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class="vector-toc-numb">2</span> <span>Algorithms based on the quantum Fourier transform</span> </div> </a> <button aria-controls="toc-Algorithms_based_on_the_quantum_Fourier_transform-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algorithms based on the quantum Fourier transform subsection</span> </button> <ul id="toc-Algorithms_based_on_the_quantum_Fourier_transform-sublist" class="vector-toc-list"> <li id="toc-Deutsch–Jozsa_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deutsch–Jozsa_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Deutsch–Jozsa algorithm</span> </div> </a> <ul id="toc-Deutsch–Jozsa_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bernstein–Vazirani_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bernstein–Vazirani_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Bernstein–Vazirani algorithm</span> </div> </a> <ul id="toc-Bernstein–Vazirani_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simon's_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simon's_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Simon's algorithm</span> </div> </a> <ul id="toc-Simon's_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_phase_estimation_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_phase_estimation_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Quantum phase estimation algorithm</span> </div> </a> <ul id="toc-Quantum_phase_estimation_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Shor's_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shor's_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Shor's algorithm</span> </div> </a> <ul id="toc-Shor's_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hidden_subgroup_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hidden_subgroup_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Hidden subgroup problem</span> </div> </a> <ul id="toc-Hidden_subgroup_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Estimating_Gauss_sums" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Estimating_Gauss_sums"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Estimating Gauss sums</span> </div> </a> <ul id="toc-Estimating_Gauss_sums-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fourier_fishing_and_Fourier_checking" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fourier_fishing_and_Fourier_checking"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Fourier fishing and Fourier checking</span> </div> </a> <ul id="toc-Fourier_fishing_and_Fourier_checking-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algorithms_based_on_amplitude_amplification" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algorithms_based_on_amplitude_amplification"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algorithms based on amplitude amplification</span> </div> </a> <button aria-controls="toc-Algorithms_based_on_amplitude_amplification-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algorithms based on amplitude amplification subsection</span> </button> <ul id="toc-Algorithms_based_on_amplitude_amplification-sublist" class="vector-toc-list"> <li id="toc-Grover's_algorithm" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grover's_algorithm"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Grover's algorithm</span> </div> </a> <ul id="toc-Grover's_algorithm-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_counting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_counting"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Quantum counting</span> </div> </a> <ul id="toc-Quantum_counting-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algorithms_based_on_quantum_walks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algorithms_based_on_quantum_walks"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algorithms based on quantum walks</span> </div> </a> <button aria-controls="toc-Algorithms_based_on_quantum_walks-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Algorithms based on quantum walks subsection</span> </button> <ul id="toc-Algorithms_based_on_quantum_walks-sublist" class="vector-toc-list"> <li id="toc-Boson_sampling_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Boson_sampling_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Boson sampling problem</span> </div> </a> <ul id="toc-Boson_sampling_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Element_distinctness_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Element_distinctness_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Element distinctness problem</span> </div> </a> <ul id="toc-Element_distinctness_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangle-finding_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Triangle-finding_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Triangle-finding problem</span> </div> </a> <ul id="toc-Triangle-finding_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formula_evaluation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formula_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Formula evaluation</span> </div> </a> <ul id="toc-Formula_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_commutativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_commutativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Group commutativity</span> </div> </a> <ul id="toc-Group_commutativity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-BQP-complete_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#BQP-complete_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>BQP-complete problems</span> </div> </a> <button aria-controls="toc-BQP-complete_problems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle BQP-complete problems subsection</span> </button> <ul id="toc-BQP-complete_problems-sublist" class="vector-toc-list"> <li id="toc-Computing_knot_invariants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computing_knot_invariants"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Computing knot invariants</span> </div> </a> <ul id="toc-Computing_knot_invariants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_simulation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_simulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Quantum simulation</span> </div> </a> <ul id="toc-Quantum_simulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solving_a_linear_system_of_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solving_a_linear_system_of_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Solving a linear system of equations</span> </div> </a> <ul id="toc-Solving_a_linear_system_of_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hybrid_quantum/classical_algorithms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Hybrid_quantum/classical_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Hybrid quantum/classical algorithms</span> </div> </a> <button aria-controls="toc-Hybrid_quantum/classical_algorithms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Hybrid quantum/classical algorithms subsection</span> </button> <ul id="toc-Hybrid_quantum/classical_algorithms-sublist" class="vector-toc-list"> <li id="toc-QAOA" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#QAOA"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>QAOA</span> </div> </a> <ul id="toc-QAOA-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variational_quantum_eigensolver" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variational_quantum_eigensolver"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Variational