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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Code quantique</span></h1>
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<p>Les <b>codes quantiques</b> sont l'équivalent <a href="Informatique_quantique" title="Informatique quantique">quantique</a> des <a href="Code_correcteur" title="Code correcteur">codes correcteurs</a>. La théorie des codes quantiques est donc une branche de l'<a href="Information_quantique" title="Information quantique">information quantique</a> qui s'applique à protéger l'information quantique des effets de la <a href="D%C3%A9coh%C3%A9rence" class="mw-redirect" title="Décohérence">décohérence</a>. La correction d'erreur quantique est un élément essentiel du calcul tolérant aux fautes qui doit gérer non seulement les erreurs dans l'information stockée, mais aussi dans l'application des portes quantiques, la préparation de nouveaux états ainsi que dans les opérations de mesure.
</p><p>De manière analogue aux codes correcteurs classiques utilisant la redondance pour protéger l'information, les codes quantiques utilisent l'<a href="Intrication_quantique" title="Intrication quantique">intrication</a> pour délocaliser sur plusieurs systèmes physiques l'information encodée.
</p>

<div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2></div>
<p>Étant donné un système physique dont l'espace des états est <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span>, un sous-espace <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
<mo>⊆<!-- ⊆ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81c9cba7ff1022ea5550fc3bbb6878e7281b1ce1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}}" loading="lazy"></span> est appelé code quantique pour l'ensemble d'erreurs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=\{E_{a}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=\{E_{a}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/882acf6db39835408e7ebd82f5a461c471561101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.552ex; height:2.843ex;" alt="{\displaystyle {\mathcal {E}}=\{E_{a}\}}" loading="lazy"></span>, s'il existe une opération quantique <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">R</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy"></span>, dite de correction ou de décodage, telle que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {R}}\circ E_{a})(\left|\psi \right\rangle )=\left|\psi \right\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">R</mi>
</mrow>
</mrow>
<mo>∘<!-- ∘ --></mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow>
<mo>|</mo>
<mi>ψ<!-- ψ --></mi>
<mo>⟩</mo>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mi>ψ<!-- ψ --></mi>
<mo>⟩</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {R}}\circ E_{a})(\left|\psi \right\rangle )=\left|\psi \right\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/759699cc921ea4c327e1495f19ba9d03ad0f8941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.829ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {R}}\circ E_{a})(\left|\psi \right\rangle )=\left|\psi \right\rangle }" loading="lazy"></span> pour tout <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{a}\in {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{a}\in {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c4039791581e381cb40ac936a8c197e3ecf24901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.969ex; height:2.509ex;" alt="{\displaystyle E_{a}\in {\mathcal {E}}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\psi \right\rangle \in C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>|</mo>
<mi>ψ<!-- ψ --></mi>
<mo>⟩</mo>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left|\psi \right\rangle \in C}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79c9b0121e9fa6b31605f27dd7ebfdaff5a52cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.672ex; height:2.843ex;" alt="{\displaystyle \left|\psi \right\rangle \in C}" loading="lazy"></span>.
</p><p>De manière opérationnelle, on peut utiliser la condition suivante: un code <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"></span> corrige les erreurs <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c298ed828ff778065aeb5f0f305097f55bb9ae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}" loading="lazy"></span> si et seulement si
</p>
<div class="center"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle \psi _{i}\right|E_{a}^{\dagger }E_{b}\left|\psi _{j}\right\rangle =C_{ab}\delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow>
<mo>⟨</mo>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>|</mo>
</mrow>
<msubsup>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>†<!-- † --></mo>
</mrow>
</msubsup>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mrow>
<mo>|</mo>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>⟩</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
<mi>b</mi>
</mrow>
</msub>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left\langle \psi _{i}\right|E_{a}^{\dagger }E_{b}\left|\psi _{j}\right\rangle =C_{ab}\delta _{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12d686ff995c22936f9e3d291b3e6c8bf34cd066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.16ex; height:3.509ex;" alt="{\displaystyle \left\langle \psi _{i}\right|E_{a}^{\dagger }E_{b}\left|\psi _{j}\right\rangle =C_{ab}\delta _{ij}}" loading="lazy"></span></div>
<p>où <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{a},E_{b}\in {\mathcal {E}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">E</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{a},E_{b}\in {\mathcal {E}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d375c40d2470e86278bce431e017358fa0346f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.656ex; height:2.509ex;" alt="{\displaystyle E_{a},E_{b}\in {\mathcal {E}}}" loading="lazy"></span> et <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\left|\psi _{i}\right\rangle \}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<mo>|</mo>
<msub>
<mi>ψ<!-- ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>⟩</mo>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \{\left|\psi _{i}\right\rangle \}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/884e80438d35d62cbf1ab419191006bd60664a61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.189ex; height:2.843ex;" alt="{\displaystyle \{\left|\psi _{i}\right\rangle \}}" loading="lazy"></span> forment une base orthonormée de <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"></span>.
</p><p>Deux points importants sont à noter. Tout d'abord deux états orthogonaux le sont toujours après l'application d'une erreur. Ensuite, le préfacteur <span class="mwe-math-element mwe-math-element-inline"><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_{ab}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
<mi>b</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C_{ab}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9efe491ebce0b4574f14c29871c092eb867bed6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.469ex; height:2.509ex;" alt="{\displaystyle C_{ab}}" loading="lazy"></span> est indépendant des choix de mots codes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> et <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>.
