer+Society+Press&rft.date=1994&rft.aulast=Simon&rft.aufirst=Daniel+R.&rft_id=http%3A%2F%2Fciteseer.ist.psu.edu%2Fsimon94power.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Quantum_computing&action=edit&section=39" title="Edit section: External links">edit</a>]</div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Quantum_computer" class="extiw" title="commons:Quantum computer">Quantum computer</a> at Wikimedia Commons</li> <li><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> Learning materials related to <a href="https://en.wikiversity.org/wiki/Quantum_computing" class="extiw" title="v:Quantum computing">Quantum computing</a> at Wikiversity</li> <li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: "<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/qt-quantcomp/">Quantum Computing</a>" by Amit Hagar and Michael E. Cuffaro.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Quantum_computation,_theory_of">"Quantum computation, theory of"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Quantum+computation%2C+theory+of&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DQuantum_computation%2C_theory_of&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+computing" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://quantum.country/qcvc">Quantum computing for the very curious</a> by Andy Matuschak and <a href="/wiki/Michael_Nielsen" title="Michael Nielsen">Michael Nielsen</a></li></ul> <dl><dt>Lectures</dt></dl> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/playlist?list=PL1826E60FD05B44E4">Quantum computing for the determined</a> – 22 video lectures by <a href="/wiki/Michael_Nielsen" title="Michael Nielsen">Michael Nielsen</a></li> <li><a rel="nofollow" class="external text" href="http://www.quiprocone.org/Protected/DD_lectures.htm">Video Lectures</a> by <a href="/wiki/David_Deutsch" title="David Deutsch">David Deutsch</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303183533/http://www.quantware.ups-tlse.fr/IHP2006/">Lectures at the Institut Henri Poincaré (slides and videos)</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130901004919/http://nanohub.org/resources/4778">Online lecture on An Introduction to Quantum Computing, Edward Gerjuoy (2008)</a></li> <li>Lomonaco, Sam. <a rel="nofollow" class="external text" href="http://www.csee.umbc.edu/~lomonaco/Lectures.html#OxfordLectures">Four Lectures on Quantum Computing given at Oxford University in July 2006</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist 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a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Processor_technologies" title="Template:Processor technologies"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Processor_technologies" title="Template talk:Processor technologies"><abbr title="Discuss this template">t</abbr></a></li><li 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class="mw-redirect" title="Finite-state machine with datapath">with datapath</a></li> <li><a href="/wiki/Hierarchical_state_machine" class="mw-redirect" title="Hierarchical state machine">Hierarchical</a></li> <li><a href="/wiki/Deterministic_finite_automaton" title="Deterministic finite automaton">Deterministic finite automaton</a></li> <li><a href="/wiki/Queue_automaton" title="Queue automaton">Queue automaton</a></li> <li><a href="/wiki/Cellular_automaton" title="Cellular automaton">Cellular automaton</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automaton</a></li></ul></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> <ul><li><a href="/wiki/Alternating_Turing_machine" title="Alternating Turing machine">Alternating Turing machine</a></li> <li><a href="/wiki/Universal_Turing_machine" title="Universal Turing machine">Universal</a></li> <li><a href="/wiki/Post%E2%80%93Turing_machine" title="Post–Turing machine">Post–Turing</a></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum</a></li> <li><a href="/wiki/Nondeterministic_Turing_machine" title="Nondeterministic Turing machine">Nondeterministic Turing machine</a></li> <li><a href="/wiki/Probabilistic_Turing_machine" title="Probabilistic Turing machine">Probabilistic Turing machine</a></li> <li><a href="/wiki/Hypercomputation" title="Hypercomputation">Hypercomputation</a></li> <li><a href="/wiki/Zeno_machine" title="Zeno machine">Zeno machine</a></li></ul></li> <li><a href="/wiki/History_of_general-purpose_CPUs#Belt_machine_architecture" title="History of general-purpose CPUs">Belt machine</a></li> <li><a href="/wiki/Stack_machine" title="Stack machine">Stack machine</a></li> <li><a href="/wiki/Register_machine" title="Register machine">Register machines</a> <ul><li><a href="/wiki/Counter_machine" title="Counter machine">Counter</a></li> <li><a href="/wiki/Pointer_machine" title="Pointer machine">Pointer</a></li> <li><a href="/wiki/Random-access_machine" title="Random-access machine">Random-access</a></li> <li><a href="/wiki/Random-access_stored-program_machine" title="Random-access stored-program machine">Random-access stored program</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_architecture" title="Computer architecture">Architecture</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Microarchitecture" title="Microarchitecture">Microarchitecture</a></li> <li><a href="/wiki/Von_Neumann_architecture" title="Von Neumann architecture">Von Neumann</a></li> <li><a href="/wiki/Harvard_architecture" title="Harvard architecture">Harvard</a> <ul><li><a href="/wiki/Modified_Harvard_architecture" title="Modified Harvard architecture">modified</a></li></ul></li> <li><a href="/wiki/Dataflow_architecture" title="Dataflow architecture">Dataflow</a></li> <li><a href="/wiki/Transport_triggered_architecture" title="Transport triggered architecture">Transport-triggered</a></li> <li><a href="/wiki/Cellular_architecture" title="Cellular architecture">Cellular</a></li> <li><a href="/wiki/Endianness" title="Endianness">Endianness</a></li> <li><a href="/wiki/Computer_data_storage" title="Computer data storage">Memory access</a> <ul><li><a href="/wiki/Non-uniform_memory_access" title="Non-uniform memory access">NUMA</a></li> <li><a href="/wiki/Uniform_memory_access" title="Uniform memory access">HUMA</a></li> <li><a href="/wiki/Load%E2%80%93store_architecture" title="Load–store architecture">Load–store</a></li> <li><a href="/wiki/Register%E2%80%93memory_architecture" title="Register–memory architecture">Register/memory</a></li></ul></li> <li><a href="/wiki/Cache_hierarchy" title="Cache hierarchy">Cache hierarchy</a></li> <li><a href="/wiki/Memory_hierarchy" title="Memory hierarchy">Memory hierarchy</a> <ul><li><a href="/wiki/Virtual_memory" title="Virtual memory">Virtual memory</a></li> <li><a href="/wiki/Secondary_storage" class="mw-redirect" title="Secondary storage">Secondary storage</a></li></ul></li> <li><a href="/wiki/Heterogeneous_System_Architecture" title="Heterogeneous System Architecture">Heterogeneous</a></li> <li><a href="/wiki/Fabric_computing" title="Fabric computing">Fabric</a></li> <li><a href="/wiki/Multiprocessing" title="Multiprocessing">Multiprocessing</a></li> <li><a href="/wiki/Cognitive_computing" title="Cognitive computing">Cognitive</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Instruction_set_architecture" title="Instruction set architecture">Instruction set