quantum eigensolver</span> </div> </a> <ul id="toc-Variational_quantum_eigensolver-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contracted_quantum_eigensolver" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contracted_quantum_eigensolver"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Contracted quantum eigensolver</span> </div> </a> <ul id="toc-Contracted_quantum_eigensolver-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"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle External links subsection</span> </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Surveys" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surveys"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Surveys</span> </div> </a> <ul id="toc-Surveys-sublist" class="vector-toc-list"> </ul> </li> </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" title="Table of Contents" > <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" ><span class="vector-icon mw-ui-icon-listBullet 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Available in 22 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-22" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">22 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AE%D9%88%D8%A7%D8%B1%D8%B2%D9%85%D9%8A%D8%A9_%D8%A7%D9%84%D9%83%D9%85" title="خوارزمية الكم – Arabic" lang="ar" hreflang="ar" data-title="خوارزمية الكم" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></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%85%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B2%E0%A6%97%E0%A7%8B%E0%A6%B0%E0%A6%BF%E0%A6%A6%E0%A6%AE" title="কোয়ান্টাম অ্যালগোরিদম – Bangla" lang="bn" hreflang="bn" data-title="কোয়ান্টাম অ্যালগোরিদম" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Li%C5%8Dng-ch%C3%AD_i%C3%A1n-s%C7%B9g-hoat" title="Liōng-chí ián-sǹg-hoat – Minnan" lang="nan" hreflang="nan" data-title="Liōng-chí ián-sǹg-hoat" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Algorisme_qu%C3%A0ntic" title="Algorisme quàntic – Catalan" lang="ca" hreflang="ca" data-title="Algorisme quàntic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs badge-Q70894304 mw-list-item" title=""><a href="https://cs.wikipedia.org/wiki/Kvantov%C3%BD_algoritmus" title="Kvantový algoritmus – Czech" lang="cs" hreflang="cs" data-title="Kvantový algoritmus" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quantenalgorithmus" title="Quantenalgorithmus – German" lang="de" hreflang="de" data-title="Quantenalgorithmus" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kvantalgoritm" title="Kvantalgoritm – Estonian" lang="et" hreflang="et" data-title="Kvantalgoritm" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Algoritmo_cu%C3%A1ntico" title="Algoritmo cuántico – Spanish" lang="es" hreflang="es" data-title="Algoritmo cuántico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%DB%8C%D8%AA%D9%85_%DA%A9%D9%88%D8%A7%D9%86%D8%AA%D9%88%D9%85%DB%8C" title="الگوریتم کوانتومی – Persian" lang="fa" hreflang="fa" data-title="الگوریتم کوانتومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%96%91%EC%9E%90_%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98" title="양자 알고리즘 – Korean" lang="ko" hreflang="ko" data-title="양자 알고리즘" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algoritmo_quantistico" title="Algoritmo quantistico – Italian" lang="it" hreflang="it" data-title="Algoritmo quantistico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%A3%E0%B5%8D%E0%B4%9F%E0%B4%82_%E0%B4%85%E0%B5%BD%E0%B4%97%E0%B5%8B%E0%B4%B0%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ക്വാണ്ടം അൽഗോരിതം – Malayalam" lang="ml" hreflang="ml" data-title="ക്വാണ്ടം അൽഗോരിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwantumalgoritme" title="Kwantumalgoritme – Dutch" lang="nl" hreflang="nl" data-title="Kwantumalgoritme" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algorytm_kwantowy" title="Algorytm kwantowy – Polish" lang="pl" hreflang="pl" data-title="Algorytm kwantowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a 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searchaux" style="display:none">Algorithm to be run on quantum computers</div> <p> In <a href="/wiki/Quantum_computing" title="Quantum computing">quantum computing</a>, a <b>quantum algorithm</b> is an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> that runs on a realistic model of <a href="/wiki/Quantum_computation" class="mw-redirect" title="Quantum computation">quantum computation</a>, the most commonly used model being the <a href="/wiki/Quantum_circuit" title="Quantum circuit">quantum circuit</a> model of computation.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A classical (or non-quantum) algorithm is a finite sequence of instructions, or a step-by-step procedure for solving a problem, where each step or instruction can be performed on a classical <a href="/wiki/Computer" title="Computer">computer</a>. Similarly, a quantum algorithm is a step-by-step procedure, where each of the steps can be performed on a <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computer</a>. Although all classical algorithms can also be performed on a quantum computer,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 126">: 126 </span></sup> the term quantum algorithm is generally reserved for algorithms that seem inherently quantum, or use some essential feature of quantum computation such as <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superposition</a> or <a href="/wiki/Quantum_entanglement" title="Quantum entanglement">quantum entanglement</a>. </p><p>Problems that are <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a> using classical computers remain undecidable using quantum computers.<sup id="cite_ref-nielchuan_4-0" class="reference"><a href="#cite_note-nielchuan-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 127">: 127 </span></sup> What makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms because the quantum superposition and quantum entanglement that quantum algorithms exploit generally cannot be efficiently simulated on classical computers (see <a href="/wiki/Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a>). </p><p>The best-known algorithms are <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a> for factoring and <a href="/wiki/Grover%27s_algorithm" title="Grover's algorithm">Grover's algorithm</a> for searching an unstructured database or an unordered list. Shor's algorithm runs much (almost exponentially) faster than the best-known classical algorithm for factoring, the <a href="/wiki/General_number_field_sieve" title="General number field sieve">general number field sieve</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Grover's algorithm runs quadratically faster than the best possible classical algorithm for the same task,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> a <a href="/wiki/Linear_search" title="Linear search">linear search</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quantum algorithms are usually described, in the commonly used circuit model of quantum computation, by a <a href="/wiki/Quantum_circuit" title="Quantum circuit">quantum circuit</a> that acts on some input <a href="/wiki/Qubit" title="Qubit">qubits</a> and terminates with a <a href="/wiki/Measurement" title="Measurement">measurement</a>. A quantum circuit consists of simple <a href="/wiki/Quantum_gate" class="mw-redirect" title="Quantum gate">quantum gates</a>, each of which acts on some finite number of qubits. Quantum algorithms may also be stated in other models of quantum computation, such as the <a href="/w/index.php?title=Hamiltonian_oracle_model&action=edit&redlink=1" class="new" title="Hamiltonian oracle model (page does not exist)">Hamiltonian oracle model</a>.