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xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="computation">Computation</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, <a class="existingWikiWord" href="/nlab/show/realizability">realizability</a>, <a class="existingWikiWord" href="/nlab/show/computability">computability</a> <a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a> <a class="existingWikiWord" href="/nlab/show/propositions+as+types">propositions as types</a>, <a class="existingWikiWord" href="/nlab/show/proofs+as+programs">proofs as programs</a>, <a class="existingWikiWord" href="/nlab/show/computational+trinitarianism">computational trinitarianism</a> <h3 id="constructive_mathematics">Constructive mathematics</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">homotopy topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/canonical+form">canonical form</a>, <a class="existingWikiWord" href="/nlab/show/univalence">univalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a>, <a class="existingWikiWord" href="/nlab/show/h-set">h-set</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable equality</a>, <a class="existingWikiWord" href="/nlab/show/decidable+subset">decidable subset</a>, <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited set</a>, <a class="existingWikiWord" href="/nlab/show/subsingleton">subsingleton</a> </li> </ul> <h3 id="realizability">Realizability</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/realizability+topos">realizability topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/realizability+model">realizability model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/realizability+interpretation">realizability interpretation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+topos">effective topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kleene%27s+first+algebra">Kleene's first algebra</a>, <a class="existingWikiWord" href="/nlab/show/Kleene%27s+second+algebra">Kleene's second algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/function+realizability">function realizability</a> </li> </ul> <h3 id="computability">Computability</h3> <ul> <li> <a class="existingWikiWord" href="/nlab/show/computability">computability</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computation">computation</a>, <a class="existingWikiWord" href="/nlab/show/computational+type+theory">computational type theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+function">computable function</a>, <a class="existingWikiWord" href="/nlab/show/partial+recursive+function">partial recursive function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a>, <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Type+Two+Theory+of+Effectivity">Type Two Theory of Effectivity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+function+%28analysis%29">computable function (analysis)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/exact+real+computer+arithmetic">exact real computer arithmetic</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+set">computable set</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/persistent+homology">persistent homology</a>, <a class="existingWikiWord" href="/nlab/show/effective+homology">effective homology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Church-Turing+thesis">Church-Turing thesis</a> </li> </ul> </div></div> <h4 id="quantum_systems">Quantum systems</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a>, <a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">dependent</a> <a class="existingWikiWord" href="/nlab/show/linear+type+theory">linear type theory</a> <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> in <a class="existingWikiWord" href="/nlab/show/quantum+information+theory+via+dagger-compact+categories">†-compact categories</a> <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+systems">quantum systems</a> (<a class="existingWikiWord" href="/nlab/show/parameterized+quantum+systems">parameterized</a>, <a class="existingWikiWord" href="/nlab/show/open+quantum+system">open</a>) <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/quantum+state+collapse">quantum state collapse</a> <a class="existingWikiWord" href="/nlab/show/quantum+decoherence">quantum decoherence</a> <a class="existingWikiWord" href="/nlab/show/quantum+adiabatic+theorem">quantum adiabatic theorem</a> <a class="existingWikiWord" href="/nlab/show/Berry+phases">Berry phases</a> <a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+many-body+physics">quantum many-body physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (<a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-</a>)<a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a> <a class="existingWikiWord" href="/nlab/show/quantum+material">quantum material</a> (<a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological</a>) <a class="existingWikiWord" href="/nlab/show/phases+of+matter">phases of matter</a> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/quantum+probability+theory">quantum probability theory</a> – <a class="existingWikiWord" href="/nlab/show/observables">observables</a> and <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/states">states</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+state">classical state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/quasi-state">quasi-state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+state+preparation">quantum state preparation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/probability+amplitude">probability amplitude</a>, <a class="existingWikiWord" href="/nlab/show/quantum+fluctuation">quantum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bra-ket">bra-ket</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell+state">Bell state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+superposition">quantum superposition</a>, <a class="existingWikiWord" href="/nlab/show/quantum+interference">quantum interference</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a> <a class="existingWikiWord" href="/nlab/show/wave+function+collapse">wave function collapse</a> <a class="existingWikiWord" href="/nlab/show/Born+rule">Born rule</a> <a class="existingWikiWord" href="/nlab/show/deferred+measurement+principle">deferred measurement principle</a> <a class="existingWikiWord" href="/nlab/show/quantum+reader+monad">quantum reader monad</a> <a class="existingWikiWord" href="/nlab/show/measurement+problem">measurement problem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/superselection+sector">superselection sector</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coherent+quantum+state">coherent quantum state</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/ground+state">ground state</a>, <a class="existingWikiWord" href="/nlab/show/excited+state">excited state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a>, <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+diagram">vacuum diagram</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+expectation+value">vacuum expectation value</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+amplitude">vacuum amplitude</a>, <a class="existingWikiWord" href="/nlab/show/vacuum+fluctuation">vacuum fluctuation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+polarization">vacuum polarization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/thermal+vacuum">thermal vacuum</a>, <a class="existingWikiWord" href="/nlab/show/KMS+state">KMS state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/false+vacuum">false vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a>, <a class="existingWikiWord" href="/nlab/show/Coleman-De+Luccia+instanton">Coleman-De Luccia instanton</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/theta+vacuum">theta vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory+vacuum">perturbative string theory vacuum</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/non-geometric+string+theory+vacuum">non-geometric string theory vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/entangled+state">entangled state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tensor+network+state">tensor network state</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/matrix+product+state">matrix product state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tree+tensor+network+state">tree tensor network state</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/observables">observables</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+observable">quantum observable</a>, <a class="existingWikiWord" href="/nlab/show/beable">beable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/star-algebra">star-algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operator+%28in+geometric+quantization%29">quantum operator (in geometric quantization)</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+effect">quantum effect</a>, <a class="existingWikiWord" href="/nlab/show/effect+algebra">effect algebra</a> </li> <li> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/local+observable">local observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/polynomial+observable">polynomial observable</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/linear+observable">linear observable</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/regular+observable">regular observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microcausal+observable">microcausal observable</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/normal-ordered+product">normal-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/time-ordered+products">time-ordered products</a>, <a class="existingWikiWord" href="/nlab/show/retarded+product">retarded product</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/interacting+field+algebra+of+observables">interacting field algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/theorems">theorems</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Nuiten%27s+lemma">Nuiten's lemma</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wigner%27s+theorem">Wigner's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/no-cloning+theorem">no-cloning theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> </ul> </li> </ul> </div> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information+via+dagger-compact+categories">quantum information via dagger-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+operation">quantum