SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Stabilizer code</title>

</head>

Stabilizer code

</h1>

</header>

/* start https://en.wikipedia.org/ */

.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style>

In <a href="Quantum_computing" title="Quantum computing">quantum computing</a> and <a href="Quantum_communication" class="mw-redirect" title="Quantum communication">quantum communication</a>, a stabilizer code is a class of quantum <a href="Code" title="Code">codes</a> for performing <a href="Quantum_error_correction" title="Quantum error correction">quantum error correction</a>. The <a href="Toric_code" title="Toric code">toric code</a>, and <a href="Toric_code#Generalizations" title="Toric code">surface codes</a> more generally,<a href="#cite_note-1">[1]</a> are types of stabilizer codes considered very important for the practical realization of quantum information processing.

<div class="mw-heading mw-heading2"><h2 id="Conceptual_background">Conceptual background</h2></div>

Quantum error-correcting codes restore a noisy,

<a href="Decoherence" class="mw-redirect" title="Decoherence">decohered</a> <a href="Quantum_state" title="Quantum state">quantum state</a> to a pure quantum state. A

<a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">stabilizer</a> quantum error-correcting code appends <a href="Ancilla_(quantum_computing)" class="mw-redirect" title="Ancilla (quantum computing)">ancilla qubits</a>

to qubits that we want to protect. A unitary encoding circuit rotates the

global state into a subspace of a larger <a href="Hilbert_space" title="Hilbert space">Hilbert space</a>. This highly <a href="Quantum_entanglement" title="Quantum entanglement">entangled</a>,

encoded state corrects for local noisy errors. A quantum error-correcting code makes <a href="Quantum_computation" class="mw-redirect" title="Quantum computation">quantum computation</a>

and <a href="Quantum_communication" class="mw-redirect" title="Quantum communication">quantum communication</a> practical by providing a way for a sender and

receiver to simulate a noiseless qubit channel given a <a href="Noisy_qubit_channel" class="mw-redirect" title="Noisy qubit channel">noisy qubit channel</a>

whose noise conforms to a particular error model. The first quantum error-correcting codes are strikingly similar to <a href="Error_correction" class="mw-redirect" title="Error correction">classical block codes</a> in their operation and performance.

The stabilizer theory of <a href="Quantum_error_correction" title="Quantum error correction">quantum error correction</a> allows one to import some

classical binary or quaternary codes for use as a quantum code. However, when importing the

classical code, it must satisfy the <a href="Dual_code" title="Dual code">dual-containing</a> (or self-orthogonality)

constraint. Researchers have found many examples of classical codes satisfying

this constraint, but most classical codes do not. Nevertheless, it is still useful to import classical codes in this way (though, see how the <a href="Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">entanglement-assisted stabilizer formalism</a> overcomes this difficulty).

<div class="mw-heading mw-heading2"><h2 id="Mathematical_background">Mathematical background</h2></div>

The stabilizer formalism exploits elements of

the <a href="Pauli_group" title="Pauli group">Pauli group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> in formulating quantum error-correcting codes. The set

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi =\left\{I,X,Y,Z\right\}}</annotation>

</semantics>

</math><img src="./790df61b6a87e290ba4765096bda413ad4dffad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.874ex; height:2.843ex;" alt="{\displaystyle \Pi =\left\{I,X,Y,Z\right\}}" loading="lazy"> consists of the <a href="Pauli_operators" class="mw-redirect" title="Pauli operators">Pauli operators</a>:

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\equiv {\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ X\equiv {\begin{bmatrix}0&1\\1&0\end{bmatrix}},\ Y\equiv {\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\ Z\equiv {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}</annotation>

</semantics>

</math><img src="./b5df74725db01c953c3fd10a4734da2cf5a57e82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.162ex; height:6.176ex;" alt="{\displaystyle I\equiv {\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ X\equiv {\begin{bmatrix}0&1\\1&0\end{bmatrix}},\ Y\equiv {\begin{bmatrix}0&-i\\i&0\end{bmatrix}},\ Z\equiv {\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}" loading="lazy"></dd></dl>

The above operators act on a single <a href="Qubit" title="Qubit">qubit</a> – a state represented by a vector in a two-dimensional

<a href="Hilbert_space" title="Hilbert space">Hilbert space</a>. Operators in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> have <a href="Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation>

</semantics>

</math><img src="./0bfeaa85da53ad1947d8000926cfea33827ef1e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}" loading="lazy"> and either <a href="Commutative_property" title="Commutative property">commute</a>

or <a href="Anti-commute" class="mw-redirect" title="Anti-commute">anti-commute</a>. The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy"> consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-fold <a href="Tensor_product" title="Tensor product">tensor products</a> of

<a href="Pauli_operator" class="mw-redirect" title="Pauli operator">Pauli operators</a>:

<msup>

</mrow>

</msup>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}=\left\{{\begin{array}{c}e^{i\phi }A_{1}\otimes \cdots \otimes A_{n}:\forall j\in \left\{1,\ldots ,n\right\}A_{j}\in \Pi ,\ \ \phi \in \left\{0,\pi /2,\pi ,3\pi /2\right\}\end{array}}\right\}.}</annotation>

</semantics>

</math><img src="./c39684e0afcf964e39a5edde4aaa1567360a1a4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:75.113ex; height:3.176ex;" alt="{\displaystyle \Pi ^{n}=\left\{{\begin{array}{c}e^{i\phi }A_{1}\otimes \cdots \otimes A_{n}:\forall j\in \left\{1,\ldots ,n\right\}A_{j}\in \Pi ,\ \ \phi \in \left\{0,\pi /2,\pi ,3\pi /2\right\}\end{array}}\right\}.}" loading="lazy"></dd></dl>

Elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy"> act on a <a href="Quantum_register" title="Quantum register">quantum register</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> qubits. We

occasionally omit <a href="Tensor_product" title="Tensor product">tensor product</a> symbols in what follows so that

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{1}\cdots A_{n}\equiv A_{1}\otimes \cdots \otimes A_{n}.}</annotation>

</semantics>

</math><img src="./eb66c6239d77ab88ff3b09e008440f540b748f54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.165ex; height:2.509ex;" alt="{\displaystyle A_{1}\cdots A_{n}\equiv A_{1}\otimes \cdots \otimes A_{n}.}" loading="lazy"></dd></dl>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-fold <a href="Pauli_group" title="Pauli group">Pauli group</a>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy"> plays an important role for both the encoding circuit and the

error-correction procedure of a quantum stabilizer code over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> qubits.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Let us define an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[n,k\right]}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[n,k\right]}</annotation>

</semantics>

</math><img src="./39cfbc83653d1574008de25dbe429486a03a8ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.934ex; height:2.843ex;" alt="{\displaystyle \left[n,k\right]}" loading="lazy"> stabilizer quantum error-correcting

code to encode <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" 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}" loading="lazy"> logical qubits into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> physical qubits. The rate of such a

code is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k/n}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k/n}</annotation>