architectures</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonal_instruction_set" title="Orthogonal instruction set">Orthogonal instruction set</a></li> <li><a href="/wiki/Complex_instruction_set_computer" title="Complex instruction set computer">CISC</a></li> <li><a href="/wiki/Reduced_instruction_set_computer" title="Reduced instruction set computer">RISC</a></li> <li><a href="/wiki/Application-specific_instruction_set_processor" title="Application-specific instruction set processor">Application-specific</a></li> <li><a href="/wiki/Explicit_data_graph_execution" title="Explicit data graph execution">EDGE</a> <ul><li><a href="/wiki/TRIPS_architecture" title="TRIPS architecture">TRIPS</a></li></ul></li> <li><a href="/wiki/Very_long_instruction_word" title="Very long instruction word">VLIW</a> <ul><li><a href="/wiki/Explicitly_parallel_instruction_computing" title="Explicitly parallel instruction computing">EPIC</a></li></ul></li> <li><a href="/wiki/Minimal_instruction_set_computer" title="Minimal instruction set computer">MISC</a></li> <li><a href="/wiki/One-instruction_set_computer" title="One-instruction set computer">OISC</a></li> <li><a href="/wiki/No_instruction_set_computing" title="No instruction set computing">NISC</a></li> <li><a href="/wiki/Zero_instruction_set_computer" class="mw-redirect" title="Zero instruction set computer">ZISC</a></li> <li><a href="/wiki/VISC_architecture" title="VISC architecture">VISC architecture</a></li> <li><a class="mw-selflink selflink">Quantum computing</a></li> <li><a href="/wiki/Comparison_of_instruction_set_architectures" title="Comparison of instruction set architectures">Comparison</a> <ul><li><a href="/wiki/Addressing_mode" title="Addressing mode">Addressing modes</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Instruction sets</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Motorola_68000_series" title="Motorola 68000 series">Motorola 68000 series</a></li> <li><a href="/wiki/VAX" title="VAX">VAX</a></li> <li><a href="/wiki/PDP-11_architecture" title="PDP-11 architecture">PDP-11</a></li> <li><a href="/wiki/X86" title="X86">x86</a></li> <li><a href="/wiki/ARM_architecture_family" title="ARM architecture family">ARM</a></li> <li><a href="/wiki/Stanford_MIPS" title="Stanford MIPS">Stanford MIPS</a></li> <li><a href="/wiki/MIPS_architecture" title="MIPS architecture">MIPS</a></li> <li><a href="/wiki/MIPS-X" title="MIPS-X">MIPS-X</a></li> <li>Power <ul><li><a href="/wiki/IBM_POWER_architecture" title="IBM POWER architecture">POWER</a></li> <li><a href="/wiki/PowerPC" title="PowerPC">PowerPC</a></li> <li><a href="/wiki/Power_ISA" title="Power ISA">Power ISA</a></li></ul></li> <li><a href="/wiki/Clipper_architecture" title="Clipper architecture">Clipper architecture</a></li> <li><a href="/wiki/SPARC" title="SPARC">SPARC</a></li> <li><a href="/wiki/SuperH" title="SuperH">SuperH</a></li> <li><a href="/wiki/DEC_Alpha" title="DEC Alpha">DEC Alpha</a></li> <li><a href="/wiki/ETRAX_CRIS" title="ETRAX CRIS">ETRAX CRIS</a></li> <li><a href="/wiki/M32R" title="M32R">M32R</a></li> <li><a href="/wiki/Unicore" title="Unicore">Unicore</a></li> <li><a href="/wiki/IA-64" title="IA-64">Itanium</a></li> <li><a href="/wiki/OpenRISC" title="OpenRISC">OpenRISC</a></li> <li><a href="/wiki/RISC-V" title="RISC-V">RISC-V</a></li> <li><a href="/wiki/MicroBlaze" title="MicroBlaze">MicroBlaze</a></li> <li><a href="/wiki/Little_man_computer" title="Little man computer">LMC</a></li> <li>System/3x0 <ul><li><a href="/wiki/IBM_System/360_architecture" title="IBM System/360 architecture">S/360</a></li> <li><a href="/wiki/IBM_System/370" title="IBM System/370">S/370</a></li> <li><a href="/wiki/IBM_System/390" title="IBM System/390">S/390</a></li> <li><a href="/wiki/Z/Architecture" title="Z/Architecture">z/Architecture</a></li></ul></li> <li>Tilera ISA</li> <li><a href="/wiki/VISC_architecture" title="VISC architecture">VISC architecture</a></li> <li><a href="/wiki/Adapteva#Products" class="mw-redirect" title="Adapteva">Epiphany architecture</a></li> <li><a href="/wiki/Comparison_of_instruction_set_architectures" title="Comparison of instruction set architectures">Others</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Instruction_cycle" title="Instruction cycle">Execution</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Instruction_pipelining" title="Instruction pipelining">Instruction pipelining</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pipeline_stall" title="Pipeline stall">Pipeline stall</a></li> <li><a href="/wiki/Operand_forwarding" title="Operand forwarding">Operand forwarding</a></li> <li><a href="/wiki/Classic_RISC_pipeline" title="Classic RISC pipeline">Classic RISC pipeline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hazard_(computer_architecture)" title="Hazard (computer architecture)">Hazards</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Data_dependency" title="Data dependency">Data dependency</a></li> <li><a href="/wiki/Structural_hazard" class="mw-redirect" title="Structural hazard">Structural</a></li> <li><a href="/wiki/Control_hazard" class="mw-redirect" title="Control hazard">Control</a></li> <li><a href="/wiki/False_sharing" title="False sharing">False sharing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Out-of-order_execution" title="Out-of-order execution">Out-of-order</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scoreboarding" title="Scoreboarding">Scoreboarding</a></li> <li><a href="/wiki/Tomasulo%27s_algorithm" title="Tomasulo's algorithm">Tomasulo's algorithm</a> <ul><li><a href="/wiki/Reservation_station" title="Reservation station">Reservation station</a></li> <li><a href="/wiki/Re-order_buffer" title="Re-order buffer">Re-order buffer</a></li></ul></li> <li><a href="/wiki/Register_renaming" title="Register renaming">Register renaming</a></li> <li><a href="/wiki/Wide-issue" title="Wide-issue">Wide-issue</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Speculative_execution" title="Speculative execution">Speculative</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Branch_predictor" title="Branch predictor">Branch prediction</a></li> <li><a href="/wiki/Memory_dependence_prediction" title="Memory dependence prediction">Memory dependence prediction</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Parallel_computing" title="Parallel computing">Parallelism</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Level</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit-level_parallelism" title="Bit-level parallelism">Bit</a> <ul><li><a href="/wiki/Bit-serial_architecture" title="Bit-serial architecture">Bit-serial</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li></ul></li> <li><a href="/wiki/Instruction-level_parallelism" title="Instruction-level parallelism">Instruction</a></li> <li><a href="/wiki/Instruction_pipelining" title="Instruction pipelining">Pipelining</a> <ul><li><a href="/wiki/Scalar_processor" title="Scalar processor">Scalar</a></li> <li><a href="/wiki/Superscalar_processor" title="Superscalar processor">Superscalar</a></li></ul></li> <li><a href="/wiki/Task_parallelism" title="Task parallelism">Task</a> <ul><li><a href="/wiki/Thread_(computing)" title="Thread (computing)">Thread</a></li> <li><a href="/wiki/Process_(computing)" title="Process (computing)">Process</a></li></ul></li> <li><a href="/wiki/Data_parallelism" title="Data