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<title>Quantum convolutional code</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Quantum convolutional code</span></span>

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<p><a href="Stabilizer_code" title="Stabilizer code">Quantum block codes</a> are useful in <a href="Quantum_computing" title="Quantum computing">quantum computing</a> and in <a href="Quantum_communication" class="mw-redirect" title="Quantum communication">quantum communications</a>. The encoding circuit for a large block code typically has a high complexity although those for modern codes do have lower complexity.

</p><p>Quantum convolutional <a href="Coding_theory" title="Coding theory">coding theory</a> offers a different paradigm for coding quantum information. The convolutional structure is useful for a <a href="Quantum_communication" class="mw-redirect" title="Quantum communication">quantum communication</a> scenario where a sender possesses a stream of <a href="Qubit" title="Qubit">qubits</a> to send to a receiver. The encoding circuit for a quantum convolutional code has a much lower complexity than an encoding circuit needed for a large block code. It also has a repetitive pattern so that the same physical devices or the same routines can manipulate the stream of quantum information.

</p><p>Quantum convolutional stabilizer codes borrow heavily from the structure of their <a href="Convolutional_code" title="Convolutional code">classical counterparts</a>. Quantum convolutional codes are similar because some of the qubits feed back into a repeated encoding unitary and give the code a memory structure like that of a classical convolutional code. The quantum codes feature online encoding and decoding of qubits. This feature gives quantum convolutional codes both their low encoding and decoding complexity and their ability to correct a larger set of errors than a block code with similar parameters.

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

<p>A quantum convolutional stabilizer code acts on a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}},}">

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

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</math></span><img src="./857d7329b4af1234b5b95c622ec77eafbd9850a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.611ex; height:2.509ex;" alt="{\displaystyle {\mathcal {H}},}" loading="lazy"></span>

which is a <a href="Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> <a href="Tensor_product" title="Tensor product">tensor product</a> of two-dimensional <a href="Qubit" title="Qubit">qubit</a> <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a> indexed over integers ≥ 0

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<annotation encoding="application/x-tex">{\displaystyle \left\{{\mathcal {H}}_{i}\right\}_{i\in \mathbb {Z} ^{+}}}</annotation>

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</math></span><img src="./8625e80ab537cd271ab0db961ee7daddd0252477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.283ex; height:3.009ex;" alt="{\displaystyle \left\{{\mathcal {H}}_{i}\right\}_{i\in \mathbb {Z} ^{+}}}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}={\displaystyle \bigotimes \limits _{i=0}^{\infty }}\ {\mathcal {H}}_{i}.}</annotation>

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</math></span><img src="./330da21e5e243b79a63d0931b743820cfce80d8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.564ex; height:6.843ex;" alt="{\displaystyle {\mathcal {H}}={\displaystyle \bigotimes \limits _{i=0}^{\infty }}\ {\mathcal {H}}_{i}.}" loading="lazy"></span></dd></dl>

<p>A sequence

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

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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> of <a href="Pauli_matrices" title="Pauli matrices">Pauli matrices</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{A_{i}\right\}_{i\in \mathbb {Z} ^{+}}}">

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<annotation encoding="application/x-tex">{\displaystyle \left\{A_{i}\right\}_{i\in \mathbb {Z} ^{+}}}</annotation>

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</math></span><img src="./c1676ee6811130b1a2f30a7e0e95418181f2afe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.062ex; height:3.009ex;" alt="{\displaystyle \left\{A_{i}\right\}_{i\in \mathbb {Z} ^{+}}}" loading="lazy"></span>, where

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\displaystyle \bigotimes \limits _{i=0}^{\infty }}\ A_{i},}</annotation>

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</math></span><img src="./5fc1182642b5212fc7d6b2c9634a6c382753bb4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.399ex; height:6.843ex;" alt="{\displaystyle \mathbf {A} ={\displaystyle \bigotimes \limits _{i=0}^{\infty }}\ A_{i},}" loading="lazy"></span></dd></dl>

<p>can act on states in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

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

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</math></span><img src="./19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span>. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi ^{\mathbb {Z} ^{+}}}">

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

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</math></span><img src="./727f5826f217349918e4c4aed66012eb6d8a61b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.274ex; height:3.009ex;" alt="{\displaystyle \Pi ^{\mathbb {Z} ^{+}}}" loading="lazy"></span> denote the set of all Pauli sequences. The support supp<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {A} \right)}">