</semantics>

</math><img src="./df39d254fa80db61cf468fed3a8c4dbb10e7e7d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.768ex; height:2.843ex;" alt="{\displaystyle k/n}" loading="lazy">. Its stabilizer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> is an <a href="Abelian_group" title="Abelian group">abelian</a> <a href="Subgroup" title="Subgroup">subgroup</a> of the

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-fold Pauli group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">

does not contain the operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -I^{\otimes n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -I^{\otimes n}}</annotation>

</semantics>

</math><img src="./ded6f0d78ea6c581d4be395e0d618147d02d2381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.522ex; height:2.676ex;" alt="{\displaystyle -I^{\otimes n}}" loading="lazy">. The simultaneous

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +1}</annotation>

</semantics>

</math><img src="./d04cf05c67d41d9f39dabf6a90722ce860a76958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}" loading="lazy">-<a href="Eigenspace" class="mw-redirect" title="Eigenspace">eigenspace</a> of the operators constitutes the codespace. The

codespace has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{k}}</annotation>

</semantics>

</math><img src="./2d82641ae2702b0db07dd11830af27b9ee0cd196.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}" loading="lazy"> so that we can encode <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" 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}" loading="lazy"> qubits into it. The

stabilizer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> has a minimal <a href="Representation_(mathematics)" title="Representation (mathematics)">representation</a> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>

</semantics>

</math><img src="./b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy">

independent generators

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{g_{1},\ldots ,g_{n-k}\ |\ \forall i\in \left\{1,\ldots ,n-k\right\},\ g_{i}\in {\mathcal {S}}\right\}.}</annotation>

</semantics>

</math><img src="./e2e9590516b3045a8fea85d4c09d7ec129280b6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.262ex; height:2.843ex;" alt="{\displaystyle \left\{g_{1},\ldots ,g_{n-k}\ |\ \forall i\in \left\{1,\ldots ,n-k\right\},\ g_{i}\in {\mathcal {S}}\right\}.}" loading="lazy"></dd></dl>

The generators are

independent in the sense that none of them are a product of any other two (up

to a <a href="Quantum_state" title="Quantum state">global phase</a>). The operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1},\ldots ,g_{n-k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1},\ldots ,g_{n-k}}</annotation>

</semantics>

</math><img src="./99935ebce823b0afeaa735c2bae431217dc181a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.804ex; height:2.009ex;" alt="{\displaystyle g_{1},\ldots ,g_{n-k}}" loading="lazy"> function in the same

way as a <a href="Parity_check_matrix" class="mw-redirect" title="Parity check matrix">parity check matrix</a> does for a classical <a href="Linear_block_code" class="mw-redirect" title="Linear block code">linear block code</a>.

<div class="mw-heading mw-heading2"><h2 id="Stabilizer_error-correction_conditions">Stabilizer error-correction conditions</h2></div>

One of the fundamental notions in quantum error correction theory is that it

suffices to correct a <a href="Discrete_set" class="mw-redirect" title="Discrete set">discrete</a> error set with <a href="Support_(mathematics)" title="Support (mathematics)">support</a> in the Pauli group

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy">. Suppose that the errors affecting an

encoded quantum state are a subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./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"> of the Pauli group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy">:

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}\subset \Pi ^{n}.}</annotation>

</semantics>

</math><img src="./9dfe956a2210f3a39b5dead4cfb31d650676f680.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.018ex; height:2.343ex;" alt="{\displaystyle {\mathcal {E}}\subset \Pi ^{n}.}" loading="lazy"></dd></dl>

Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> are both subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy">, an error <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\in {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E\in {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./d22a239ba253499c81794d2285d678391fb6af06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.927ex; height:2.176ex;" alt="{\displaystyle E\in {\mathcal {E}}}" loading="lazy"> that affects an

encoded quantum state either <a href="Commutative_property" title="Commutative property">commutes</a> or <a href="Anticommute" class="mw-redirect" title="Anticommute">anticommutes</a> with any particular

element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">. The error <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is correctable if it

anticommutes with an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">. An anticommuting error

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is detectable by <a href="Quantum_measurement" class="mw-redirect" title="Quantum measurement">measuring</a> each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> and

computing a syndrome <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> identifying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy">. The syndrome is a binary

vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> with length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>

</semantics>

</math><img src="./b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy"> whose elements identify whether the

error <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> commutes or anticommutes with each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./36f12b1d87577dc8a25b2e5107ba7f3d76167ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.449ex; height:2.509ex;" alt="{\displaystyle g\in {\mathcal {S}}}" loading="lazy">. An error

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> that commutes with every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> is correctable if

and only if it is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">. It corrupts the encoded state if it

commutes with every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> but does not lie in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">. So we compactly summarize the stabilizer error-correcting conditions: a

stabilizer code can correct any errors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1},E_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1},E_{2}}</annotation>

</semantics>

</math><img src="./c9c035f17c20c94cf377a808a0244d360a7954c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle E_{1},E_{2}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation>

</semantics>

</math><img src="./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"> if

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}^{\dagger }E_{2}\notin {\mathcal {Z}}\left({\mathcal {S}}\right)}</annotation>

</semantics>

</math><img src="./8986fb8141192bef387ad494b04a905857de70a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.851ex; height:3.509ex;" alt="{\displaystyle E_{1}^{\dagger }E_{2}\notin {\mathcal {Z}}\left({\mathcal {S}}\right)}" loading="lazy"></dd></dl>

or

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1}^{\dagger }E_{2}\in {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./ab44027aa1af6a5ded6cd36a1751c9268b8fbe3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.872ex; height:3.509ex;" alt="{\displaystyle E_{1}^{\dagger }E_{2}\in {\mathcal {S}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Z}}\left({\mathcal {S}}\right)}">

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {Z}}\left({\mathcal {S}}\right)}</annotation>

</semantics>

</math><img src="./546022671cafebf3e5fa12060469d98cdc1f1a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.471ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Z}}\left({\mathcal {S}}\right)}" loading="lazy"> is the <a href="Centralizer" class="mw-redirect" title="Centralizer">centralizer</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy"> (i.e., the subgroup of elements that commute with all members of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}}</annotation>

</semantics>

</math><img src="./2302a18e269dbecc43c57c0c2aced3bfae15278d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.492ex; height:2.176ex;" alt="{\displaystyle {\mathcal {S}}}" loading="lazy">, also known as the commutant).