parallelism">Data</a> <ul><li><a href="/wiki/Vector_processor" title="Vector processor">Vector</a></li></ul></li> <li><a href="/wiki/Memory-level_parallelism" title="Memory-level parallelism">Memory</a></li> <li><a href="/wiki/Distributed_architecture" class="mw-redirect" title="Distributed architecture">Distributed</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multithreading_(computer_architecture)" title="Multithreading (computer architecture)">Multithreading</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Temporal_multithreading" title="Temporal multithreading">Temporal</a></li> <li><a href="/wiki/Simultaneous_multithreading" title="Simultaneous multithreading">Simultaneous</a> <ul><li><a href="/wiki/Hyper-threading" title="Hyper-threading">Hyperthreading</a></li> <li><a href="/wiki/Simultaneous_and_heterogeneous_multithreading" title="Simultaneous and heterogeneous multithreading">Simultaneous and heterogenous</a></li></ul></li> <li><a href="/wiki/Speculative_multithreading" title="Speculative multithreading">Speculative</a></li> <li><a href="/wiki/Preemption_(computing)" title="Preemption (computing)">Preemptive</a></li> <li><a href="/wiki/Cooperative_multitasking" title="Cooperative multitasking">Cooperative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Flynn%27s_taxonomy" title="Flynn's taxonomy">Flynn's taxonomy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single_instruction,_single_data" title="Single instruction, single data">SISD</a></li> <li><a href="/wiki/Single_instruction,_multiple_data" title="Single instruction, multiple data">SIMD</a> <ul><li><a href="/wiki/Single_instruction,_multiple_threads" title="Single instruction, multiple threads">Array processing (SIMT)</a></li> <li><a href="/wiki/Flynn%27s_taxonomy#Pipelined_processor" title="Flynn's taxonomy">Pipelined processing</a></li> <li><a href="/wiki/Flynn%27s_taxonomy#Associative_processor" title="Flynn's taxonomy">Associative processing</a></li> <li><a href="/wiki/SWAR" title="SWAR">SWAR</a></li></ul></li> <li><a href="/wiki/Multiple_instruction,_single_data" title="Multiple instruction, single data">MISD</a></li> <li><a href="/wiki/Multiple_instruction,_multiple_data" title="Multiple instruction, multiple data">MIMD</a> <ul><li><a href="/wiki/Single_program,_multiple_data" title="Single program, multiple data">SPMD</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computer_performance" title="Computer performance">Processor performance</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Transistor_count" title="Transistor count">Transistor count</a></li> <li><a href="/wiki/Instructions_per_cycle" title="Instructions per cycle">Instructions per cycle</a> (IPC) <ul><li><a href="/wiki/Cycles_per_instruction" title="Cycles per instruction">Cycles per instruction</a> (CPI)</li></ul></li> <li><a href="/wiki/Instructions_per_second" title="Instructions per second">Instructions per second</a> (IPS)</li> <li><a href="/wiki/FLOPS" class="mw-redirect" title="FLOPS">Floating-point operations per second</a> (FLOPS)</li> <li><a href="/wiki/Transactions_per_second" title="Transactions per second">Transactions per second</a> (TPS)</li> <li><a href="/wiki/SUPS" title="SUPS">Synaptic updates per second</a> (SUPS)</li> <li><a href="/wiki/Performance_per_watt" title="Performance per watt">Performance per watt</a> (PPW)</li> <li><a href="/wiki/Cache_performance_measurement_and_metric" title="Cache performance measurement and metric">Cache performance metrics</a></li> <li><a href="/wiki/Computer_performance_by_orders_of_magnitude" title="Computer performance by orders of magnitude">Computer performance by orders of magnitude</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Processor_(computing)" title="Processor (computing)">Types</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_processing_unit" title="Central processing unit">Central processing unit</a> (CPU)</li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a> (GPU) <ul><li><a href="/wiki/General-purpose_computing_on_graphics_processing_units" title="General-purpose computing on graphics processing units">GPGPU</a></li></ul></li> <li><a href="/wiki/Vector_processor" title="Vector processor">Vector</a></li> <li><a href="/wiki/Barrel_processor" title="Barrel processor">Barrel</a></li> <li><a href="/wiki/Stream_processing" title="Stream processing">Stream</a></li> <li><a href="/wiki/Tile_processor" title="Tile processor">Tile processor</a></li> <li><a href="/wiki/Coprocessor" title="Coprocessor">Coprocessor</a></li> <li><a href="/wiki/Programmable_Array_Logic" title="Programmable Array Logic">PAL</a></li> <li><a href="/wiki/Application-specific_integrated_circuit" title="Application-specific integrated circuit">ASIC</a></li> <li><a href="/wiki/Field-programmable_gate_array" title="Field-programmable gate array">FPGA</a></li> <li><a href="/wiki/Field-programmable_object_array" title="Field-programmable object array">FPOA</a></li> <li><a href="/wiki/Complex_programmable_logic_device" title="Complex programmable logic device">CPLD</a></li> <li><a href="/wiki/Multi-chip_module" title="Multi-chip module">Multi-chip module</a> (MCM)</li> <li><a href="/wiki/System_in_a_package" title="System in a package">System in a package</a> (SiP)</li> <li><a href="/wiki/Package_on_a_package" title="Package on a package">Package on a package</a> (PoP)</li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">By application</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Embedded_system" title="Embedded system">Embedded system</a></li> <li><a href="/wiki/Microprocessor" title="Microprocessor">Microprocessor</a></li> <li><a href="/wiki/Microcontroller" title="Microcontroller">Microcontroller</a></li> <li><a href="/wiki/Mobile_processor" title="Mobile processor">Mobile</a></li> <li><a href="/wiki/Ultra-low-voltage_processor" title="Ultra-low-voltage processor">Ultra-low-voltage</a></li> <li><a href="/wiki/Application-specific_instruction_set_processor" title="Application-specific instruction set processor">ASIP</a></li> <li><a href="/wiki/Soft_microprocessor" title="Soft microprocessor">Soft microprocessor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Systems on chip</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/System_on_a_chip" title="System on a chip">System on a chip</a> (SoC)</li> <li><a href="/wiki/Multiprocessor_system_on_a_chip" class="mw-redirect" title="Multiprocessor system on a chip">Multiprocessor</a> (MPSoC)</li> <li><a href="/wiki/Cypress_PSoC" title="Cypress PSoC">Cypress PSoC</a></li> <li><a href="/wiki/Network_on_a_chip" title="Network on a chip">Network on a chip</a> (NoC)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hardware_acceleration" title="Hardware acceleration">Hardware accelerators</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coprocessor" title="Coprocessor">Coprocessor</a></li> <li><a href="/wiki/AI_accelerator" title="AI accelerator">AI accelerator</a></li> <li><a href="/wiki/Graphics_processing_unit" title="Graphics processing unit">Graphics processing unit</a> (GPU)</li> <li><a href="/wiki/Image_processor" title="Image processor">Image processor</a></li> <li><a href="/wiki/Vision_processing_unit" title="Vision processing unit">Vision processing unit</a> (VPU)</li> <li><a href="/wiki/Physics_processing_unit" title="Physics processing