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</math></span><img src="./71040b8977d8f4ce6fccefd0081e8e714ccd30fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.829ex; height:2.843ex;" alt="{\displaystyle \left(\mathbf {A} \right)}" loading="lazy"></span> of a Pauli sequence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

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

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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> is the set of indices of the entries in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

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

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</math></span><img src="./0795cc96c75d81520a120482662b90f024c9a1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }" loading="lazy"></span> that are not equal to the identity. The weight of a sequence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }">

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

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<annotation encoding="application/x-tex">{\displaystyle \left\vert {\text{supp}}\left(\mathbf {A} \right)\right\vert }</annotation>

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

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identity. The degree deg<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {A} \right)}">

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

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{cccccccc}I&X&I&Y&Z&I&I&\cdots \end{array}},}</annotation>

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<p>has support <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{1,3,4\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle \left\{1,3,4\right\}}</annotation>

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</math></span><img src="./ae54db4cafe4327844dadd0d3cce2cae6b3291ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.88ex; height:2.843ex;" alt="{\displaystyle \left\{1,3,4\right\}}" loading="lazy"></span>, weight three, delay one, and degree four. A sequence has finite support if its weight is finite. Let

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<annotation encoding="application/x-tex">{\displaystyle F(\Pi ^{\mathbb {Z} ^{+}})}</annotation>

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</math></span><img src="./b3a595f2ea53a95376bd97da360aae3742c61340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.824ex; height:3.509ex;" alt="{\displaystyle F(\Pi ^{\mathbb {Z} ^{+}})}" loading="lazy"></span> denote the set of Pauli sequences with finite support. The following definition for a quantum convolutional code utilizes the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\Pi ^{\mathbb {Z} ^{+}})}">

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</p><p>A rate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k/n}">

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

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</math></span><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"></span>-convolutional stabilizer code with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq k\leq n}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq k\leq n}</annotation>

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</math></span><img src="./e7b429c3c44b3dc10332285272eee6f754dbf985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0\leq k\leq n}" loading="lazy"></span> is a commuting set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

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

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

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</math></span><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"></span>-qubit shifts of a basic generator set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{0}}</annotation>

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</math></span><img src="./55ddfab42df2cd382a424550abbee1692b38d5fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.437ex; height:2.509ex;" alt="{\displaystyle {\mathcal {G}}_{0}}" loading="lazy"></span>. The basic generator set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{0}}">

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</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{0}}</annotation>

</semantics>

</math></span><img src="./55ddfab42df2cd382a424550abbee1692b38d5fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.437ex; height:2.509ex;" alt="{\displaystyle {\mathcal {G}}_{0}}" loading="lazy"></span> has

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> Pauli sequences of finite support:

</p>

<msub>

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</msub>

<mrow>

<mrow>

<msub>

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</msub>

<msup>

<msup>

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</msup>

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</msup>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{0}=\left\{\mathbf {G} _{i}\in F(\Pi ^{\mathbb {Z} ^{+}}):1\leq i\leq n-k\right\}.}</annotation>

</semantics>

</math></span><img src="./7b7473893460e3d0abcd1b6f62c8073f5f97e2a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.781ex; height:4.843ex;" alt="{\displaystyle {\mathcal {G}}_{0}=\left\{\mathbf {G} _{i}\in F(\Pi ^{\mathbb {Z} ^{+}}):1\leq i\leq n-k\right\}.}" loading="lazy"></span></dd></dl>

<p>The constraint length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span> of the code is the maximum degree of the

generators in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{0}}</annotation>

</semantics>

</math></span><img src="./55ddfab42df2cd382a424550abbee1692b38d5fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.437ex; height:2.509ex;" alt="{\displaystyle {\mathcal {G}}_{0}}" loading="lazy"></span>. A frame of the code consists of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> qubits.