<div class="mw-heading mw-heading2"><h2 id="Simple_example_of_a_stabilizer_code">Simple example of a stabilizer code</h2></div>

A simple example of a stabilizer code is a three qubit

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[[3,1,3\right]]}</annotation>

</semantics>

</math><img src="./7def4a3240b80dad385f0948ecc1e6b3548f7d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.143ex; height:2.843ex;" alt="{\displaystyle \left[[3,1,3\right]]}" loading="lazy"> stabilizer code. It encodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

</semantics>

</math><img src="./6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy"> logical qubit

into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=3}</annotation>

</semantics>

</math><img src="./1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy"> physical qubits and protects against a single-bit flip

error in the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{i}\right\}}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{X_{i}\right\}}</annotation>

</semantics>

</math><img src="./d9964f7145dbc74e4e87538b1094faf90db98bda.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.049ex; height:2.843ex;" alt="{\displaystyle \left\{X_{i}\right\}}" loading="lazy">. This does not protect against other Pauli errors such as phase flip errors in the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{Y_{i}\right\}}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{Y_{i}\right\}}</annotation>

</semantics>

</math><img src="./a5ee0ad20f0ae60babbe8645d2d18c029f4800a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.475ex; height:2.843ex;" alt="{\displaystyle \left\{Y_{i}\right\}}" loading="lazy">.or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{Z_{i}\right\}}">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{Z_{i}\right\}}</annotation>

</semantics>

</math><img src="./540157895266de6da48f0c659ce637354940808a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.712ex; height:2.843ex;" alt="{\displaystyle \left\{Z_{i}\right\}}" loading="lazy">. This has code distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=3}</annotation>

</semantics>

</math><img src="./58a4a6128b5cc1e25fcba3a16cbd9126b2d3e2fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=3}" loading="lazy">. Its stabilizer consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-k=2}</annotation>

</semantics>

</math><img src="./eb9daf97ffa7767f7a129a5adcd759afd7c50758.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.707ex; height:2.343ex;" alt="{\displaystyle n-k=2}" loading="lazy"> Pauli operators:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccc}g_{1}&=&Z&Z&I\\g_{2}&=&I&Z&Z\\\end{array}}}</annotation>

</semantics>

</math><img src="./6c7288d037f5bcca412db58df7281ff9535ff865.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.054ex; height:6.176ex;" alt="{\displaystyle {\begin{array}{ccc}g_{1}&=&Z&Z&I\\g_{2}&=&I&Z&Z\\\end{array}}}" loading="lazy"></dd></dl>

If there are no bit-flip errors, both operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

</math><img src="./3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> commute, the syndrome is +1,+1, and no errors are detected.

If there is a bit-flip error on the first encoded qubit, operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

</math><img src="./3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> will anti-commute and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> commute, the syndrome is -1,+1, and the error is detected. If there is a bit-flip error on the second encoded qubit, operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

</math><img src="./3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> will anti-commute and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> anti-commute, the syndrome is -1,-1, and the error is detected. If there is a bit-flip error on the third encoded qubit, operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{1}}</annotation>

</semantics>

</math><img src="./3755e3e04ec295992b2b5331655ef83a500a05c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{1}}" loading="lazy"> will commute and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

</semantics>

</math><img src="./f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> anti-commute, the syndrome is +1,-1, and the error is detected.

<div class="mw-heading mw-heading2"><h2 id="Example_of_a_stabilizer_code">Example of a stabilizer code</h2></div>

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Five-qubit_error_correcting_code" title="Five-qubit error correcting code">Five-qubit error correcting code</a></div>

An example of a stabilizer code is the five qubit

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[[5,1,3\right]]}</annotation>

</semantics>

</math><img src="./6e2afda14da428b8c638ba17ec205289ae6b2f3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.143ex; height:2.843ex;" alt="{\displaystyle \left[[5,1,3\right]]}" loading="lazy"> stabilizer code. It encodes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

</semantics>

</math><img src="./6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy"> logical qubit

into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=5}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=5}</annotation>

</semantics>

</math><img src="./cdb41e9a10a8fd7179b9170149a8d70949ba5d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}" loading="lazy"> physical qubits and protects against an arbitrary single-qubit

error. It has code distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d=3}</annotation>

</semantics>

</math><img src="./58a4a6128b5cc1e25fcba3a16cbd9126b2d3e2fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=3}" loading="lazy">. Its stabilizer consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k=4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-k=4}</annotation>

</semantics>

</math><img src="./a4cc9bbb79c2bddb6c10ae8e6ab77af72b4af04a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.707ex; height:2.343ex;" alt="{\displaystyle n-k=4}" loading="lazy"> Pauli operators:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccccccc}g_{1}&=&X&Z&Z&X&I\\g_{2}&=&I&X&Z&Z&X\\g_{3}&=&X&I&X&Z&Z\\g_{4}&=&Z&X&I&X&Z\end{array}}}</annotation>

</semantics>

</math><img src="./125f52a7c34a1e4d09c93c9b3fa307cc0064863e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:28.558ex; height:12.509ex;" alt="{\displaystyle {\begin{array}{ccccccc}g_{1}&=&X&Z&Z&X&I\\g_{2}&=&I&X&Z&Z&X\\g_{3}&=&X&I&X&Z&Z\\g_{4}&=&Z&X&I&X&Z\end{array}}}" loading="lazy"></dd></dl>

The above operators commute. Therefore, the codespace is the simultaneous

+1-eigenspace of the above operators. Suppose a single-qubit error occurs on

the encoded quantum register. A single-qubit error is in the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{X_{i},Y_{i},Z_{i}\right\}}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{X_{i},Y_{i},Z_{i}\right\}}</annotation>

</semantics>

</math><img src="./434f0d08482716006041fddbe67c4c952a84b5e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.654ex; height:2.843ex;" alt="{\displaystyle \left\{X_{i},Y_{i},Z_{i}\right\}}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation>

</semantics>

</math><img src="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> denotes a Pauli error on qubit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./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">.

It is straightforward to verify that any arbitrary single-qubit error has a

unique syndrome. The receiver corrects any single-qubit error by identifying

the syndrome via a <a href="Parity_measurement" title="Parity measurement">parity measurement</a> and applying a corrective operation.

<div class="mw-heading mw-heading2"><h2 id="Relation_between_Pauli_group_and_binary_vectors">Relation between Pauli group and binary vectors</h2></div>

A simple but useful mapping exists between elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> and the binary

<a href="Vector_space" title="Vector space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Z} _{2}\right)^{2}}">

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Z} _{2}\right)^{2}}</annotation>

</semantics>

</math><img src="./c3e37a7fccc4c24ac333021545f7dcc25054762d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.468ex; height:3.343ex;" alt="{\displaystyle \left(\mathbb {Z} _{2}\right)^{2}}" loading="lazy">. This mapping gives a

simplification of quantum error correction theory. It represents quantum codes

with <a href="Bit_vector" class="mw-redirect" title="Bit vector">binary vectors</a> and <a href="Binary_operation" title="Binary operation">binary operations</a> rather than with <a href="Pauli_operator" class="mw-redirect" title="Pauli operator">Pauli operators</a> and

<a href="Matrix_operation" class="mw-redirect" title="Matrix operation">matrix operations</a> respectively.