unit">Physics processing unit</a> (PPU)</li> <li><a href="/wiki/Digital_signal_processor" title="Digital signal processor">Digital signal processor</a> (DSP)</li> <li><a href="/wiki/Tensor_Processing_Unit" title="Tensor Processing Unit">Tensor Processing Unit</a> (TPU)</li> <li><a href="/wiki/Secure_cryptoprocessor" title="Secure cryptoprocessor">Secure cryptoprocessor</a></li> <li><a href="/wiki/Network_processor" title="Network processor">Network processor</a></li> <li><a href="/wiki/Baseband_processor" title="Baseband processor">Baseband processor</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word size</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1-bit_computing" title="1-bit computing">1-bit</a></li> <li><a href="/wiki/4-bit_computing" title="4-bit computing">4-bit</a></li> <li><a href="/wiki/8-bit_computing" title="8-bit computing">8-bit</a></li> <li><a href="/wiki/12-bit_computing" title="12-bit computing">12-bit</a></li> <li><a href="/wiki/Apollo_Guidance_Computer" title="Apollo Guidance Computer">15-bit</a></li> <li><a href="/wiki/16-bit_computing" title="16-bit computing">16-bit</a></li> <li><a href="/wiki/24-bit_computing" title="24-bit computing">24-bit</a></li> <li><a href="/wiki/32-bit_computing" title="32-bit computing">32-bit</a></li> <li><a href="/wiki/48-bit_computing" title="48-bit computing">48-bit</a></li> <li><a href="/wiki/64-bit_computing" title="64-bit computing">64-bit</a></li> <li><a href="/wiki/128-bit_computing" title="128-bit computing">128-bit</a></li> <li><a href="/wiki/256-bit_computing" title="256-bit computing">256-bit</a></li> <li><a href="/wiki/512-bit_computing" title="512-bit computing">512-bit</a></li> <li><a href="/wiki/Bit_slicing" title="Bit slicing">bit slicing</a></li> <li><a href="/wiki/Word_(computer_architecture)#Table_of_word_sizes" title="Word (computer architecture)">others</a> <ul><li><a href="/wiki/Word_(computer_architecture)#Variable-word_architectures" title="Word (computer architecture)">variable</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Core count</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single-core" title="Single-core">Single-core</a></li> <li><a href="/wiki/Multi-core_processor" title="Multi-core processor">Multi-core</a></li> <li><a href="/wiki/Manycore_processor" title="Manycore processor">Manycore</a></li> <li><a href="/wiki/Heterogeneous_computing" title="Heterogeneous computing">Heterogeneous architecture</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Components</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_processing_unit" title="Central processing unit">Core</a></li> <li><a href="/wiki/Cache_(computing)" title="Cache (computing)">Cache</a> <ul><li><a href="/wiki/CPU_cache" title="CPU cache">CPU cache</a></li> <li><a href="/wiki/Scratchpad_memory" title="Scratchpad memory">Scratchpad memory</a></li> <li><a href="/wiki/Data_cache" class="mw-redirect" title="Data cache">Data cache</a></li> <li><a href="/wiki/Instruction_cache" class="mw-redirect" title="Instruction cache">Instruction cache</a></li> <li><a href="/wiki/Cache_replacement_policies" title="Cache replacement policies">replacement policies</a></li> <li><a href="/wiki/Cache_coherence" title="Cache coherence">coherence</a></li></ul></li> <li><a href="/wiki/Bus_(computing)" title="Bus (computing)">Bus</a></li> <li><a href="/wiki/Clock_rate" title="Clock rate">Clock rate</a></li> <li><a href="/wiki/Clock_signal" title="Clock signal">Clock signal</a></li> <li><a href="/wiki/FIFO_(computing_and_electronics)" title="FIFO (computing and electronics)">FIFO</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Execution_unit" title="Execution unit">Functional units</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_logic_unit" title="Arithmetic logic unit">Arithmetic logic unit</a> (ALU)</li> <li><a href="/wiki/Address_generation_unit" title="Address generation unit">Address generation unit</a> (AGU)</li> <li><a href="/wiki/Floating-point_unit" title="Floating-point unit">Floating-point unit</a> (FPU)</li> <li><a href="/wiki/Memory_management_unit" title="Memory management unit">Memory management unit</a> (MMU) <ul><li><a href="/wiki/Load%E2%80%93store_unit" title="Load–store unit">Load–store unit</a></li> <li><a href="/wiki/Translation_lookaside_buffer" title="Translation lookaside buffer">Translation lookaside buffer</a> (TLB)</li></ul></li> <li><a href="/wiki/Branch_predictor" title="Branch predictor">Branch predictor</a></li> <li><a href="/wiki/Branch_target_predictor" title="Branch target predictor">Branch target predictor</a></li> <li><a href="/wiki/Memory_controller" title="Memory controller">Integrated memory controller</a> (IMC) <ul><li><a href="/wiki/Memory_management_unit" title="Memory management unit">Memory management unit</a></li></ul></li> <li><a href="/wiki/Instruction_decoder" class="mw-redirect" title="Instruction decoder">Instruction decoder</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Logic_gate" title="Logic gate">Logic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinational_logic" title="Combinational logic">Combinational</a></li> <li><a href="/wiki/Sequential_logic" title="Sequential logic">Sequential</a></li> <li><a href="/wiki/Glue_logic" title="Glue logic">Glue</a></li> <li><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum</a></li> <li><a href="/wiki/Gate_array" title="Gate array">Array</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hardware_register" title="Hardware register">Registers</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Processor_register" title="Processor register">Processor register</a></li> <li><a href="/wiki/Status_register" title="Status register">Status register</a></li> <li><a href="/wiki/Stack_register" title="Stack register">Stack register</a></li> <li><a href="/wiki/Register_file" title="Register file">Register file</a></li> <li><a href="/wiki/Memory_buffer_register" title="Memory buffer register">Memory buffer</a></li> <li><a href="/wiki/Memory_address_register" title="Memory address register">Memory address register</a></li> <li><a href="/wiki/Program_counter" title="Program counter">Program counter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_unit" title="Control unit">Control unit</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hardwired_control_unit" class="mw-redirect" title="Hardwired control unit">Hardwired control unit</a></li> <li><a href="/wiki/Instruction_unit" title="Instruction unit">Instruction unit</a></li> <li><a href="/wiki/Data_buffer" title="Data buffer">Data buffer</a></li> <li><a href="/wiki/Write_buffer" title="Write buffer">Write buffer</a></li> <li><a href="/wiki/Microcode" title="Microcode">Microcode</a> <a href="/wiki/ROM_image" title="ROM image">ROM</a></li> <li><a href="/wiki/Counter_(digital)" title="Counter (digital)">Counter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Datapath" title="Datapath">Datapath</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Multiplexer" title="Multiplexer">Multiplexer</a></li> <li><a href="/wiki/Demultiplexer" class="mw-redirect" title="Demultiplexer">Demultiplexer</a></li> <li><a href="/wiki/Adder_(electronics)" title="Adder (electronics)">Adder</a></li> <li><a href="/wiki/Binary_multiplier" title="Binary multiplier">Multiplier</a> <ul><li><a href="/wiki/CPU_multiplier" title="CPU