<sup id="cite_ref-Hamiltonian_NAND_Tree_7-0" class="reference"><a href="#cite_note-Hamiltonian_NAND_Tree-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Quantum algorithms can be categorized by the main techniques involved in the algorithm. Some commonly used techniques/ideas in quantum algorithms include <a href="/w/index.php?title=Phase_kick-back&action=edit&redlink=1" class="new" title="Phase kick-back (page does not exist)">phase kick-back</a>, <a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">phase estimation</a>, the <a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">quantum Fourier transform</a>, <a href="/wiki/Quantum_walk" title="Quantum walk">quantum walks</a>, <a href="/wiki/Amplitude_amplification" title="Amplitude amplification">amplitude amplification</a> and <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a>. Quantum algorithms may also be grouped by the type of problem solved; see, e.g., the survey on quantum algorithms for algebraic problems.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algorithms_based_on_the_quantum_Fourier_transform">Algorithms based on the quantum Fourier transform</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=2" title="Edit section: Algorithms based on the quantum Fourier transform"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Quantum_Fourier_transform" title="Quantum Fourier transform">quantum Fourier transform</a> is the quantum analogue of the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a>, and is used in several quantum algorithms. The <a href="/wiki/Hadamard_transform" title="Hadamard transform">Hadamard transform</a> is also an example of a quantum Fourier transform over an n-dimensional vector space over the field <a href="/wiki/GF(2)" title="GF(2)"><b>F</b><sub>2</sub></a>. The quantum Fourier transform can be efficiently implemented on a quantum computer using only a polynomial number of <a href="/wiki/Quantum_logic_gate#Hadamard_transform" title="Quantum logic gate">quantum gates</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2018)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Deutsch–Jozsa_algorithm"><span id="Deutsch.E2.80.93Jozsa_algorithm"></span>Deutsch–Jozsa algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=3" title="Edit section: Deutsch–Jozsa algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa algorithm</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Deutsch-Jozsa-algorithm-quantum-circuit.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Deutsch-Jozsa-algorithm-quantum-circuit.png/250px-Deutsch-Jozsa-algorithm-quantum-circuit.png" decoding="async" width="220" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Deutsch-Jozsa-algorithm-quantum-circuit.png/330px-Deutsch-Jozsa-algorithm-quantum-circuit.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Deutsch-Jozsa-algorithm-quantum-circuit.png/500px-Deutsch-Jozsa-algorithm-quantum-circuit.png 2x" data-file-width="947" data-file-height="462" /></a><figcaption>Deutsch-Jozsa algorithm</figcaption></figure> <p>The Deutsch–Jozsa algorithm solves a <a href="/wiki/Black-box" class="mw-redirect" title="Black-box">black-box</a> problem that requires exponentially many queries to the black box for any deterministic classical computer, but can be done with a single query by a quantum computer. However, when comparing bounded-error classical and quantum algorithms, there is no speedup, since a classical probabilistic algorithm can solve the problem with a constant number of queries with small probability of error. The algorithm determines whether a function <i>f</i> is either constant (0 on all inputs or 1 on all inputs) or balanced (returns 1 for half of the input domain and 0 for the other half). </p> <div class="mw-heading mw-heading3"><h3 id="Bernstein–Vazirani_algorithm"><span id="Bernstein.E2.80.93Vazirani_algorithm"></span>Bernstein–Vazirani algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=4" title="Edit section: Bernstein–Vazirani algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani algorithm</a></div> <p>The Bernstein–Vazirani algorithm is the first quantum algorithm that solves a problem more efficiently than the best known classical algorithm. It was designed to create an <a href="/w/index.php?title=Oracle_separation&action=edit&redlink=1" class="new" title="Oracle separation (page does not exist)">oracle separation</a> between <a href="/wiki/BQP" title="BQP">BQP</a> and <a href="/wiki/BPP_(complexity)" title="BPP (complexity)">BPP</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Simon's_algorithm"><span id="Simon.27s_algorithm"></span>Simon's algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=5" title="Edit section: Simon's algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Simon%27s_algorithm" class="mw-redirect" title="Simon's algorithm">Simon's algorithm</a></div> <p>Simon's algorithm solves a black-box problem exponentially faster than any classical algorithm, including bounded-error probabilistic algorithms. This algorithm, which achieves an exponential speedup over all classical algorithms that we consider efficient, was the motivation for <a href="/wiki/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a> for factoring. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_phase_estimation_algorithm">Quantum phase estimation algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=6" title="Edit section: Quantum phase estimation algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation algorithm</a></div> <p>The <a href="/wiki/Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">quantum phase estimation algorithm</a> is used to determine the eigenphase of an eigenvector of a unitary gate, given a quantum state proportional to the eigenvector and access to the gate. The algorithm is frequently used as a subroutine in other algorithms. </p> <div class="mw-heading mw-heading3"><h3 id="Shor's_algorithm"><span id="Shor.27s_algorithm"></span>Shor's algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=7" title="Edit section: Shor's algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Shor%27s_algorithm" title="Shor's algorithm">Shor's algorithm</a></div> <p>Shor's algorithm solves the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> problem and the <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a> problem in polynomial time,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> whereas the best known classical algorithms take super-polynomial time. It is unknown whether these problems are in <a href="/wiki/P_(complexity)" title="P (complexity)">P</a> or <a href="/wiki/NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>. It is also one of the few quantum algorithms that solves a non-black-box problem in polynomial time, where the best known classical algorithms run in super-polynomial time. </p> <div class="mw-heading mw-heading3"><h3 id="Hidden_subgroup_problem">Hidden subgroup problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=8" title="Edit section: Hidden subgroup problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Hidden_subgroup_problem" title="Hidden subgroup problem">hidden subgroup problem</a> is a generalization of many problems that can be solved by a quantum computer, such as Simon's problem, solving <a href="/wiki/Pell%27s_equation" title="Pell's equation">Pell's equation</a>, testing the <a href="/wiki/Principal_ideal" title="Principal ideal">principal ideal</a> of a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> R and <a href="/wiki/Integer_factorization" title="Integer factorization">factoring</a>. There are efficient quantum algorithms known for the Abelian hidden subgroup problem.