We first give the mapping for the one-qubit case. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[A\right]}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[A\right]}</annotation>

</semantics>

</math><img src="./408e5e41e60df49212d5206ede3da5fb0de7b83f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle \left[A\right]}" loading="lazy">

is a set of <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> of an <a href="Operator_(physics)" title="Operator (physics)">operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> that have the same <a href="Phase_(waves)" title="Phase (waves)">phase</a>:

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[A\right]=\left\{\beta A\ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.}</annotation>

</semantics>

</math><img src="./fd49593693fe70823827513bbb79051321af4131.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.116ex; height:2.843ex;" alt="{\displaystyle \left[A\right]=\left\{\beta A\ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.}" loading="lazy"></dd></dl>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\Pi \right]}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\Pi \right]}</annotation>

</semantics>

</math><img src="./d51d1a878cb85b1f8616bd772efa3fa8a13751cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle \left[\Pi \right]}" loading="lazy"> be the set of phase-free Pauli operators where

<mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\Pi \right]=\left\{\left[A\right]\ |\ A\in \Pi \right\}}</annotation>

</semantics>

</math><img src="./0abfb4baea5c08a488b162507a3649c0b0d86dae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.019ex; height:2.843ex;" alt="{\displaystyle \left[\Pi \right]=\left\{\left[A\right]\ |\ A\in \Pi \right\}}" loading="lazy">.

Define the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \Pi }">

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \Pi }</annotation>

</semantics>

</math><img src="./fb3f7c8c1b0e7f05dc060662ef24cdab1a212d6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.826ex; height:3.343ex;" alt="{\displaystyle N:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \Pi }" loading="lazy"> as

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 00\to I,\,\,01\to X,\,\,11\to Y,\,\,10\to Z}</annotation>

</semantics>

</math><img src="./b8d665094a1d48f19a01ddbee60feadf40b7d051.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.786ex; height:2.509ex;" alt="{\displaystyle 00\to I,\,\,01\to X,\,\,11\to Y,\,\,10\to Z}" loading="lazy"></dd></dl>

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in \left(\mathbb {Z} _{2}\right)^{2}}">

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v\in \left(\mathbb {Z} _{2}\right)^{2}}</annotation>

</semantics>

</math><img src="./c44d5c0f4461ea3369421b1981116c87fcc2d098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.8ex; height:3.343ex;" alt="{\displaystyle u,v\in \left(\mathbb {Z} _{2}\right)^{2}}" loading="lazy">. Let us employ the

shorthand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\left(z|x\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=\left(z|x\right)}</annotation>

</semantics>

</math><img src="./f3bd6c73f106f33f6eb91595edfd2d2431b58467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.302ex; height:2.843ex;" alt="{\displaystyle u=\left(z|x\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=\left(z^{\prime }|x^{\prime }\right)}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v=\left(z^{\prime }|x^{\prime }\right)}</annotation>

</semantics>

</math><img src="./ccc2667d78eeacb37553abd6f1d9999fa16c4bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.472ex; height:3.009ex;" alt="{\displaystyle v=\left(z^{\prime }|x^{\prime }\right)}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{\prime }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{\prime }}</annotation>

</semantics>

</math><img src="./326e362e139045ade5c9e914a0d33bb62d4609fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle z^{\prime }}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\prime }\in \mathbb {Z} _{2}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{\prime }\in \mathbb {Z} _{2}}</annotation>

</semantics>

</math><img src="./58c3949666b936dcb8f1cd0989dcd1eb47eef0f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.46ex; height:2.843ex;" alt="{\displaystyle x^{\prime }\in \mathbb {Z} _{2}}" loading="lazy">. For

example, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=\left(0|1\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=\left(0|1\right)}</annotation>

</semantics>

</math><img src="./d67992dd090bcb8dd92b471fc3b537889b7cf344.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.209ex; height:2.843ex;" alt="{\displaystyle u=\left(0|1\right)}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\left(u\right)=X}">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\left(u\right)=X}</annotation>

</semantics>

</math><img src="./be9762eb1adcb281e0b6213b89bbf512c1111ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.668ex; height:2.843ex;" alt="{\displaystyle N\left(u\right)=X}" loading="lazy">. The

map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" 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}" loading="lazy"> induces an <a href="Isomorphism" title="Isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[N\right]:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \left[\Pi \right]}">

<mrow>

</mrow>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[N\right]:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \left[\Pi \right]}</annotation>

</semantics>

</math><img src="./d0677e952d512bdd01f1de0aefe594234406efa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.413ex; height:3.343ex;" alt="{\displaystyle \left[N\right]:\left(\mathbb {Z} _{2}\right)^{2}\rightarrow \left[\Pi \right]}" loading="lazy"> because addition of vectors

in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Z} _{2}\right)^{2}}">

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Z} _{2}\right)^{2}}</annotation>

</semantics>

</math><img src="./c3e37a7fccc4c24ac333021545f7dcc25054762d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.468ex; height:3.343ex;" alt="{\displaystyle \left(\mathbb {Z} _{2}\right)^{2}}" loading="lazy"> is equivalent to multiplication of

Pauli operators up to a global phase:

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[N\left(u+v\right)\right]=\left[N\left(u\right)\right]\left[N\left(v\right)\right].}</annotation>

</semantics>

</math><img src="./84c72186bd8d9741f99407d37dd32e804bf4a0e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.936ex; height:2.843ex;" alt="{\displaystyle \left[N\left(u+v\right)\right]=\left[N\left(u\right)\right]\left[N\left(v\right)\right].}" loading="lazy"></dd></dl>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \odot }</annotation>

</semantics>

</math><img src="./e89e009eb8a8839c82aa5c76c15e9f2d67006276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }" loading="lazy"> denote the <a href="Symplectic_product" class="mw-redirect" title="Symplectic product">symplectic product</a> between two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v\in \left(\mathbb {Z} _{2}\right)^{2}}">

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u,v\in \left(\mathbb {Z} _{2}\right)^{2}}</annotation>

</semantics>

</math><img src="./c44d5c0f4461ea3369421b1981116c87fcc2d098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.8ex; height:3.343ex;" alt="{\displaystyle u,v\in \left(\mathbb {Z} _{2}\right)^{2}}" loading="lazy">:

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\odot v\equiv zx^{\prime }-xz^{\prime }.}</annotation>

</semantics>

</math><img src="./8a22691e3ddc6839d54485ed873d2e3301b89af1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.091ex; height:2.676ex;" alt="{\displaystyle u\odot v\equiv zx^{\prime }-xz^{\prime }.}" loading="lazy"></dd></dl>

The symplectic product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \odot }</annotation>

</semantics>

</math><img src="./e89e009eb8a8839c82aa5c76c15e9f2d67006276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }" loading="lazy"> gives the <a href="Commutative_property" title="Commutative property">commutation</a> relations of elements of

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy">:

<mrow>

</mrow>

<mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\left(u\right)N\left(v\right)=\left(-1\right)^{\left(u\odot v\right)}N\left(v\right)N\left(u\right).}</annotation>

</semantics>

</math><img src="./d98c3cfb6c542334c8ec297d88c10c9e51b299dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.169ex; height:3.509ex;" alt="{\displaystyle N\left(u\right)N\left(v\right)=\left(-1\right)^{\left(u\odot v\right)}N\left(v\right)N\left(u\right).}" loading="lazy"></dd></dl>

The symplectic product and the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" 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}" loading="lazy"> thus give a useful way to phrase

Pauli relations in terms of <a href="Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">binary algebra</a>.