multiplier">CPU</a></li></ul></li> <li><a href="/wiki/Binary_decoder" title="Binary decoder">Binary decoder</a> <ul><li><a href="/wiki/Address_decoder" title="Address decoder">Address decoder</a></li> <li><a href="/wiki/Sum-addressed_decoder" title="Sum-addressed decoder">Sum-addressed decoder</a></li></ul></li> <li><a href="/wiki/Barrel_shifter" title="Barrel shifter">Barrel shifter</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Electronic_circuit" title="Electronic circuit">Circuitry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a> <ul><li><a href="/wiki/Three-dimensional_integrated_circuit" title="Three-dimensional integrated circuit">3D</a></li> <li><a href="/wiki/Mixed-signal_integrated_circuit" title="Mixed-signal integrated circuit">Mixed-signal</a></li> <li><a href="/wiki/Power_management_integrated_circuit" title="Power management integrated circuit">Power management</a></li></ul></li> <li><a href="/wiki/Boolean_circuit" title="Boolean circuit">Boolean</a></li> <li><a href="/wiki/Circuit_(computer_science)" title="Circuit (computer science)">Digital</a></li> <li><a href="/wiki/Analogue_electronics" title="Analogue electronics">Analog</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum</a></li> <li><a href="/wiki/Switch#Electronic_switches" title="Switch">Switch</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Power_management" title="Power management">Power management</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Power_Management_Unit" title="Power Management Unit">PMU</a></li> <li><a href="/wiki/Advanced_Power_Management" title="Advanced Power Management">APM</a></li> <li><a href="/wiki/ACPI" title="ACPI">ACPI</a></li> <li><a href="/wiki/Dynamic_frequency_scaling" title="Dynamic frequency scaling">Dynamic frequency scaling</a></li> <li><a href="/wiki/Dynamic_voltage_scaling" title="Dynamic voltage scaling">Dynamic voltage scaling</a></li> <li><a href="/wiki/Clock_gating" title="Clock gating">Clock gating</a></li> <li><a href="/wiki/Performance_per_watt" title="Performance per watt">Performance per watt</a> (PPW)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_general-purpose_CPUs" title="History of general-purpose CPUs">History of general-purpose CPUs</a></li> <li><a href="/wiki/Microprocessor_chronology" title="Microprocessor chronology">Microprocessor chronology</a></li> <li><a href="/wiki/Processor_design" title="Processor design">Processor design</a></li> <li><a href="/wiki/Digital_electronics" title="Digital electronics">Digital electronics</a></li> <li><a href="/wiki/Hardware_security_module" title="Hardware security module">Hardware security module</a></li> <li><a href="/wiki/Semiconductor_device_fabrication" title="Semiconductor device fabrication">Semiconductor device fabrication</a></li> <li><a href="/wiki/Tick%E2%80%93tock_model" title="Tick–tock model">Tick–tock model</a></li> <li><a href="/wiki/Pin_grid_array" title="Pin grid array">Pin grid array</a></li> <li><a href="/wiki/Chip_carrier" title="Chip carrier">Chip carrier</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_information_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_information" title="Template:Quantum information"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_information" title="Template talk:Quantum information"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_information" title="Special:EditPage/Template:Quantum information"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_information_science" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_information_science" title="Quantum information science">Quantum information science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DiVincenzo%27s_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li> <li><a href="/wiki/Noisy_intermediate-scale_quantum_era" title="Noisy intermediate-scale quantum era">NISQ era</a></li> <li><a class="mw-selflink selflink">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li> <li><a href="/wiki/Qubit" title="Qubit">Qubit</a> <ul><li><a href="/wiki/Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li> <li><a href="/wiki/List_of_quantum_processors" title="List of quantum processors">Quantum processors</a> <ul><li><a href="/wiki/Cloud-based_quantum_computing" title="Cloud-based quantum computing">cloud-based</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell%27s_theorem" title="Bell's theorem">Bell's</a></li> <li><a href="/wiki/Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li> <li><a href="/wiki/Gleason%27s_theorem" title="Gleason's theorem">Gleason's</a></li> <li><a href="/wiki/Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li> <li><a href="/wiki/Holevo%27s_theorem" title="Holevo's theorem">Holevo's</a></li> <li><a href="/wiki/No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li> <li><a href="/wiki/No-cloning_theorem" title="No-cloning theorem">No-cloning</a></li> <li><a href="/wiki/No-communication_theorem" title="No-communication theorem">No-communication</a></li> <li><a href="/wiki/No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li> <li><a href="/wiki/No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li> <li><a href="/wiki/No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li> <li><a href="/wiki/PBR_theorem" class="mw-redirect" title="PBR theorem">PBR</a></li> <li><a href="/wiki/Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li> <li><a href="/wiki/Threshold_theorem" title="Threshold theorem">Threshold</a></li> <li><a href="/wiki/Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li> <li><a href="/wiki/Schr%C3%B6dinger%E2%80%93HJW_theorem" title="Schrödinger–HJW theorem">Purification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_capacity" title="Classical capacity">Classical capacity</a> <ul><li><a href="/wiki/Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li> <li><a href="/wiki/Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li> <li><a href="/wiki/Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li> <li><a href="/wiki/Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li> <li><a href="/wiki/LOCC" title="LOCC">LOCC</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a> <ul><li><a href="/wiki/Quantum_network" title="Quantum network">quantum network</a></li></ul></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a> <ul><li><a href="/wiki/Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li> <li><a href="/wiki/Superdense_coding" title="Superdense coding">Superdense coding</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Quantum_cryptography" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li> <li><a href="/wiki/Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li> <li><a href="/wiki/Quantum_money" title="Quantum money">Quantum money</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a> <ul><li><a href="/wiki/BB84" title="BB84">BB84</a></li> <li><a href="/wiki/SARG04" title="SARG04">SARG04</a></li> <li><a href="/wiki/List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li> <li><a href="/wiki/Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li> <li><a href="/wiki/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li> <li><a href="/wiki/BHT_algorithm" title="BHT algorithm">BHT</a></li> <li><a href="/wiki/Boson_sampling" title="Boson sampling">Boson