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> The more general hidden subgroup problem, where the group is not necessarily abelian, is a generalization of the previously mentioned problems, as well as <a href="/wiki/Graph_isomorphism" title="Graph isomorphism">graph isomorphism</a> and certain <a href="/wiki/Lattice_problems" class="mw-redirect" title="Lattice problems">lattice problems</a>. Efficient quantum algorithms are known for certain non-abelian groups. However, no efficient algorithms are known for the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a>, which would give an efficient algorithm for graph isomorphism<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a>, which would solve certain lattice problems.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Estimating_Gauss_sums">Estimating Gauss sums</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=9" title="Edit section: Estimating Gauss sums"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Gauss_sum" title="Gauss sum">Gauss sum</a> is a type of <a href="/wiki/Exponential_sum" title="Exponential sum">exponential sum</a>. The best known classical algorithm for estimating these sums takes exponential time. Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely. However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fourier_fishing_and_Fourier_checking">Fourier fishing and Fourier checking</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=10" title="Edit section: Fourier fishing and Fourier checking"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider an <a href="/wiki/Oracle_machine" title="Oracle machine">oracle</a> consisting of <i>n</i> random Boolean functions mapping <i>n</i>-bit strings to a Boolean value, with the goal of finding n <i>n</i>-bit strings <i>z</i><sub>1</sub>,..., <i>z<sub>n</sub></i> such that for the Hadamard-Fourier transform, at least 3/4 of the strings satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\tilde {f}}(z_{i})|\geqslant 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩾</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\tilde {f}}(z_{i})|\geqslant 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a405b4f83aaca5c38eb79cc213b3bf26f669cb99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.944ex; height:3.176ex;" alt="{\displaystyle |{\tilde {f}}(z_{i})|\geqslant 1}" /></span></dd></dl> <p>and at least 1/4 satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\tilde {f}}(z_{i})|\geqslant 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⩾</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\tilde {f}}(z_{i})|\geqslant 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae7dc1c5230edcd5921c61d69299397973e499cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.591ex; height:3.176ex;" alt="{\displaystyle |{\tilde {f}}(z_{i})|\geqslant 2.}" /></span></dd></dl> <p>This can be done in <a href="/wiki/BQP" title="BQP">bounded-error quantum polynomial time</a> (BQP).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Algorithms_based_on_amplitude_amplification">Algorithms based on amplitude amplification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=11" title="Edit section: Algorithms based on amplitude amplification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a> is a technique that allows the amplification of a chosen subspace of a quantum state. Applications of amplitude amplification usually lead to quadratic speedups over the corresponding classical algorithms. It can be considered as a generalization of Grover's algorithm.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Grover's_algorithm"><span id="Grover.27s_algorithm"></span>Grover's algorithm</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=12" title="Edit section: Grover's algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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> <p>Grover's algorithm searches an unstructured database (or an unordered list) with N entries for a marked entry, using only <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8566840a0a5d4480a0b8ed63d0717c3a89ec9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:3.176ex;" alt="{\displaystyle O({\sqrt {N}})}" /></span> queries instead of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({N})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f61c898f43f99824d919717c4298fa89018dd74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle O({N})}" /></span> queries required classically.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Classically, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({N})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({N})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f61c898f43f99824d919717c4298fa89018dd74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle O({N})}" /></span> queries are required even allowing bounded-error probabilistic algorithms. </p><p>Theorists have considered a hypothetical generalization of a standard quantum computer that could access the histories of the hidden variables in <a href="/wiki/De_Broglie%E2%80%93Bohm_theory" title="De Broglie–Bohm theory">Bohmian mechanics</a>. (Such a computer is completely hypothetical and would <i>not</i> be a standard quantum computer, or even possible under the standard theory of quantum mechanics.) Such a hypothetical computer could implement a search of an N-item database in at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({\sqrt[{3}]{N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O({\sqrt[{3}]{N}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a953364313112d7f3243feb0c59146295a65263d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:3.176ex;" alt="{\displaystyle O({\sqrt[{3}]{N}})}" /></span> steps. This is slightly faster than the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8566840a0a5d4480a0b8ed63d0717c3a89ec9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.582ex; height:3.176ex;" alt="{\displaystyle O({\sqrt {N}})}" /></span> steps taken by Grover's algorithm. However, neither search method would allow either model of quantum computer to solve <a href="/wiki/NP-completeness" title="NP-completeness">NP-complete</a> problems in polynomial time.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_counting">Quantum counting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=13" title="Edit section: Quantum counting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Quantum_counting" class="mw-redirect" title="Quantum counting">Quantum counting</a> solves a generalization of the search problem. It solves the problem of counting the number of marked entries in an unordered list, instead of just detecting whether one exists. Specifically, it counts the number of marked entries in an <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span>-element list with an error of at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" /></span> by making only <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta \left(\varepsilon ^{-1}{\sqrt {N/k}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta \left(\varepsilon ^{-1}{\sqrt {N/k}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4677c7ede02476962977040638ff2a2f77e51bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.148ex; height:4.843ex;" alt="{\displaystyle \Theta \left(\varepsilon ^{-1}{\sqrt {N/k}}\right)}" /></span> queries, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span> is the number of marked elements in the list.