The extension of the above definitions and mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" 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}" loading="lazy"> to multiple qubits is

straightforward. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =A_{1}\otimes \cdots \otimes A_{n}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =A_{1}\otimes \cdots \otimes A_{n}}</annotation>

</semantics>

</math><img src="./700862bf7e0c1ea51e93666811bc61d21213088b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.281ex; height:2.509ex;" alt="{\displaystyle \mathbf {A} =A_{1}\otimes \cdots \otimes A_{n}}" loading="lazy"> denote an

arbitrary element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi ^{n}}</annotation>

</semantics>

</math><img src="./543e53efbf4ee17735d71307e0637421564f2520.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.962ex; height:2.343ex;" alt="{\displaystyle \Pi ^{n}}" loading="lazy">. We can similarly define the phase-free

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">-qubit Pauli group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\Pi ^{n}\right]=\left\{\left[\mathbf {A} \right]\ |\ \mathbf {A} \in \Pi ^{n}\right\}}">

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\Pi ^{n}\right]=\left\{\left[\mathbf {A} \right]\ |\ \mathbf {A} \in \Pi ^{n}\right\}}</annotation>

</semantics>

</math><img src="./0196a05f4266b0080f0f8e5c2fab3e9de2101b21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.009ex; height:2.843ex;" alt="{\displaystyle \left[\Pi ^{n}\right]=\left\{\left[\mathbf {A} \right]\ |\ \mathbf {A} \in \Pi ^{n}\right\}}" loading="lazy"> where

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\mathbf {A} \right]=\left\{\beta \mathbf {A} \ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.}</annotation>

</semantics>

</math><img src="./6552c351391c46e4a10ec670ffdcea07011c2a83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.669ex; height:2.843ex;" alt="{\displaystyle \left[\mathbf {A} \right]=\left\{\beta \mathbf {A} \ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.}" loading="lazy"></dd></dl>

The <a href="Group_operation" class="mw-redirect" title="Group operation">group operation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

</semantics>

</math><img src="./f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy"> for the above equivalence class is as follows:

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\mathbf {A} \right]\ast \left[\mathbf {B} \right]\equiv \left[A_{1}\right]\ast \left[B_{1}\right]\otimes \cdots \otimes \left[A_{n}\right]\ast \left[B_{n}\right]=\left[A_{1}B_{1}\right]\otimes \cdots \otimes \left[A_{n}B_{n}\right]=\left[\mathbf {AB} \right].}</annotation>

</semantics>

</math><img src="./ce39a8417c2e35337f743ae919d79a2141e7c490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.325ex; height:2.843ex;" alt="{\displaystyle \left[\mathbf {A} \right]\ast \left[\mathbf {B} \right]\equiv \left[A_{1}\right]\ast \left[B_{1}\right]\otimes \cdots \otimes \left[A_{n}\right]\ast \left[B_{n}\right]=\left[A_{1}B_{1}\right]\otimes \cdots \otimes \left[A_{n}B_{n}\right]=\left[\mathbf {AB} \right].}" loading="lazy"></dd></dl>

The equivalence class <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\Pi ^{n}\right]}">

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\Pi ^{n}\right]}</annotation>

</semantics>

</math><img src="./4113cf3ec21b188bccc0869f835d59811abd472b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle \left[\Pi ^{n}\right]}" loading="lazy"> forms a <a href="Commutative_group" class="mw-redirect" title="Commutative group">commutative group</a>

under operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ast }</annotation>

</semantics>

</math><img src="./f1858484bef51b1435c2b986c728a81788051803.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }" loading="lazy">. Consider the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n}</annotation>

</semantics>

</math><img src="./134afa8ff09fdddd24b06f289e92e3a045092bd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.557ex; height:2.176ex;" alt="{\displaystyle 2n}" loading="lazy">-dimensional <a href="Vector_space" title="Vector space">vector space</a>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Z} _{2}\right)^{2n}=\left\{\left(\mathbf {z,x} \right):\mathbf {z} ,\mathbf {x} \in \left(\mathbb {Z} _{2}\right)^{n}\right\}.}</annotation>

</semantics>

</math><img src="./26091b34e8621ced5298b753769d7503bdb20219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.492ex; height:3.343ex;" alt="{\displaystyle \left(\mathbb {Z} _{2}\right)^{2n}=\left\{\left(\mathbf {z,x} \right):\mathbf {z} ,\mathbf {x} \in \left(\mathbb {Z} _{2}\right)^{n}\right\}.}" loading="lazy"></dd></dl>

It forms the commutative group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\left(\mathbb {Z} _{2}\right)^{2n},+)}">

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\left(\mathbb {Z} _{2}\right)^{2n},+)}</annotation>

</semantics>

</math><img src="./0cae2f03c5cc31512b23005bb7f09c2758eaba45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.106ex; height:3.343ex;" alt="{\displaystyle (\left(\mathbb {Z} _{2}\right)^{2n},+)}" loading="lazy"> with

operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle +}</annotation>

</semantics>

</math><img src="./fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"> defined as binary vector addition. We employ the notation

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =\left(\mathbf {z} |\mathbf {x} \right),\mathbf {v} =\left(\mathbf {z} ^{\prime }|\mathbf {x} ^{\prime }\right)}</annotation>

</semantics>

</math><img src="./9cc027c934e5df013589748cf5654d24517025a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.994ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} =\left(\mathbf {z} |\mathbf {x} \right),\mathbf {v} =\left(\mathbf {z} ^{\prime }|\mathbf {x} ^{\prime }\right)}" loading="lazy"> to represent any vectors

</mrow>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}}</annotation>

</semantics>

</math><img src="./fcd8999b3ecc18a9ad4dc3750d194b8a23544f65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.32ex; height:3.343ex;" alt="{\displaystyle \mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}}" loading="lazy"> respectively. Each

vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} }</annotation>

</semantics>

</math><img src="./82eca5d0928078d5a61b9e7e98cc73db31070909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\displaystyle \mathbf {z} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"> has elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(z_{1},\ldots ,z_{n}\right)}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(z_{1},\ldots ,z_{n}\right)}</annotation>

</semantics>

</math><img src="./fc112b8b0323289d92d28fb67b44022c44a8884c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.843ex;" alt="{\displaystyle \left(z_{1},\ldots ,z_{n}\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(x_{1},\ldots ,x_{n}\right)}">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(x_{1},\ldots ,x_{n}\right)}</annotation>

</semantics>

</math><img src="./d374b76137f89de9a7d6481372aa0fa85752cdcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle \left(x_{1},\ldots ,x_{n}\right)}" loading="lazy"> respectively with

similar representations for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {z} ^{\prime }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {z} ^{\prime }}</annotation>

</semantics>

</math><img src="./61bc51da772501b4fb02becf2dff176848ef8fb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.873ex; height:2.509ex;" alt="{\displaystyle \mathbf {z} ^{\prime }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\prime }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{\prime }}</annotation>

</semantics>

</math><img src="./b56a7c4251e064d6a4757e2aa44a5e31aa3f7a9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.096ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} ^{\prime }}" loading="lazy">.