operation</a>, <a class="existingWikiWord" href="/nlab/show/quantum+channel">quantum channel</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+teleportation">quantum teleportation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+entanglement">quantum entanglement</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/entanglement+entropy">entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/holographic+entanglement+entropy">holographic entanglement entropy</a> <a class="existingWikiWord" href="/nlab/show/topological+entanglement+entropy">topological entanglement entropy</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+technology">quantum technology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+sensing">quantum sensing</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+communication">quantum communication</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+cryptography">quantum cryptography</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+machine+learning">quantum machine learning</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/qbit">qbit</a>, <a class="existingWikiWord" href="/nlab/show/qdit">qdit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+gate">quantum gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adiabatic+quantum+computation">adiabatic quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/measurement-based+quantum+computation">measurement-based quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+quantum+computation">topological quantum computation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+programming+language">quantum programming language</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/HaPPY+code">HaPPY code</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Majorana+dimer+code">Majorana dimer code</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a> </li> <li> quantum algorithms: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Grover%27s+algorithm">Grover's algorithm</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Shor%27s+algorithm">Shor's algorithm</a> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#in_terms_of_geometric_quantization'>In terms of geometric quantization</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#SpinResonanceQBitsReferences'>Spin resonance qbits</a></li> <li><a href='#SuperconductingQBitsReferences'>Superconducting qbits</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> In <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> and <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a>, by a q-bit (or qubit) one means a <a class="existingWikiWord" href="/nlab/show/quantum+state">quantum state</a> in a 2-<a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimensional</a> <a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">of states</a>. Hence the quantum <a class="existingWikiWord" href="/nlab/show/data+type">data type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QBit</mi></mrow><annotation encoding="application/x-tex">QBit</annotation></semantics></math> is the 2-<a class="existingWikiWord" href="/nlab/show/dimension+of+a+vector+space">dimensional</a> <a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a> equipped with its canonical <a class="existingWikiWord" href="/nlab/show/quantum+measurement">quantum measurement</a>-<a class="existingWikiWord" href="/nlab/show/linear+basis">basis</a> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mo>⋅</mo><mo stretchy="false">|</mo><mn>0</mn><mo stretchy="false">⟩</mo><mo>⊕</mo><mi>ℂ</mi><mo>⋅</mo><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{C}^2 \,\simeq\, \mathbb{C} \cdot \vert 0 \rangle \oplus \mathbb{C} \cdot \vert 1 \rangle \,. </annotation></semantics></math></div> Analogous higher- but still finite- <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional quantum data (types) are called <a class="existingWikiWord" href="/nlab/show/qudit">qdits</a> (“qtrits” for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">d = 3</annotation></semantics></math>). <h2 id="properties">Properties</h2> <h3 id="in_terms_of_geometric_quantization">In terms of geometric quantization</h3> In <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> qbits are naturally understood as the states given by the <a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a> for <a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a> (plus <a class="existingWikiWord" href="/nlab/show/metaplectic+correction">metaplectic correction</a>) being of unit <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a>. See at <a href="geometric+quantization+of+the+2-sphere#TheSpaceOfQuantumStates">geometric quantization of the 2-sphere – The space of quantum states</a>. <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/data+type">data type</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/one+clean+qbit">one clean qbit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Pauli+gate">Pauli gate</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bit">bit</a>, <a class="existingWikiWord" href="/nlab/show/truth+value">truth value</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bit+flip+channel">bit flip channel</a>, <a