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> More precisely, the algorithm outputs an estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a2b832ed184c7b6481b3926bf8172d353fa7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.896ex; height:2.509ex;" alt="{\displaystyle k'}" /></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span>, the number of marked entries, with accuracy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k-k'|\leq \varepsilon k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>k</mi> <mo>−</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mi>ε</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |k-k'|\leq \varepsilon k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af47611da48e555dbe45358c2abad0954a331b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.634ex; height:3.009ex;" alt="{\displaystyle |k-k'|\leq \varepsilon k}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Algorithms_based_on_quantum_walks">Algorithms based on quantum walks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=14" title="Edit section: Algorithms based on quantum walks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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_walk" title="Quantum walk">Quantum walk</a></div> <p>A quantum walk is the quantum analogue of a classical <a href="/wiki/Random_walk" title="Random walk">random walk</a>. A classical random walk can be described by a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> over some states, while a quantum walk can be described by a <a href="/wiki/Quantum_superposition" title="Quantum superposition">quantum superposition</a> over states. Quantum walks are known to give exponential speedups for some black-box problems.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> They also provide polynomial speedups for many problems. A framework for the creation of quantum walk algorithms exists and is a versatile tool.<sup id="cite_ref-Search_via_quantum_walk_21-0" class="reference"><a href="#cite_note-Search_via_quantum_walk-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Boson_sampling_problem">Boson sampling problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=15" title="Edit section: Boson sampling problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Boson_sampling" title="Boson sampling">Boson sampling</a></div> <p>The Boson Sampling Problem in an experimental configuration assumes<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> an input of <a href="/wiki/Boson" title="Boson">bosons</a> (e.g., photons) of moderate number that are randomly scattered into a large number of output modes, constrained by a defined <a href="/wiki/Unitarity" class="mw-redirect" title="Unitarity">unitarity</a>. When individual photons are used, the problem is isomorphic to a multi-photon quantum walk.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The problem is then to produce a fair sample of the <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> of the output that depends on the input arrangement of bosons and the unitarity.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Solving this problem with a classical computer algorithm requires computing the <a href="/wiki/Permanent_(mathematics)" title="Permanent (mathematics)">permanent</a> of the unitary transform matrix, which may take a prohibitively long time or be outright impossible. In 2014, it was proposed<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> that existing technology and standard probabilistic methods of generating single-photon states could be used as an input into a suitable quantum computable <a href="/wiki/Linear_optical_quantum_computing" title="Linear optical quantum computing">linear optical network</a> and that sampling of the output probability distribution would be demonstrably superior using quantum algorithms. In 2015, investigation predicted<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> the sampling problem had similar complexity for inputs other than <a href="/wiki/Fock_state" title="Fock state">Fock-state</a> photons and identified a transition in <a href="/wiki/Quantum_complexity_theory" title="Quantum complexity theory">computational complexity</a> from classically simulable to just as hard as the Boson Sampling Problem, depending on the size of coherent amplitude inputs. </p> <div class="mw-heading mw-heading3"><h3 id="Element_distinctness_problem">Element distinctness problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=16" title="Edit section: Element distinctness problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Element_distinctness_problem" title="Element distinctness problem">Element distinctness problem</a></div> <p>The element distinctness problem is the problem of determining whether all the elements of a list are distinct. Classically, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577727c452671a036afca43509f0da81ddd7a7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.551ex; height:2.843ex;" alt="{\displaystyle \Omega (N)}" /></span> queries are required for a list of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span>; however, it can be solved in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (N^{2/3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (N^{2/3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859de2e30e6d0f1391e7357d3f8858e830bd5039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.439ex; height:3.343ex;" alt="{\displaystyle \Theta (N^{2/3})}" /></span> queries on a quantum computer. The optimal algorithm was put forth by <a href="/wiki/Andris_Ambainis" title="Andris Ambainis">Andris Ambainis</a>,<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> and <a href="/w/index.php?title=Yaoyun_Shi&action=edit&redlink=1" class="new" title="Yaoyun Shi (page does not exist)">Yaoyun Shi</a> first proved a tight lower bound when the size of the range is sufficiently large.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Ambainis<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> and Kutin<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> independently (and via different proofs) extended that work to obtain the lower bound for all functions. </p> <div class="mw-heading mw-heading3"><h3 id="Triangle-finding_problem">Triangle-finding problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=17" title="Edit section: Triangle-finding problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Triangle_finding_problem" class="mw-redirect" title="Triangle finding problem">Triangle finding problem</a></div> <p>The triangle-finding problem is the problem of determining whether a given graph contains a triangle (a <a href="/wiki/Clique_(graph_theory)" title="Clique (graph theory)">clique</a> of size 3). The best-known lower bound for quantum algorithms is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/577727c452671a036afca43509f0da81ddd7a7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.551ex; height:2.843ex;" alt="{\displaystyle \Omega (N)}" /></span>, but the best algorithm known requires O(<i>N</i><sup>1.297</sup>) queries,<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> an improvement over the previous best O(<i>N</i><sup>1.3</sup>) queries.