The symplectic product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \odot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \odot }</annotation>

</semantics>

</math><img src="./e89e009eb8a8839c82aa5c76c15e9f2d67006276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \odot }" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation>

</semantics>

</math><img src="./261e20fe101de02a771021d9d4466c0ad3e352d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation>

</semantics>

</math><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"> is

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \odot \mathbf {v\equiv } \sum _{i=1}^{n}z_{i}x_{i}^{\prime }-x_{i}z_{i}^{\prime },}</annotation>

</semantics>

</math><img src="./9d5efb7ec30a0c68c3b5bc3c327b2bfdda2aaefa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.827ex; height:6.843ex;" alt="{\displaystyle \mathbf {u} \odot \mathbf {v\equiv } \sum _{i=1}^{n}z_{i}x_{i}^{\prime }-x_{i}z_{i}^{\prime },}" loading="lazy"></dd></dl>

or

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \odot \mathbf {v\equiv } \sum _{i=1}^{n}u_{i}\odot v_{i},}</annotation>

</semantics>

</math><img src="./b6f4ec268da5cce5ead62534a41c2ea49791121d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.863ex; height:6.843ex;" alt="{\displaystyle \mathbf {u} \odot \mathbf {v\equiv } \sum _{i=1}^{n}u_{i}\odot v_{i},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}=\left(z_{i}|x_{i}\right)}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{i}=\left(z_{i}|x_{i}\right)}</annotation>

</semantics>

</math><img src="./e8b451afbff414c8f5b28b9cafce703535a80d29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.694ex; height:2.843ex;" alt="{\displaystyle u_{i}=\left(z_{i}|x_{i}\right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}=\left(z_{i}^{\prime }|x_{i}^{\prime }\right)}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{i}=\left(z_{i}^{\prime }|x_{i}^{\prime }\right)}</annotation>

</semantics>

</math><img src="./ef34ef4cd074e38472d8fd37aaef5967f504a723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.813ex; height:3.176ex;" alt="{\displaystyle v_{i}=\left(z_{i}^{\prime }|x_{i}^{\prime }\right)}" loading="lazy">. Let us define a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} :\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \Pi ^{n}}">

</mrow>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} :\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \Pi ^{n}}</annotation>

</semantics>

</math><img src="./ce14a976e3a22877f7a67647abb5a5de5dac7bf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.059ex; height:3.343ex;" alt="{\displaystyle \mathbf {N} :\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \Pi ^{n}}" loading="lazy"> as follows:

</mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} \left(\mathbf {u} \right)\equiv N\left(u_{1}\right)\otimes \cdots \otimes N\left(u_{n}\right).}</annotation>

</semantics>

</math><img src="./8852ceac561db748c407c188a3b86571424c32e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.762ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} \left(\mathbf {u} \right)\equiv N\left(u_{1}\right)\otimes \cdots \otimes N\left(u_{n}\right).}" loading="lazy"></dd></dl>

Let

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {X} \left(\mathbf {x} \right)\equiv X^{x_{1}}\otimes \cdots \otimes X^{x_{n}},\,\,\,\,\,\,\,\mathbf {Z} \left(\mathbf {z} \right)\equiv Z^{z_{1}}\otimes \cdots \otimes Z^{z_{n}},}</annotation>

</semantics>

</math><img src="./c1747a24243cd4651f0c0110cdee2ca555395d71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.383ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} \left(\mathbf {x} \right)\equiv X^{x_{1}}\otimes \cdots \otimes X^{x_{n}},\,\,\,\,\,\,\,\mathbf {Z} \left(\mathbf {z} \right)\equiv Z^{z_{1}}\otimes \cdots \otimes Z^{z_{n}},}" loading="lazy"></dd></dl>

so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} \left(\mathbf {u} \right)}">

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} \left(\mathbf {u} \right)}</annotation>

</semantics>

</math><img src="./913725313011d0f5d468306d33cfafa2f243b3c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.773ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} \left(\mathbf {u} \right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)}">

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)}</annotation>

</semantics>

</math><img src="./52d61081bc8af9e2a1a45e12167036e3a1e9eea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.032ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)}" loading="lazy"> belong to the same

<a href="Equivalence_class" title="Equivalence class">equivalence class</a>:

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\mathbf {N} \left(\mathbf {u} \right)\right]=\left[\mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)\right].}</annotation>

</semantics>

</math><img src="./ea6db5042a87808bdf9e86fb42ecdb90f8ee3ee1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.525ex; height:2.843ex;" alt="{\displaystyle \left[\mathbf {N} \left(\mathbf {u} \right)\right]=\left[\mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)\right].}" loading="lazy"></dd></dl>

The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[\mathbf {N} \right]:\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \left[\Pi ^{n}\right]}">

<mrow>

</mrow>

</mrow>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\mathbf {N} \right]:\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \left[\Pi ^{n}\right]}</annotation>

</semantics>

</math><img src="./df244752ecde2f914536c73af895795d2d50282b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.646ex; height:3.343ex;" alt="{\displaystyle \left[\mathbf {N} \right]:\left(\mathbb {Z} _{2}\right)^{2n}\rightarrow \left[\Pi ^{n}\right]}" loading="lazy"> is an <a href="Isomorphism" title="Isomorphism">isomorphism</a> for the same

reason given as in the previous case:

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[\mathbf {N} \left(\mathbf {u+v} \right)\right]=\left[\mathbf {N} \left(\mathbf {u} \right)\right]\left[\mathbf {N} \left(\mathbf {v} \right)\right],}</annotation>

</semantics>

</math><img src="./891c51f5b7994f30b80c859f641f9f4dd2d0301a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.167ex; height:2.843ex;" alt="{\displaystyle \left[\mathbf {N} \left(\mathbf {u+v} \right)\right]=\left[\mathbf {N} \left(\mathbf {u} \right)\right]\left[\mathbf {N} \left(\mathbf {v} \right)\right],}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}}">

</mrow>

<msup>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}}</annotation>

</semantics>

</math><img src="./fcd8999b3ecc18a9ad4dc3750d194b8a23544f65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.32ex; height:3.343ex;" alt="{\displaystyle \mathbf {u,v} \in \left(\mathbb {Z} _{2}\right)^{2n}}" loading="lazy">. The <a href="Symplectic_product" class="mw-redirect" title="Symplectic product">symplectic product</a>

captures the commutation relations of any operators <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} \left(\mathbf {u} \right)}">

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} \left(\mathbf {u} \right)}</annotation>

</semantics>

</math><img src="./913725313011d0f5d468306d33cfafa2f243b3c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.773ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} \left(\mathbf {u} \right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} \left(\mathbf {v} \right)}">

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} \left(\mathbf {v} \right)}</annotation>

</semantics>

</math><img src="./83482671416a2102dbb9acc131c7af3cded65791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.699ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} \left(\mathbf {v} \right)}" loading="lazy">:

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N\left(\mathbf {u} \right)N} \left(\mathbf {v} \right)=\left(-1\right)^{\left(\mathbf {u} \odot \mathbf {v} \right)}\mathbf {N} \left(\mathbf {v} \right)\mathbf {N} \left(\mathbf {u} \right).}</annotation>

</semantics>

</math><img src="./eb8419fa705adefe6eb458387ae9f7ffaf9b5129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.469ex; height:3.509ex;" alt="{\displaystyle \mathbf {N\left(\mathbf {u} \right)N} \left(\mathbf {v} \right)=\left(-1\right)^{\left(\mathbf {u} \odot \mathbf {v} \right)}\mathbf {N} \left(\mathbf {v} \right)\mathbf {N} \left(\mathbf {u} \right).}" loading="lazy"></dd></dl>

The above binary representation and <a href="Symplectic_algebra" class="mw-redirect" title="Symplectic algebra">symplectic algebra</a> are useful in making

the relation between classical linear <a href="Error_correction" class="mw-redirect" title="Error correction">error correction</a> and quantum error correction more explicit.

By comparing quantum error correcting codes in this language to <a href="Symplectic_vector_space" title="Symplectic vector space">symplectic vector spaces</a>, we can see the following. A <a href="Symplectic_vector_space#Subspaces" title="Symplectic vector space">symplectic</a> subspace corresponds to a <a href="Direct_sum" title="Direct sum">direct sum</a> of Pauli algebras (i.e., encoded qubits), while an <a href="Symplectic_vector_space#Subspaces" title="Symplectic vector space">isotropic</a> subspace corresponds to a set of stabilizers.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.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}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://quantumcomputing.stackexchange.com/questions/2106/what-is-the-surface-code-in-the-context-of-quantum-error-correction">"What is the "surface code" in the context of quantum error correction?"</a>. Quantum Computing Stack Exchange. Retrieved 2024-01-12.</cite>

</li>

</ol></div></div>

<ul><li>D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. <a rel="nofollow" class="external free" href="https://arxiv.org/abs/quant-ph/9705052">https://arxiv.org/abs/quant-ph/9705052</a></li>

<li><cite id="CITEREFShor1995" class="citation journal cs1">Shor, Peter W. (1995-10-01). "Scheme for reducing decoherence in quantum computer memory". Physical Review A. 52 (4). American Physical Society (APS): R2493 – R2496. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995PhRvA..52.2493S">1995PhRvA..52.2493S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.52.r2493">10.1103/physreva.52.r2493</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9912632">9912632</a>.</cite></li>

<li><cite id="CITEREFCalderbankShor1996" class="citation journal cs1">Calderbank, A. R.; Shor, Peter W. (1996-08-01). "Good quantum error-correcting codes exist". Physical Review A. 54 (2). American Physical Society (APS): 1098–1105. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/9512032">quant-ph/9512032</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvA..54.1098C">1996PhRvA..54.1098C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.54.1098">10.1103/physreva.54.1098</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9913578">9913578</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11524969">11524969</a>.</cite></li>

<li><cite id="CITEREFSteane1996" class="citation journal cs1">Steane, A. M. (1996-07-29). "Error Correcting Codes in Quantum Theory". Physical Review Letters. 77 (5). American Physical Society (APS): 793–797. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvL..77..793S">1996PhRvL..77..793S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevlett.77.793">10.1103/physrevlett.77.793</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10062908">10062908</a>.</cite></li>

<li>A. Calderbank, E. Rains, P. Shor, and N. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, pp. 1369–1387, 1998. Available at <a rel="nofollow" class="external free" href="https://arxiv.org/abs/quant-ph/9608006">https://arxiv.org/abs/quant-ph/9608006</a></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Quantum_information_science667" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .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}}

/* end https://en.wikipedia.org/ */

</style><div id="Quantum_information_science667" style="font-size:114%;margin:0 4em"><a href="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="DiVincenzo's_criteria" title="DiVincenzo's criteria">DiVincenzo's criteria</a></li>

<li><a href="Quantum_computing" title="Quantum computing">Quantum computing</a>

<ul><li><a href="Timeline_of_quantum_computing_and_communication" title="Timeline of quantum computing and communication">timeline</a></li></ul></li>

<li><a href="Quantum_information" title="Quantum information">Quantum information</a></li>

<li><a href="Quantum_programming" title="Quantum programming">Quantum programming</a></li>

<li><a href="Quantum_simulator" title="Quantum simulator">Quantum simulation</a></li>

<li><a href="Qubit" title="Qubit">Qubit</a>

<ul><li><a href="Physical_and_logical_qubits" title="Physical and logical qubits">physical vs. logical</a></li></ul></li>

<li><a href="List_of_quantum_processors" title="List of quantum processors">Quantum processors</a>

<ul><li><a href="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">

<li><a href="Eastin%E2%80%93Knill_theorem" title="Eastin–Knill theorem">Eastin–Knill</a></li>

<li><a href="Gleason's_theorem" title="Gleason's theorem">Gleason's</a></li>

<li><a href="Gottesman%E2%80%93Knill_theorem" title="Gottesman–Knill theorem">Gottesman–Knill</a></li>

<li><a href="Holevo's_theorem" title="Holevo's theorem">Holevo's</a></li>

<li><a href="No-broadcasting_theorem" title="No-broadcasting theorem">No-broadcasting</a></li>

<li><a href="No-cloning_theorem" title="No-cloning theorem">No-cloning</a></li>

<li><a href="No-communication_theorem" title="No-communication theorem">No-communication</a></li>

<li><a href="No-deleting_theorem" title="No-deleting theorem">No-deleting</a></li>

<li><a href="No-hiding_theorem" title="No-hiding theorem">No-hiding</a></li>

<li><a href="No-teleportation_theorem" title="No-teleportation theorem">No-teleportation</a></li>

<li><a href="Quantum_speed_limit_theorems" class="mw-redirect" title="Quantum speed limit theorems">Quantum speed limit</a></li>

<li><a href="Threshold_theorem" title="Threshold theorem">Threshold</a></li>

<li><a href="Solovay%E2%80%93Kitaev_theorem" title="Solovay–Kitaev theorem">Solovay–Kitaev</a></li>

<li><a href="Schr%C3%B6dinger%E2%80%93HJW_theorem" class="mw-redirect" title="Schrödinger–HJW theorem">Schrödinger-HJW</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum communication</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_capacity" title="Classical capacity">Classical capacity</a>

<ul><li><a href="Entanglement-assisted_classical_capacity" title="Entanglement-assisted classical capacity">entanglement-assisted</a></li>

<li><a href="Quantum_capacity" title="Quantum capacity">quantum capacity</a></li></ul></li>

<li><a href="Entanglement_distillation" title="Entanglement distillation">Entanglement distillation</a></li>

<li><a href="Entanglement_swapping" title="Entanglement swapping">Entanglement swapping</a></li>

<li><a href="Monogamy_of_entanglement" title="Monogamy of entanglement">Monogamy of entanglement</a></li>