sampling</a></li> <li><a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li> <li><a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's</a></li> <li><a href="/wiki/HHL_algorithm" title="HHL algorithm">HHL</a></li> <li><a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li> <li><a href="/wiki/Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li> <li><a href="/wiki/Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li> <li><a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li> <li><a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li> <li><a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's</a></li> <li><a href="/wiki/Simon%27s_problem" title="Simon's problem">Simon's</a></li> <li><a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">VQE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/BQP" title="BQP">BQP</a></li> <li><a href="/wiki/Exact_quantum_polynomial_time" title="Exact quantum polynomial time">EQP</a></li> <li><a href="/wiki/QIP_(complexity)" title="QIP (complexity)">QIP</a></li> <li><a href="/wiki/QMA" title="QMA">QMA</a></li> <li><a href="/wiki/PostBQP" title="PostBQP">PostBQP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li> <li><a href="/wiki/Quantum_volume" title="Quantum volume">Quantum volume</a></li> <li><a href="/wiki/Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a> <ul><li><a href="/wiki/Cross-entropy_benchmarking" title="Cross-entropy benchmarking">XEB</a></li></ul></li> <li><a href="/wiki/Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a> <ul><li><a href="/wiki/Spin%E2%80%93lattice_relaxation" title="Spin–lattice relaxation">T1</a></li> <li><a href="/wiki/Spin%E2%80%93spin_relaxation" title="Spin–spin relaxation">T2</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <a href="/wiki/Model_of_computation" title="Model of computation">computing models</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li> <li><a href="/wiki/Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li> <li><a href="/wiki/One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a> <ul><li><a href="/wiki/Cluster_state" title="Cluster state">cluster state</a></li></ul></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a> <ul><li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a> <ul><li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li> <li><a href="/wiki/Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li> <li><a href="/wiki/Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Codes <ul><li><a href="/wiki/CSS_code" title="CSS code">CSS</a></li> <li><a href="/wiki/Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li> <li><a href="/wiki/Stabilizer_code" title="Stabilizer code">stabilizer</a></li> <li><a href="/wiki/Shor_code" class="mw-redirect" title="Shor code">Shor</a></li> <li><a href="/wiki/Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li> <li><a href="/wiki/Steane_code" title="Steane code">Steane</a></li> <li><a href="/wiki/Toric_code" title="Toric code">Toric</a></li> <li><a href="/wiki/Gnu_code" title="Gnu code">gnu</a></li></ul></li> <li><a href="/wiki/Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li> <li><a href="/wiki/Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li> <li><a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li> <li><a href="/wiki/KLM_protocol" title="KLM protocol">KLM protocol</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ultracold_atom" title="Ultracold atom">Ultracold atoms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li> <li><a href="/wiki/Trapped-ion_quantum_computer" title="Trapped-ion quantum computer">Trapped-ion QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spin_(physics)" title="Spin (physics)">Spin</a>-based</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kane_quantum_computer" title="Kane quantum computer">Kane QC</a></li> <li><a href="/wiki/Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li> <li><a href="/wiki/Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li> <li><a href="/wiki/Nuclear_magnetic_resonance_quantum_computer" title="Nuclear magnetic resonance quantum computer">NMR QC</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Superconducting_quantum_computing" title="Superconducting quantum computing">Superconducting</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Charge_qubit" title="Charge qubit">Charge qubit</a></li> <li><a href="/wiki/Flux_qubit" title="Flux qubit">Flux qubit</a></li> <li><a href="/wiki/Phase_qubit" title="Phase qubit">Phase qubit</a></li> <li><a href="/wiki/Transmon" title="Transmon">Transmon</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="/wiki/Qiskit" title="Qiskit">Qiskit</a>–<a href="/wiki/IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li> <li><a href="/wiki/Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="/wiki/Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li> <li><a href="/wiki/Cirq" title="Cirq">Cirq</a></li> <li><a href="/wiki/Q_Sharp" title="Q Sharp">Q#</a></li> <li><a href="/wiki/Libquantum" title="Libquantum">libquantum</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">many others...</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Quantum_information_science" title="Category:Quantum information science">Quantum information science</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/23px-Symbol_template_class_pink.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/31px-Symbol_template_class_pink.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics">Quantum mechanics topics</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Emerging_technologies" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align: center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Emerging_technologies" title="Template:Emerging technologies"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Emerging_technologies" title="Template talk:Emerging technologies"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Emerging_technologies" title="Special:EditPage/Template:Emerging technologies"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Emerging_technologies" style="font-size:114%;margin:0 4em"><a href="/wiki/Emerging_technologies" title="Emerging technologies">Emerging technologies</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align: center;;width:1%">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Quantum</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">cellular automata</a></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">complexity theory</a></li> <li><a class="mw-selflink selflink">computing</a></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">cryptography</a> <ul><li><a href="/wiki/Post-quantum_cryptography" title="Post-quantum cryptography">post-quantum</a></li></ul></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">dynamics</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">error correction</a></li> <li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">finite automata</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">image processing</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">imaging</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">logic</a></li> <li><a href="/wiki/Quantum_logic_clock" title="Quantum logic clock">logic clock</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">logic gate</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">metamaterial</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align: center;">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acoustic_levitation" title="Acoustic levitation">Acoustic levitation</a></li> <li><a href="/wiki/Anti-gravity" title="Anti-gravity">Anti-gravity</a></li> <li><a href="/wiki/Cloak_of_invisibility" title="Cloak of invisibility">Cloak of invisibility</a></li> <li><a href="/wiki/Digital_scent_technology" title="Digital scent technology">Digital scent technology</a></li> <li><a href="/wiki/Force_field_(technology)" title="Force field (technology)">Force field</a> <ul><li><a href="/wiki/Plasma_window" title="Plasma window">Plasma window</a></li></ul></li> <li><a href="/wiki/Immersion_(virtual_reality)" title="Immersion (virtual reality)">Immersive virtual reality</a></li> <li><a href="/wiki/Magnetic_refrigeration" title="Magnetic refrigeration">Magnetic refrigeration</a></li> <li><a href="/wiki/Phased-array_optics" title="Phased-array optics">Phased-array optics</a></li> <li><a href="/wiki/Thermoacoustic_heat_engine" title="Thermoacoustic heat engine">Thermoacoustic heat engine</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="text-align: center;"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/List_of_emerging_technologies" title="List of emerging technologies">List</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Quantum_mechanics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Quantum_mechanics_topics" title="Template:Quantum mechanics topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Quantum_mechanics_topics" title="Template talk:Quantum mechanics topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Quantum_mechanics_topics" title="Special:EditPage/Template:Quantum mechanics topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Quantum_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction</a></li> <li><a href="/wiki/History_of_quantum_mechanics" title="History of quantum mechanics">History</a> <ul><li><a href="/wiki/Timeline_of_quantum_mechanics" title="Timeline of quantum mechanics">Timeline</a></li></ul></li> <li><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a></li> <li><a href="/wiki/Old_quantum_theory" title="Old quantum theory">Old quantum theory</a></li> <li><a href="/wiki/Glossary_of_elementary_quantum_mechanics" title="Glossary of elementary quantum mechanics">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fundamentals</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Born_rule" title="Born rule">Born rule</a></li> <li><a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">Bra–ket notation</a></li> <li><a href="/wiki/Complementarity_(physics)" title="Complementarity (physics)"> Complementarity</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density matrix</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a> <ul><li><a href="/wiki/Ground_state" title="Ground state">Ground state</a></li> <li><a href="/wiki/Excited_state" title="Excited state">Excited state</a></li> <li><a href="/wiki/Degenerate_energy_levels" title="Degenerate energy levels">Degenerate levels</a></li> <li><a href="/wiki/Zero-point_energy" title="Zero-point energy">Zero-point energy</a></li></ul></li> <li><a href="/wiki/Quantum_entanglement" title="Quantum entanglement">Entanglement</a></li> <li><a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a></li> <li><a href="/wiki/Wave_interference" title="Wave interference">Interference</a></li> <li><a href="/wiki/Quantum_decoherence" title="Quantum decoherence">Decoherence</a></li> <li><a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">Measurement</a></li> <li><a href="/wiki/Quantum_nonlocality" title="Quantum nonlocality">Nonlocality</a></li> <li><a href="/wiki/Quantum_state" title="Quantum state">Quantum state</a></li> <li><a href="/wiki/Quantum_superposition" title="Quantum superposition">Superposition</a></li> <li><a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">Tunnelling</a></li> <li><a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">Scattering theory</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics</a></li> <li><a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Uncertainty</a></li> <li><a href="/wiki/Wave_function" title="Wave function">Wave function</a> <ul><li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Collapse</a></li> <li><a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">Formulations</a></li> <li><a href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg</a></li> <li><a href="/wiki/Interaction_picture" title="Interaction picture">Interaction</a></li> <li><a href="/wiki/Matrix_mechanics" title="Matrix mechanics">Matrix mechanics</a></li> <li><a href="/wiki/Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger</a></li> <li><a href="/wiki/Path_integral_formulation" title="Path integral formulation">Path integral formulation</a></li> <li><a href="/wiki/Phase-space_formulation" title="Phase-space formulation">Phase space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Equations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon</a></li> <li><a href="/wiki/Dirac_equation" title="Dirac equation">Dirac</a></li> <li><a href="/wiki/Weyl_equation" title="Weyl equation">Weyl</a></li> <li><a href="/wiki/Majorana_equation" title="Majorana equation">Majorana</a></li> <li><a href="/wiki/Rarita%E2%80%93Schwinger_equation" title="Rarita–Schwinger equation">Rarita–Schwinger</a></li> <li><a href="/wiki/Pauli_equation" title="Pauli equation">Pauli</a></li> <li><a href="/wiki/Rydberg_formula" title="Rydberg formula">Rydberg</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Interpretations_of_quantum_mechanics" title="Interpretations of quantum mechanics">Interpretations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_Bayesianism" title="Quantum Bayesianism">Bayesian</a></li> <li><a href="/wiki/Consistent_histories" title="Consistent histories">Consistent histories</a></li> <li><a href="/wiki/Copenhagen_interpretation" title="Copenhagen interpretation">Copenhagen</a></li> <li><a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">de Broglie–Bohm</a></li> <li><a href="/wiki/Ensemble_interpretation" title="Ensemble interpretation">Ensemble</a></li> <li><a href="/wiki/Hidden-variable_theory" title="Hidden-variable theory">Hidden-variable</a> <ul><li><a href="/wiki/Local_hidden-variable_theory" title="Local hidden-variable theory">Local</a> <ul><li><a href="/wiki/Superdeterminism" title="Superdeterminism">Superdeterminism</a></li></ul></li></ul></li> <li><a href="/wiki/Many-worlds_interpretation" title="Many-worlds interpretation">Many-worlds</a></li> <li><a href="/wiki/Objective-collapse_theory" title="Objective-collapse theory">Objective collapse</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Relational_quantum_mechanics" title="Relational quantum mechanics">Relational</a></li> <li><a href="/wiki/Transactional_interpretation" title="Transactional interpretation">Transactional</a></li> <li><a href="/wiki/Von_Neumann%E2%80%93Wigner_interpretation" title="Von Neumann–Wigner interpretation">Von Neumann–Wigner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Experiments</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bell_test" title="Bell test">Bell test</a></li> <li><a href="/wiki/Davisson%E2%80%93Germer_experiment" title="Davisson–Germer experiment">Davisson–Germer</a></li> <li><a