<sup id="cite_ref-Search_via_quantum_walk_21-1" class="reference"><a href="#cite_note-Search_via_quantum_walk-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Formula_evaluation">Formula evaluation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=18" title="Edit section: Formula evaluation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A formula is a tree with a gate at each internal node and an input bit at each leaf node. The problem is to evaluate the formula, which is the output of the root node, given oracle access to the input. </p><p>A well studied formula is the balanced binary tree with only NAND gates.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> This type of formula requires <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (N^{c})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (N^{c})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20941d9a78d520f67b0fbc23d092296fda15b589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.685ex; height:2.843ex;" alt="{\displaystyle \Theta (N^{c})}" /></span> queries using randomness,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=\log _{2}(1+{\sqrt {33}})/4\approx 0.754}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>33</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>≈</mo> <mn>0.754</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=\log _{2}(1+{\sqrt {33}})/4\approx 0.754}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0068e761b9aa0c0a835d2fd1d2d180128c335df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.924ex; height:3.009ex;" alt="{\displaystyle c=\log _{2}(1+{\sqrt {33}})/4\approx 0.754}" /></span>. With a quantum algorithm, however, it can be solved in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (N^{1/2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (N^{1/2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66359c4e8d11f59c4a602f9ad8d2d0216efecada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.439ex; height:3.343ex;" alt="{\displaystyle \Theta (N^{1/2})}" /></span> queries. No better quantum algorithm for this case was known until one was found for the unconventional Hamiltonian oracle model.<sup id="cite_ref-Hamiltonian_NAND_Tree_7-1" class="reference"><a href="#cite_note-Hamiltonian_NAND_Tree-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The same result for the standard setting soon followed.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p><p>Fast quantum algorithms for more complicated formulas are also known.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Group_commutativity">Group commutativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=19" title="Edit section: Group commutativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The problem is to determine if a <a href="/wiki/Black_box_group" title="Black box group">black-box group</a>, given by <i>k</i> generators, is <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutative</a>. A black-box group is a group with an oracle function, which must be used to perform the group operations (multiplication, inversion, and comparison with identity). The interest in this context lies in the query complexity, which is the number of oracle calls needed to solve the problem. The deterministic and randomized query complexities are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (k^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (k^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b880af97033724b3fca22aba20d0d404437c0bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.883ex; height:3.176ex;" alt="{\displaystyle \Theta (k^{2})}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta (k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de19a58de2d96b3f842698aa22fb1dc001851fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle \Theta (k)}" /></span>, respectively.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> A quantum algorithm requires <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega (k^{2/3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega (k^{2/3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8d7d1cdd077872fa495d54e23cbb92d72927a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.397ex; height:3.343ex;" alt="{\displaystyle \Omega (k^{2/3})}" /></span> queries, while the best-known classical algorithm uses <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(k^{2/3}\log k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mi>log</mi> <mo>⁡</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(k^{2/3}\log k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a91bf7a5ecb41e6ec13f1bee9890fe5dbfa5a47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.449ex; height:3.343ex;" alt="{\displaystyle O(k^{2/3}\log k)}" /></span> queries.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="BQP-complete_problems">BQP-complete problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=20" title="Edit section: BQP-complete problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Complexity_class" title="Complexity class">complexity class</a> <b><a href="/wiki/BQP" title="BQP">BQP</a></b> (bounded-error quantum polynomial time) is the set of <a href="/wiki/Decision_problems" class="mw-redirect" title="Decision problems">decision problems</a> solvable by a <a href="/wiki/Quantum_computer" class="mw-redirect" title="Quantum computer">quantum computer</a> in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a> with error probability of at most 1/3 for all instances.<sup id="cite_ref-Chuang2000_39-0" class="reference"><a href="#cite_note-Chuang2000-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> It is the quantum analogue to the classical complexity class <b><a href="/wiki/BPP_(complexity)" title="BPP (complexity)">BPP</a></b>. </p><p>A problem is <b>BQP</b>-complete if it is in <b>BQP</b> and any problem in <b>BQP</b> can be <a href="/wiki/Reduction_(complexity)" title="Reduction (complexity)">reduced</a> to it in <a href="/wiki/Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>. Informally, the class of <b>BQP</b>-complete problems are those that are as hard as the hardest problems in <b>BQP</b> and are themselves efficiently solvable by a quantum computer (with bounded error). </p> <div class="mw-heading mw-heading3"><h3 id="Computing_knot_invariants">Computing knot invariants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=21" title="Edit section: Computing knot invariants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Witten had shown that the <a href="/wiki/Chern-Simons" class="mw-redirect" title="Chern-Simons">Chern-Simons</a> <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">topological quantum field theory</a> (TQFT) can be solved in terms of <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomials</a>. A quantum computer can simulate a TQFT, and thereby approximate the Jones polynomial,<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> which as far as we know, is hard to compute classically in the worst-case scenario.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2014)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_simulation">Quantum simulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=22" title="Edit section: Quantum simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Hamiltonian_simulation" title="Hamiltonian simulation">Hamiltonian simulation</a></div> <p>The idea that quantum computers might be more powerful than classical computers originated in Richard Feynman's observation that classical computers seem to require exponential time to simulate many-particle quantum systems, yet quantum many-body systems are able to "solve themselves."