class="existingWikiWord" href="/nlab/show/bit+flip+code">bit flip code</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+logic+gate">quantum logic gate</a>, <a class="existingWikiWord" href="/nlab/show/quantum+circuit">quantum circuit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>, <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a>, <a class="existingWikiWord" href="/nlab/show/quantum+error+correction">quantum error correction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dimer">dimer</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+set">quantum set</a> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> The term q-bit goes back to <ul> <li><a class="existingWikiWord" href="/nlab/show/Benjamin+Schumacher">Benjamin Schumacher</a>, Quantum coding, Phys. Rev. A 51 (1995) 2738 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1103/PhysRevA.51.2738">doi:10.1103/PhysRevA.51.2738</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> and was popularized by early adoption such as in <ul> <li id="Shor95"><a class="existingWikiWord" href="/nlab/show/Peter+W.+Shor">Peter W. Shor</a>, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R) 1995 (<a href="https://doi.org/10.1103/PhysRevA.52.R2493">doi:10.1103/PhysRevA.52.R2493</a>)</li> </ul> Textbook account: <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+A.+Nielsen">Michael A. Nielsen</a>, <a class="existingWikiWord" href="/nlab/show/Isaac+L.+Chuang">Isaac L. Chuang</a>, §1.2 in: Quantum computation and quantum information, Cambridge University Press (2000) [<a href="https://doi.org/10.1017/CBO9780511976667">doi:10.1017/CBO9780511976667</a>, <a href="http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/NielsenChuangQuantumComputation.pdf" title="pdf">pdf</a>]</li> </ul> See also: <ul> <li>Wikipedia, <a href="http://en.wikipedia.org/wiki/Qubit">Qbit</a></li> </ul> Laboratoy-realizations of qbits for use in <a class="existingWikiWord" href="/nlab/show/quantum+computers">quantum computers</a>: <ul> <li><a href="https://qubitzoo.org/qubitzoo/">Qubit Zoo</a></li> </ul> <div> <h3 id="SpinResonanceQBitsReferences">Spin resonance qbits</h3> The idea of <a class="existingWikiWord" href="/nlab/show/spin+resonance+qbit">spin resonance qbits</a>, i.e. of <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a> realized on <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanical</a> <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a> (e.g. <a class="existingWikiWord" href="/nlab/show/electron">electron</a>-spin or <a class="existingWikiWord" href="/nlab/show/nucleus">nucleus</a>-spin) and <a class="existingWikiWord" href="/nlab/show/quantum+gate">manipulated</a> via <a class="existingWikiWord" href="/nlab/show/spin+resonance">spin resonance</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Loss">Daniel Loss</a>, <a class="existingWikiWord" href="/nlab/show/David+P.+DiVincenzo">David P. DiVincenzo</a>, Quantum Computation with Quantum Dots, Phys. Rev. A 57 120 (1998) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/cond-mat/9701055">arXiv:cond-mat/9701055</a>, <a href="https://doi.org/10.1103/PhysRevA.57.120">doi:10.1103/PhysRevA.57.120</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> The very first proof-of-principle <a class="existingWikiWord" href="/nlab/show/quantum+computations">quantum computations</a> were made with <a class="existingWikiWord" href="/nlab/show/nuclear+magnetic+resonance">nuclear magnetic resonance</a>-technology: <ul> <li> D. G. Cory et al, NMR Based Quantum Information Processing: Achievements and Prospects, Fortsch. Phys. 48 9-11 (2000) 875-907 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/quant-ph/0004104">arXiv:quant-ph/0004104</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Jonathan+A.+Jones">Jonathan A. Jones</a>, Quantum Computing and Nuclear Magnetic Resonance, PhysChemComm 11 (2001) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1039/b103231n">doi:10.1039/b103231n</a>, <a href="https://arxiv.org/abs/quant-ph/0106067">arXiv:quant-ph/0106067</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Jonathan+A.+Jones">Jonathan A. Jones</a>, Quantum Computing with NMR, Prog. NMR Spectrosc. 59 (2011) 91-120 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/j.pnmrs.2010.11.001">doi:10.1016/j.pnmrs.2010.11.001</a>, <a href="https://arxiv.org/abs/1011.1382">arXiv:1011.1382</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> Dorothea Golze, Maik Icker, Stefan Berger, Implementation of two-qubit and three-qubit quantum computers using liquid-state nuclear magnetic resonance, Concepts in Magnetic Resonance 40A 1 (2012) 25-37 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1002/cmr.a.21222">doi:10.1002/cmr.a.21222</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> NMR Quantum Computing (2012) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>slides <a href="https://qudev.phys.ethz.ch/static/content/QSIT12/QSIT12_NMR_L01.