href="/wiki/Delayed-choice_quantum_eraser" title="Delayed-choice quantum eraser">Delayed-choice quantum eraser</a></li> <li><a href="/wiki/Double-slit_experiment" title="Double-slit experiment">Double-slit</a></li> <li><a href="/wiki/Franck%E2%80%93Hertz_experiment" title="Franck–Hertz experiment">Franck–Hertz</a></li> <li><a href="/wiki/Mach%E2%80%93Zehnder_interferometer" title="Mach–Zehnder interferometer">Mach–Zehnder interferometer</a></li> <li><a href="/wiki/Elitzur%E2%80%93Vaidman_bomb_tester" title="Elitzur–Vaidman bomb tester">Elitzur–Vaidman</a></li> <li><a href="/wiki/Popper%27s_experiment" title="Popper's experiment">Popper</a></li> <li><a href="/wiki/Quantum_eraser_experiment" title="Quantum eraser experiment">Quantum eraser</a></li> <li><a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach</a></li> <li><a href="/wiki/Wheeler%27s_delayed-choice_experiment" title="Wheeler's delayed-choice experiment">Wheeler's delayed choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_nanoscience" class="mw-redirect" title="Quantum nanoscience">Science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_biology" title="Quantum biology">Quantum biology</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chaos" title="Quantum chaos">Quantum chaos</a></li> <li><a href="/wiki/Quantum_cosmology" title="Quantum cosmology">Quantum cosmology</a></li> <li><a href="/wiki/Quantum_differential_calculus" title="Quantum differential calculus">Quantum differential calculus</a></li> <li><a href="/wiki/Quantum_dynamics" title="Quantum dynamics">Quantum dynamics</a></li> <li><a href="/wiki/Quantum_geometry" title="Quantum geometry">Quantum geometry</a></li> <li><a href="/wiki/Measurement_problem" title="Measurement problem">Quantum measurement problem</a></li> <li><a href="/wiki/Quantum_mind" title="Quantum mind">Quantum mind</a></li> <li><a href="/wiki/Quantum_stochastic_calculus" title="Quantum stochastic calculus">Quantum stochastic calculus</a></li> <li><a href="/wiki/Quantum_spacetime" title="Quantum spacetime">Quantum spacetime</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_technology" class="mw-redirect" title="Quantum technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></li> <li><a href="/wiki/Quantum_amplifier" title="Quantum amplifier">Quantum amplifier</a></li> <li><a href="/wiki/Quantum_bus" title="Quantum bus">Quantum bus</a></li> <li><a href="/wiki/Quantum_cellular_automaton" title="Quantum cellular automaton">Quantum cellular automata</a> <ul><li><a href="/wiki/Quantum_finite_automaton" title="Quantum finite automaton">Quantum finite automata</a></li></ul></li> <li><a href="/wiki/Quantum_channel" title="Quantum channel">Quantum channel</a></li> <li><a href="/wiki/Quantum_circuit" title="Quantum circuit">Quantum circuit</a></li> <li><a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></li> <li><a class="mw-selflink selflink">Quantum computing</a> <ul><li><a href="/wiki/Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">Timeline</a></li></ul></li> <li><a href="/wiki/Quantum_cryptography" title="Quantum cryptography">Quantum cryptography</a></li> <li><a href="/wiki/Quantum_electronics" class="mw-redirect" title="Quantum electronics">Quantum electronics</a></li> <li><a href="/wiki/Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></li> <li><a href="/wiki/Quantum_imaging" title="Quantum imaging">Quantum imaging</a></li> <li><a href="/wiki/Quantum_image_processing" title="Quantum image processing">Quantum image processing</a></li> <li><a href="/wiki/Quantum_information" title="Quantum information">Quantum information</a></li> <li><a href="/wiki/Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a></li> <li><a href="/wiki/Quantum_logic" title="Quantum logic">Quantum logic</a></li> <li><a href="/wiki/Quantum_logic_gate" title="Quantum logic gate">Quantum logic gates</a></li> <li><a href="/wiki/Quantum_machine" title="Quantum machine">Quantum machine</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_metamaterial" title="Quantum metamaterial">Quantum metamaterial</a></li> <li><a href="/wiki/Quantum_metrology" title="Quantum metrology">Quantum metrology</a></li> <li><a href="/wiki/Quantum_network" title="Quantum network">Quantum network</a></li> <li><a href="/wiki/Quantum_neural_network" title="Quantum neural network">Quantum neural network</a></li> <li><a href="/wiki/Quantum_optics" title="Quantum optics">Quantum optics</a></li> <li><a href="/wiki/Quantum_programming" title="Quantum programming">Quantum programming</a></li> <li><a href="/wiki/Quantum_sensor" title="Quantum sensor">Quantum sensing</a></li> <li><a href="/wiki/Quantum_simulator" title="Quantum simulator">Quantum simulator</a></li> <li><a href="/wiki/Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Extensions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quantum_fluctuation" title="Quantum fluctuation">Quantum fluctuation</a></li> <li><a href="/wiki/Casimir_effect" title="Casimir effect">Casimir effect</a></li> <li><a href="/wiki/Quantum_statistical_mechanics" title="Quantum statistical mechanics">Quantum statistical mechanics</a></li> <li><a href="/wiki/Quantum_field_theory" title="Quantum field theory">Quantum field theory</a> <ul><li><a href="/wiki/History_of_quantum_field_theory" title="History of quantum field theory">History</a></li></ul></li> <li><a href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</a></li> <li><a href="/wiki/Relativistic_quantum_mechanics" title="Relativistic quantum mechanics">Relativistic quantum mechanics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat" title="Schrödinger's cat">Schrödinger's cat</a> <ul><li><a href="/wiki/Schr%C3%B6dinger%27s_cat_in_popular_culture" title="Schrödinger's cat in popular culture">in popular culture</a></li></ul></li> <li><a href="/wiki/Wigner%27s_friend" title="Wigner's friend">Wigner's friend</a></li> <li><a href="/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox" title="Einstein–Podolsky–Rosen paradox">EPR paradox</a></li> <li><a href="/wiki/Quantum_mysticism" title="Quantum mysticism">Quantum mysticism</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Quantum_mechanics" title="Category:Quantum mechanics">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" 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[\"CITEREFBenentiCasatiRossiniStrini2019\"] = 1,\n [\"CITEREFBenioff1980\"] = 1,\n [\"CITEREFBennett2020\"] = 1,\n [\"CITEREFBennettBernsteinBrassardVazirani1997\"] = 1,\n [\"CITEREFBennettBrassard1984\"] = 1,\n [\"CITEREFBernhardt2019\"] = 1,\n [\"CITEREFBernstein2009\"] = 1,\n [\"CITEREFBernsteinVazirani1993\"] = 1,\n [\"CITEREFBernsteinVazirani1997\"] = 1,\n [\"CITEREFBerthiaume1998\"] = 1,\n [\"CITEREFBhatta2020\"] = 1,\n [\"CITEREFBiamonteWittekPancottiRebentrost2017\"] = 1,\n [\"CITEREFBiondiHeidHenkeMohr2021\"] = 1,\n [\"CITEREFBluvsteinEveredGeimLi2023\"] = 1,\n [\"CITEREFBoixoIsakovSmelyanskiyBabbush2018\"] = 1,\n [\"CITEREFBourzac2017\"] = 1,\n [\"CITEREFBrassard2005\"] = 1,\n [\"CITEREFBrassardHøyerTapp2016\"] = 1,\n [\"CITEREFBrooks2023\"] = 1,\n [\"CITEREFBuddeVolz2019\"] = 1,\n [\"CITEREFBulmerBellChadwickJones2022\"] = 1,\n [\"CITEREFBulutaNori2009\"] = 1,\n [\"CITEREFCaoRomeroOlsonDegroote2019\"] = 1,\n [\"CITEREFCavaliereMattssonSmeets2020\"] = 1,\n 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