</p><p>Le <a href="Qubit" title="Qubit">qubit</a>, analogue quantique du bit classique, est souvent utilisé comme unité fondamentale de l'information quantique. C'est pourquoi on note souvent les propriétés d'un code [[n,k,d]], où n est le nombre de qubits physiques, k, le nombre de qubits encodés et d, la distance du code. La distance du code correspond au nombre minimal de qubits devant être affectés pour passer d'un état encodé à un autre.
</p>
<div class="mw-heading mw-heading2"><h2 id="Histoire">Histoire</h2></div>
<p>En 1996, deux groupes de recherche découvrent indépendamment d'importants codes quantiques. La classe de codes CSS a été découverte par <a href="Robert_Calderbank" title="Robert Calderbank">Calderbank</a> et <a href="Peter_Shor" title="Peter Shor">Shor</a>, puis par Steane&nbsp;<a href="https://en.wikipedia.org/wiki/Andrew_Steane" class="extiw external" title="en:Andrew Steane"><span class="indicateur-langue" title="Article en anglais&nbsp;: «&nbsp;Andrew Steane&nbsp;»">(en)</span></a>. Les <a href="Code_stabilisateur" title="Code stabilisateur">codes stabilisateurs</a> les ont maintenant remplacés. Ceux-ci ont été développés, encore une fois, indépendamment par Calderbank, Rains, <a href="Peter_Shor" title="Peter Shor">Shor</a> et <a href="Neil_Sloane" title="Neil Sloane">Sloane</a>, puis Gottesman&nbsp;<a href="https://en.wikipedia.org/wiki/Daniel_Gottesman" class="extiw external" title="en:Daniel Gottesman"><span class="indicateur-langue" title="Article en anglais&nbsp;: «&nbsp;Daniel Gottesman&nbsp;»">(en)</span></a>.
</p><p>Dès 1992, on considère aussi le problème de la protection de bits classiques avec des codes quantiques. C'est ainsi que <a href="Charles_H._Bennett" title="Charles H. Bennett">Bennett</a> et <a href="Stephen_Wiesner" title="Stephen Wiesner">Wiesner</a> développent le <a href="Codage_super-dense" class="mw-redirect" title="Codage super-dense">codage super-dense</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2></div>
<p>Les codes quantiques diffèrent de leurs homologues classiques de par les difficultés particulières au quantique&nbsp;:
</p>
<ol><li><a href="Impossibilit%C3%A9_du_clonage_quantique" class="mw-redirect" title="Impossibilité du clonage quantique">Impossibilité du clonage quantique</a></li>
<li><a href="Continuit%C3%A9_des_erreurs_quantiques" title="Continuité des erreurs quantiques">Continuité des erreurs quantiques</a></li>
<li><a href="Fragilit%C3%A9_de_l'information_quantique_face_aux_mesures" title="Fragilité de l'information quantique face aux mesures">Fragilité de l'information quantique face aux mesures</a></li></ol>
<p>Néanmoins, l'informatique quantique fournit des outils pour les gérer. Par exemple, pour le canal d'inversion de bit, un code quantique semblable au <a href="Code_de_r%C3%A9p%C3%A9tition" title="Code de répétition">code de répétition</a> classique existe.
</p>
<div class="mw-heading mw-heading2"><h2 id="Exemples">Exemples</h2></div>
<p>Autres codes quantiques&nbsp;:
</p>
<ul><li><a href="Code_de_Kitaev" title="Code de Kitaev">Code de Kitaev</a></li>
<li><a href="Code_stabilisateur" title="Code stabilisateur">Code stabilisateur</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>
<p>Sécurisation des transactions financières mobiles<sup id="cite_ref-HQK1_1-0" class="reference"><a href="#cite_note-HQK1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2></div>
<div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references">
<li id="cite_note-HQK1-1"><span class="mw-cite-backlink"><a href="#cite_ref-HQK1_1-0">↑</a> </span><span class="reference-text"><span class="ouvrage" id="2017"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> «&nbsp;<cite style="font-style:normal" lang="en">Handheld quantum key system makes mobile tranactions more secure</cite>&nbsp;», <i><span class="lang-en" lang="en">Electric Optics</span></i>,‎ <time class="nowrap" datetime="2017-04" data-sort-value="2017-04">avril 2017</time>, <abbr class="abbr" title="page">p.</abbr>&nbsp;34 <small style="line-height:1em;">(<a href="International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a>&nbsp;<span class=" noarchive"><a rel="nofollow" class="external text" href="https://portal.issn.org/resource/issn/0013-4589">0013-4589</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Handheld+quantum+key+system+makes+mobile+tranactions+more+secure&amp;rft.jtitle=Electric+Optics&amp;rft.date=2017-04&amp;rft.pages=34&amp;rft.issn=0013-4589&amp;rfr_id=info%3Asid%2Ffr.wikipedia.org%3ACode+quantique"></span></span></span>
</li>
</ol></div>
</div>
<div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3></div>
<p><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Michael Nielsen&nbsp;<a href="https://en.wikipedia.org/wiki/Michael_Nielsen" class="extiw external" title="en:Michael Nielsen"><span class="indicateur-langue" title="Article en anglais&nbsp;: «&nbsp;Michael Nielsen&nbsp;»">(en)</span></a> et Isaac Chuang&nbsp;<a href="https://en.wikipedia.org/wiki/Isaac_Chuang" class="extiw external" title="en:Isaac Chuang"><span class="indicateur-langue" title="Article en anglais&nbsp;: «&nbsp;Isaac Chuang&nbsp;»">(en)</span></a>, <i>Quantum Computation and Quantum Information</i>, Cambridge University Press, 2000 <small style="line-height:1em;">(<a href="International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<span class="nowrap">978-0-521-63503-5</span>)</small>
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