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> Tao Xin et al., Nuclear magnetic resonance for quantum computing: Techniques and recent achievements (Topic Review - Solid-state quantum information processing), Chinese Physics B 27 020308 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://iopscience.iop.org/article/10.1088/1674-1056/27/2/020308">doi:10.1088/1674-1056/27/2/020308</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> See also: <ul> <li>Lieven Vandersypen, Mark Eriksson: Quantum computing with semiconductor spins, Physics Today 72 8 (2019) 38 [<a href="https://doi.org/10.1063/PT.3.4270">doi:10.1063/PT.3.4270</a>]</li> </ul> Monograph: <ul> <li>Chen, Church, Englert, Henkel, Rohwedder, Scully, Zubairy, section 10 of: Quantum Computing Devices – Principles, Designs, and Analysis, Routledge (2007) [<a href="https://www.routledge.com/Quantum-Computing-Devices-Principles-Designs-and-Analysis/Chen-Church-Englert-Henkel-Rohwedder-Scully-Zubairy/p/book/9780367390372?srsltid=AfmBOooGiSmyK4mzxjmR1LBJz4zU9cgCpp_z4cPiCzEhM6SJo4b2GxRp">ISBN:9780367390372</a>]</li> </ul> Exposition, review and outlook: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Raymond+Laflamme">Raymond Laflamme</a>, <a class="existingWikiWord" href="/nlab/show/Emanuel+Knill">Emanuel Knill</a>, et al., Introduction to NMR Quantum Information Processing, Proceedings of the International School of Physics “Enrico Fermi” 148 Experimental Quantum Computation and Information [<a href="https://arxiv.org/abs/quant-ph/0207172">arXiv:quant-ph/0207172</a>] </li> <li id="EqubalNov22"> <a class="existingWikiWord" href="/nlab/show/Asif+Equbal">Asif Equbal</a>, Molecular spin qubits for future quantum technology, <a href="CQTS#EqubalNov22">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Nov 2022) [slides: <a class="existingWikiWord" href="/nlab/files/Equbal-CQTS-Nov2022.pdf" title="pdf">pdf</a>, video: <a href="https://nyu.zoom.us/rec/play/YTjIGL-Bevb1H44UuL-ZimXdxph5cffddWpP3H4ZsuRT2xu3OrnTbC0NZLsKedUGwS68DJ8onVFPAETb.rreoi7Wt6uXFyaYN?continueMode=true&_x_zm_rtaid=e0VPIMlfT9KlVd_wiaOq6A.1669794985784.11f8cd37091ebf6bdd2a878668e26cd6&_x_zm_rhtaid=404">rec</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Jonathan+A.+Jones">Jonathan A. Jones</a>, Controlling NMR spin systems for quantum computation, Spectroscopy 140–141 (2024) 49-85 [<a href="https://doi.org/10.1016/j.pnmrs.2024.02.002">doi:10.1016/j.pnmrs.2024.02.002</a>, <a href="https://arxiv.org/abs/2402.01308">arXiv:2402.01308</a>] <blockquote> “Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor’s quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. […] the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.” </blockquote> </li> </ul> See also: <ul> <li> Wikipedia, <a href="https://en.wikipedia.org/wiki/Spin_qubit_quantum_computer">Spin qbit quantum computer</a> </li> <li> Wikipedia, <a href="https://en.wikipedia.org/wiki/Nuclear_magnetic_resonance_quantum_computer">Nuclear magnetic resonance quantum computer</a> </li> </ul> More on implementation of <a class="existingWikiWord" href="/nlab/show/quantum+logic+gates">quantum logic gates</a> on <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a> realized on <a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a>-<a class="existingWikiWord" href="/nlab/show/spin">spin</a>, via pulse protocols in NMR-technology: <ul> <li>Price, Somaroo, Tseng, Gore, Fahmy,, Havel, Cory: Construction and Implementation of NMR Quantum Logic Gates for Two Spin Systems, Journal of Magnetic Resonance 140 2 (1999) 371-378 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1006/jmre.1999.1851">doi;10.1006/jmre.1999.1851</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> and analogously on <a class="existingWikiWord" href="/nlab/show/electron">electron</a>-<a class="existingWikiWord" href="/nlab/show/spin">spin</a>: <ul> <li>M. Yu. Volkov and K. M. Salikhov, Pulse Protocols for Quantum Computing with Electron Spins as Qubits, Appl Magn Reson 41 (2011) 145–154 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/s00723-011-0297-2">doi:10.1007/s00723-011-0297-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> For references on spin resonance qbits realized on a <a class="existingWikiWord" href="/nlab/show/nitrogen-vacancy+center+in+diamond">nitrogen-vacancy center in diamond</a>, see there. There exist toy desktop <a class="existingWikiWord" href="/nlab/show/quantum+computers">quantum computers</a> for educational purposes, operating on a couple of nuclear magnetic resonance <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a> at room temperature : <ul> <li> <a href="https://en.ciqtek.com/">CIQTEK</a>: <a href="https://en.ciqtek.com/products/detail/diamond-quantum-computer-for-education.html?id=644102ee-942e-4568-aacd-20241a1a2469&id=644102ee-942e-4568-aacd-20241a1a2469">Diamond Quantum Computer for Education</a> </li> <li> <a href="https://www.spinquanta.com/">SpinQ</a>: SpinQ Triangulum: a commercial three-qubit desktop quantum computer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2202.02983">arXiv:2202.02983</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> </div><div> <h3 id="SuperconductingQBitsReferences">Superconducting qbits</h3> On realizing <a class="existingWikiWord" href="/nlab/show/qbits">qbits</a> and <a class="existingWikiWord" href="/nlab/show/quantum+gates">quantum gates</a> (hence <a class="existingWikiWord" href="/nlab/show/quantum+computation">quantum computation</a>) via <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> of <a class="existingWikiWord" href="/nlab/show/magnetic+flux">magnetic flux</a> through (<a class="existingWikiWord" href="/nlab/show/Josephson+junctions">Josephson junctions</a> in) <a class="existingWikiWord" href="/nlab/show/superconductors">superconductors</a>, manipulated via <a class="existingWikiWord" href="/nlab/show/electromagnetic+radiation">electromagnetic pulses</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Michel+H.