<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> Since then, the idea that quantum computers can simulate quantum physical processes exponentially faster than classical computers has been greatly fleshed out and elaborated. Efficient (i.e., polynomial-time) quantum algorithms have been developed for simulating both Bosonic and Fermionic systems,<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> as well as the simulation of chemical reactions beyond the capabilities of current classical supercomputers using only a few hundred qubits.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> Quantum computers can also efficiently simulate topological quantum field theories.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> In addition to its intrinsic interest, this result has led to efficient quantum algorithms for estimating <a href="/wiki/Quantum_invariant" title="Quantum invariant">quantum topological invariants</a> such as <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones</a><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/HOMFLY_polynomial" title="HOMFLY polynomial">HOMFLY polynomials</a>,<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> and the <a href="/w/index.php?title=Turaev-Viro_invariant&action=edit&redlink=1" class="new" title="Turaev-Viro invariant (page does not exist)">Turaev-Viro invariant</a> of three-dimensional manifolds.<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Solving_a_linear_system_of_equations">Solving a linear system of equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=23" title="Edit section: Solving a linear system of equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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_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></div> <p>In 2009, <a href="/wiki/Aram_Harrow" title="Aram Harrow">Aram Harrow</a>, Avinatan Hassidim, and <a href="/wiki/Seth_Lloyd" title="Seth Lloyd">Seth Lloyd</a>, formulated a quantum algorithm for solving <a href="/wiki/System_of_linear_equations" title="System of linear equations">linear systems</a>. The <a href="/wiki/Quantum_algorithm_for_linear_systems_of_equations" class="mw-redirect" title="Quantum algorithm for linear systems of equations">algorithm</a> estimates the result of a scalar measurement on the solution vector to a given linear system of equations.<sup id="cite_ref-Quantum_algorithm_for_solving_linear_systems_of_equations_by_Harrow_et_al._48-0" class="reference"><a href="#cite_note-Quantum_algorithm_for_solving_linear_systems_of_equations_by_Harrow_et_al.-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>Provided that the linear system is <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse</a> and has a low <a href="/wiki/Condition_number" title="Condition number">condition number</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" /></span>, and that the user is interested in the result of a scalar measurement on the solution vector (instead of the values of the solution vector itself), then the algorithm has a runtime of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\log(N)\kappa ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\log(N)\kappa ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cdea440dd6924fced14a411afe7ce0c829665f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.82ex; height:3.176ex;" alt="{\displaystyle O(\log(N)\kappa ^{2})}" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> is the number of variables in the linear system. This offers an exponential speedup over the fastest classical algorithm, which runs in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(N\kappa )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mi>κ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(N\kappa )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942716bad6c7090a9922a5eaaf0da1a8ea093bd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.985ex; height:2.843ex;" alt="{\displaystyle O(N\kappa )}" /></span> (or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(N{\sqrt {\kappa }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>κ</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(N{\sqrt {\kappa }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31d931a01da41e9383d34b62f615777ab835d51d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.921ex; height:3.009ex;" alt="{\displaystyle O(N{\sqrt {\kappa }})}" /></span> for positive semidefinite matrices). </p> <div class="mw-heading mw-heading2"><h2 id="Hybrid_quantum/classical_algorithms"><span id="Hybrid_quantum.2Fclassical_algorithms"></span>Hybrid quantum/classical algorithms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=24" title="Edit section: Hybrid quantum/classical algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hybrid Quantum/Classical Algorithms combine quantum state preparation and measurement with classical optimization.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> These algorithms generally aim to determine the ground-state eigenvector and eigenvalue of a Hermitian operator. </p> <div class="mw-heading mw-heading3"><h3 id="QAOA">QAOA</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=25" title="Edit section: QAOA"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Quantum_approximate_optimization_algorithm" class="mw-redirect" title="Quantum approximate optimization algorithm">quantum approximate optimization algorithm</a> takes inspiration from quantum annealing, performing a discretized approximation of quantum annealing using a quantum circuit. It can be used to solve problems in graph theory.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> The algorithm makes use of classical optimization of quantum operations to maximize an "objective function." </p> <div class="mw-heading mw-heading3"><h3 id="Variational_quantum_eigensolver">Variational quantum eigensolver</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=26" title="Edit section: Variational quantum eigensolver"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Variational_quantum_eigensolver" title="Variational quantum eigensolver">variational quantum eigensolver</a> (VQE) algorithm applies classical optimization to minimize the energy expectation value of an <a href="/wiki/Ansatz" title="Ansatz">ansatz state</a> to find the ground state of a Hermitian operator, such as a molecule's Hamiltonian.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> It can also be extended to find excited energies of molecular Hamiltonians.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Contracted_quantum_eigensolver">Contracted quantum eigensolver</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=27" title="Edit section: Contracted quantum eigensolver"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The contracted quantum eigensolver (CQE) algorithm minimizes the residual of a contraction (or projection) of the Schrödinger equation onto the space of two (or more) electrons to find the ground- or excited-state energy and two-electron reduced density matrix of a molecule.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> It is based on classical methods for solving energies and two-electron reduced density matrices directly from the anti-Hermitian contracted Schrödinger equation.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=28" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization algorithms</a></li> <li><a href="/wiki/Quantum_sort" title="Quantum sort">Quantum sort</a></li> <li><a href="/wiki/Primality_test" title="Primality test">Primality test</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Quantum_algorithm&action=edit&section=29" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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 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"Variational Quantum Computation of Excited States". <i>Quantum</i>. <b>3</b>: 156. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1805.08138">1805.08138</a></span>. <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/2019Quant...3..156H">2019Quant...3..156H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.22331%2Fq-2019-07-01-156">10.22331/q-2019-07-01-156</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:119185497">119185497</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum&rft.atitle=Variational+Quantum+Computation+of+Excited+States&rft.volume=3&rft.pages=156&rft.date=2019&rft_id=info%3Aarxiv%2F1805.08138&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119185497%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.22331%2Fq-2019-07-01-156&rft_id=info%3Abibcode%2F2019Quant...3..156H&rft.aulast=Higgott&rft.aufirst=Oscar&rft.au=Wang%2C+Daochen&rft.au=Brierley%2C+Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmartMazziotti2021" class="citation journal cs1">Smart, Scott; Mazziotti, David (18 February 2021). "Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices". <i>Phys. Rev. Lett</i>. <b>125</b> (7): 070504. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2004.11416">2004.11416</a></span>. <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/2021PhRvL.126g0504S">2021PhRvL.126g0504S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.126.070504">10.1103/PhysRevLett.126.070504</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33666467">33666467</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:216144443">216144443</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+Lett.&rft.atitle=Quantum+Solver+of+Contracted+Eigenvalue+Equations+for+Scalable+Molecular+Simulations+on+Quantum+Computing+Devices&rft.volume=125&rft.issue=7&rft.pages=070504&rft.date=2021-02-18&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A216144443%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2021PhRvL.126g0504S&rft_id=info%3Aarxiv%2F2004.11416&rft_id=info%3Apmid%2F33666467&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.126.070504&rft.aulast=Smart&rft.aufirst=Scott&rft.au=Mazziotti%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+algorithm" class="Z3988"></span></span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMazziotti2006" class="citation journal cs1">Mazziotti, David (6 October 2006). 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(2023). "Quantum algorithms: A survey of applications and end-to-end complexities". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2310.03011">2310.03011</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/quant-ph">quant-ph</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Quantum+algorithms%3A+A+survey+of+applications+and+end-to-end+complexities&rft.date=2023&rft_id=info%3Aarxiv%2F2310.03011&rft.aulast=Dalzell&rft.aufirst=Alexander+M.&rft.au=McArdle%2C+Sam&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+algorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmithMosca2012" class="citation book cs1">Smith, J.; Mosca, M. 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"Algorithms for Quantum Computers". <i>Handbook of Natural Computing</i>. pp. <span class="nowrap">1451–</span>1492. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1001.0767">1001.0767</a></span>. <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%2F978-3-540-92910-9_43">10.1007/978-3-540-92910-9_43</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-92909-3" title="Special:BookSources/978-3-540-92909-3"><bdi>978-3-540-92909-3</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:16565723">16565723</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algorithms+for+Quantum+Computers&rft.btitle=Handbook+of+Natural+Computing&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1451-%3C%2Fspan%3E1492&rft.date=2012&rft_id=info%3Aarxiv%2F1001.0767&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16565723%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F978-3-540-92910-9_43&rft.isbn=978-3-540-92909-3&rft.aulast=Smith&rft.aufirst=J.&rft.au=Mosca%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AQuantum+algorithm" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChildsVan_Dam2010" class="citation journal cs1">Childs, A. 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.navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar 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: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_science667" 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 href="/wiki/Quantum_computing" title="Quantum computing">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<br />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/Entanglement_swapping" title="Entanglement swapping">Entanglement swapping</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_cryptography24" 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 class="mw-selflink selflink">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<br />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 <br /> 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"><i>T</i><sub>1</sub></a></li> <li><a href="/wiki/Spin%E2%80%93spin_relaxation" title="Spin–spin relaxation"><i>T</i><sub>2</sub></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum<br /><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> <li><a href="/wiki/Hamiltonian_quantum_computation" title="Hamiltonian quantum computation">Hamiltonian quantum computation</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<br />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"><i>gnu</i></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<br />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<br />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><span class="noviewer" typeof="mw:File"><span title="Category"><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" /></span></span> <a href="/wiki/Category:Quantum_information_science" title="Category:Quantum information science">Quantum information science</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/20px-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/40px-Symbol_template_class_pink.svg.png 1.5x" data-file-width="180" data-file-height="185" /></span></span> <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="Quantum_mechanics332" 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_mechanics332" 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 class="mw-selflink selflink">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 href="/wiki/Quantum_computing" title="Quantum computing">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><span class="noviewer" typeof="mw:File"><span title="Category"><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" /></span></span> <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" /></div><div role="navigation" class="navbox" aria-labelledby="Emerging_technologies167" 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_technologies167" 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 class="mw-selflink selflink">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 href="/wiki/Quantum_computing" title="Quantum computing">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" class="mw-redirect" title="Magnetic refrigeration">Magnetic 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