<li><a href="Quantum_channel" title="Quantum channel">Quantum channel</a>

<ul><li><a href="Quantum_network" title="Quantum network">quantum network</a></li></ul></li>

<li><a href="Quantum_state_purification" title="Quantum state purification">State purification</a></li>

<li><a href="Quantum_teleportation" title="Quantum teleportation">Quantum teleportation</a>

<ul><li><a href="Quantum_energy_teleportation" title="Quantum energy teleportation">quantum energy teleportation</a></li>

<li><a href="Quantum_gate_teleportation" title="Quantum gate teleportation">quantum gate teleportation</a></li></ul></li>

<li><a href="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="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="Post-quantum_cryptography" title="Post-quantum cryptography">Post-quantum cryptography</a></li>

<li><a href="Quantum_coin_flipping" title="Quantum coin flipping">Quantum coin flipping</a></li>

<li><a href="Quantum_money" title="Quantum money">Quantum money</a></li>

<li><a href="Quantum_key_distribution" title="Quantum key distribution">Quantum key distribution</a>

<li><a href="List_of_quantum_key_distribution_protocols" title="List of quantum key distribution protocols">other protocols</a></li></ul></li>

<li><a href="Quantum_secret_sharing" title="Quantum secret sharing">Quantum secret sharing</a></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_algorithm" title="Quantum algorithm">Quantum algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithmic_cooling" title="Algorithmic cooling">Algorithmic cooling</a></li>

<li><a href="Amplitude_amplification" title="Amplitude amplification">Amplitude amplification</a></li>

<li><a href="Bernstein%E2%80%93Vazirani_algorithm" title="Bernstein–Vazirani algorithm">Bernstein–Vazirani</a></li>

<li><a href="Boson_sampling" title="Boson sampling">Boson sampling</a></li>

<li><a href="Deutsch%E2%80%93Jozsa_algorithm" title="Deutsch–Jozsa algorithm">Deutsch–Jozsa</a></li>

<li><a href="Grover's_algorithm" title="Grover's algorithm">Grover's</a></li>

<li><a href="Hidden_subgroup_problem" title="Hidden subgroup problem">Hidden subgroup</a></li>

<li><a href="Magic_state_distillation" title="Magic state distillation">Magic state distillation</a></li>

<li><a href="Quantum_annealing" title="Quantum annealing">Quantum annealing</a></li>

<li><a href="Quantum_counting_algorithm" title="Quantum counting algorithm">Quantum counting</a></li>

<li><a href="Quantum_Fourier_transform" title="Quantum Fourier transform">Quantum Fourier transform</a></li>

<li><a href="Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization</a></li>

<li><a href="Quantum_phase_estimation_algorithm" title="Quantum phase estimation algorithm">Quantum phase estimation</a></li>

<li><a href="Simon's_problem" title="Simon's problem">Simon's</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quantum_complexity_theory" title="Quantum complexity theory">Quantum complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="PostBQP" title="PostBQP">PostBQP</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum processor benchmarks</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Quantum_supremacy" title="Quantum supremacy">Quantum supremacy</a></li>

<li><a href="Quantum_volume" title="Quantum volume">Quantum volume</a></li>

<li><a href="Randomized_benchmarking" title="Randomized benchmarking">Randomized benchmarking</a>

<li><a href="Relaxation_(NMR)" title="Relaxation (NMR)">Relaxation times</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantum <a href="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="Adiabatic_quantum_computation" title="Adiabatic quantum computation">Adiabatic quantum computation</a></li>

<li><a href="Continuous-variable_quantum_information" title="Continuous-variable quantum information">Continuous-variable quantum information</a></li>

<li><a href="One-way_quantum_computer" title="One-way quantum computer">One-way quantum computer</a>

<ul><li><a href="Cluster_state" title="Cluster state">cluster state</a></li></ul></li>

<li><a href="Quantum_circuit" title="Quantum circuit">Quantum circuit</a>

<ul><li><a href="Quantum_logic_gate" title="Quantum logic gate">quantum logic gate</a></li></ul></li>

<li><a href="Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a>

<ul><li><a href="Quantum_neural_network" title="Quantum neural network">quantum neural network</a></li></ul></li>

<li><a href="Quantum_Turing_machine" title="Quantum Turing machine">Quantum Turing machine</a></li>

<li><a href="Topological_quantum_computer" title="Topological quantum computer">Topological quantum computer</a></li>

<li><a href="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="Quantum_error_correction" title="Quantum error correction">Quantum error correction</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>Codes

<ul><li><a href="Five-qubit_error_correcting_code" title="Five-qubit error correcting code">5 qubit</a></li>

<li><a href="Quantum_convolutional_code" title="Quantum convolutional code">quantum convolutional</a></li>

<li><a href="Bacon%E2%80%93Shor_code" title="Bacon–Shor code">Bacon–Shor</a></li>

<li><a href="Steane_code" title="Steane code">Steane</a></li>

<li><a href="Toric_code" title="Toric code">Toric</a></li>

<li><a href="Entanglement-assisted_stabilizer_formalism" title="Entanglement-assisted stabilizer formalism">Entanglement-assisted</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Physical implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Cavity_quantum_electrodynamics" title="Cavity quantum electrodynamics">Cavity QED</a></li>

<li><a href="Circuit_quantum_electrodynamics" title="Circuit quantum electrodynamics">Circuit QED</a></li>

<li><a href="Linear_optical_quantum_computing" title="Linear optical quantum computing">Linear optical QC</a></li>

<li><a href="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="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="Neutral_atom_quantum_computer" title="Neutral atom quantum computer">Neutral atom QC</a></li>

<li><a href="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="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">

<li><a href="Spin_qubit_quantum_computer" title="Spin qubit quantum computer">Spin qubit QC</a></li>

<li><a href="Nitrogen-vacancy_center" title="Nitrogen-vacancy center">NV center</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Charge_qubit" title="Charge qubit">Charge qubit</a></li>

<li><a href="Flux_qubit" title="Flux qubit">Flux qubit</a></li>

<li><a href="Phase_qubit" title="Phase qubit">Phase qubit</a></li>

<li><a href="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="Quantum_programming" title="Quantum programming">Quantum programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="OpenQASM" title="OpenQASM">OpenQASM</a>–<a href="Qiskit" title="Qiskit">Qiskit</a>–<a href="IBM_Quantum_Experience" class="mw-redirect" title="IBM Quantum Experience">IBM QX</a></li>

<li><a href="Quil_(instruction_set_architecture)" title="Quil (instruction set architecture)">Quil</a>–<a href="Rigetti_Computing" title="Rigetti Computing">Forest/Rigetti QCS</a></li>

<li><a href="Libquantum" title="Libquantum">libquantum</a></li>

<li><a href="Quantum_programming" title="Quantum programming">many others...</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Quantum information science</li>

<li> Quantum mechanics topics</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-08" href="https://en.wikipedia.org/wiki/?title=Stabilizer_code&oldid=1304823437">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>