+Devoret">Michel H. Devoret</a>, A. Wallraff, J. M. Martinis, Superconducting Qubits: A Short Review [<a href="https://arxiv.org/abs/cond-mat/0411174">arXiv:cond-mat/0411174</a>] </li> <li> John Clarke, Frank K. Wilhelm, Superconducting quantum bits, Nature 453 (2008) 1031–1042 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1038/nature07128">doi:10.1038/nature07128</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> Jerry Moy Chow, Quantum Information Processing with Superconducting Qubits (2010) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://rsl.yale.edu/sites/default/files/files/RSL_Theses/jmcthesis.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Michel+H.+Devoret">Michel H. Devoret</a>, R. J. Schoelkopf, Superconducting Circuits for Quantum Information: An Outlook, Science 339 6124 (2013) 1169-1174 [<a href="https://doi.org/10.1126/science.1231930">doi:10.1126/science.1231930</a>] </li> <li> Jay M. Gambetta, Jerry M. Chow, Matthias Steffen, Building logical qubits in a superconducting quantum computing system, npj Quantum Information 3 2 (2017) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1038/s41534-016-0004-0">doi:10.1038/s41534-016-0004-0</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> Morten Kjaergaard et al. Superconducting Qubits: Current State of Play, Annual Review of Condensed Matter Physics 11 (2019) 369-395 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1146/annurev-conmatphys-031119-050605">doi:10.1146/annurev-conmatphys-031119-050605</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> He-Liang Huang, Dachao Wu, Daojin Fan, Xiaobo Zhu, Superconducting Quantum Computing: A Review, Science China Information Sciences 63 8 (2020) 1-32 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2006.10433">arXiv:2006.10433</a>, <a href="https://doi.org/10.1007/s11432-020-2881-9">doi:10.1007/s11432-020-2881-9</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> S. Kwon et al., Gate-based superconducting quantum computing, Journal of Applied Physics 129 (2021) 041102 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1063/5.0029735">doi:10.1063/5.0029735</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> <a class="existingWikiWord" href="/nlab/show/Olivier+Ezratty">Olivier Ezratty</a>, Perspective on superconducting qubit quantum computing, Eur. Phys. J. A 59 94 (2023) [<a href="https://doi.org/10.1140/epja/s10050-023-01006-7">doi:10.1140/epja/s10050-023-01006-7</a>] </li> </ul> Monograph: <ul> <li>Chen, Church, Englert, Henkel, Rohwedder, Scully, Zubairy, section 9 of: Quantum Computing Devices Principles, Designs, and Analysis, Routledge (2007) [<a href="https://www.routledge.com/Quantum-Computing-Devices-Principles-Designs-and-Analysis/Chen-Church-Englert-Henkel-Rohwedder-Scully-Zubairy/p/book/9780367390372?srsltid=AfmBOooGiSmyK4mzxjmR1LBJz4zU9cgCpp_z4cPiCzEhM6SJo4b2GxRp">ISBN:9780367390372</a>]</li> </ul> Fine detail of the pulse control: <ul> <li> M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, S. Filipp: Leakage reduction in fast superconducting qubit gates via optimal control, npj Quantum Information 7 14 (2021) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1038/s41534-020-00346-2">doi:10.1038/s41534-020-00346-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer & A. Kandala. Dynamics of superconducting qubit relaxation times, npj Quantum Information 8 132 (2022) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1038/s41534-022-00643-y">doi:10.1038/s41534-022-00643-y</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> <li> Elisha Siddiqui Matekole, Yao-Lung L. Fang, Meifeng Lin, Methods and Results for Quantum Optimal Pulse Control on Superconducting Qubit Systems, 2022 IEEE International Parallel and Distributed Processing Symposium Workshops (2022) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2202.03260">arXiv:2202.03260</a>, <a href="https://doi.org/10.1109/IPDPSW55747.2022.00102">doi:10.1109/IPDPSW55747.2022.00102</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math> </li> </ul> Corrections due to <a class="existingWikiWord" href="/nlab/show/quasiparticle">quasiparticle</a>-excitations: <ul> <li><a class="existingWikiWord" href="/nlab/show/Leonid+I.+Glazman">Leonid I. Glazman</a>, <a class="existingWikiWord" href="/nlab/show/Gianluigi+Catelani">Gianluigi Catelani</a>, Bogoliubov Quasiparticles in Superconducting Qubits, SciPost Phys. Lect. Notes 31 (2021) &lbrack;<a href="https://arxiv.org/abs/2003.04366">arXiv:2003.04366</a>, <a href="https://doi.org/10.21468/SciPostPhysLectNotes.31">doi:10.21468/SciPostPhysLectNotes.31</a>&rbrack;</li> </ul> </div></body></html> </div> <div class="revisedby"> Last revised on January 20, 2024 at 13:21:54. 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