</p><p>A quantum convolutional code admits an equivalent definition in terms of the delay transform or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>-transform. The <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>-transform captures shifts of the basic generator set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{0}}</annotation>

</semantics>

</math></span><img src="./55ddfab42df2cd382a424550abbee1692b38d5fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.437ex; height:2.509ex;" alt="{\displaystyle {\mathcal {G}}_{0}}" loading="lazy"></span>. Let us define the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>-qubit delay operator <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> acting on any Pauli sequence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} \in \Pi ^{\mathbb {Z} ^{+}}}">

</mrow>

<msup>

<msup>

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</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \in \Pi ^{\mathbb {Z} ^{+}}}</annotation>

</semantics>

</math></span><img src="./d1cfc59c68916d46ab317d76b4d0f4247d4caf20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.134ex; height:3.009ex;" alt="{\displaystyle \mathbf {A} \in \Pi ^{\mathbb {Z} ^{+}}}" loading="lazy"></span> as follows:

</p>

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<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\left(\mathbf {A} \right)=I^{\otimes n}\otimes \mathbf {A.} }</annotation>

</semantics>

</math></span><img src="./193aef901462eddda9317e3b6ae3015037ad35b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.554ex; height:3.009ex;" alt="{\displaystyle D\left(\mathbf {A} \right)=I^{\otimes n}\otimes \mathbf {A.} }" loading="lazy"></span></dd></dl>

<p>We can write <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> repeated applications of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span> as a power of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></span>:

</p>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle D^{j}\left(\mathbf {A} \right)=I^{\otimes jn}\otimes \mathbf {A.} }</annotation>

</semantics>

</math></span><img src="./c990f0533b0ba3844b84ecf3b4f417af79ba56e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.141ex; height:3.176ex;" alt="{\displaystyle D^{j}\left(\mathbf {A} \right)=I^{\otimes jn}\otimes \mathbf {A.} }" loading="lazy"></span></dd></dl>

<msup>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle D^{j}\left({\mathcal {G}}_{0}\right)}</annotation>

</semantics>

</math></span><img src="./67905f342c75e1c70a48c20359818bf58bd5e03e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.468ex; height:3.176ex;" alt="{\displaystyle D^{j}\left({\mathcal {G}}_{0}\right)}" loading="lazy"></span> be the set of shifts of elements

of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}_{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}_{0}}</annotation>

</semantics>

</math></span><img src="./55ddfab42df2cd382a424550abbee1692b38d5fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.437ex; height:2.509ex;" alt="{\displaystyle {\mathcal {G}}_{0}}" loading="lazy"></span> by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>. Then the full stabilizer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"></span> for the

convolutional stabilizer code is

</p>

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<msup>

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</msup>

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</munder>

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<msup>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {G}}={\textstyle \bigcup \limits _{j\in \mathbb {Z} ^{+}}}D^{j}\left({\mathcal {G}}_{0}\right).}</annotation>

</semantics>

</math></span><img src="./043f9f608494eda1fab4e1252407d0891077db2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:17.065ex; height:5.509ex;" alt="{\displaystyle {\mathcal {G}}={\textstyle \bigcup \limits _{j\in \mathbb {Z} ^{+}}}D^{j}\left({\mathcal {G}}_{0}\right).}" loading="lazy"></span></dd></dl>

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

<p>The operation of a convolutional stabilizer code is as follows. The protocol begins with the sender encoding a stream of qubits with an online encoding circuit such as that given in (Grassl and Roetteler 2006). The encoding circuit is <i>online</i> if it acts on a few blocks of qubits at a time. The sender transmits a set of qubits as soon as the first unitary finishes processing them. The receiver measures all the generators in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"></span> and corrects for errors as he receives the online encoded qubits. He finally decodes the encoded qubits with a decoding circuit. The qubits decoded from this convolutional procedure should be error free and ready for quantum computation at the receiving end.

</p><p>A <i>finite-depth</i> circuit maps a Pauli sequence with finite weight to one with finite weight (Ollivier and Tillich 2004). It does not map a Pauli sequence with finite weight to one with infinite weight. This property is important because we do not want the decoding circuit to propagate uncorrected errors into the information qubit stream (Johannesson and Zigangirov 1999). A finite-depth decoding circuit corresponding to the <a href="Stabilizer_code" title="Stabilizer code">stabilizer</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {G}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b8a980c59d42c003fd07fdf3646e1fb95ff82f99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.392ex; height:2.343ex;" alt="{\displaystyle {\mathcal {G}}}" loading="lazy"></span> exists by the algorithm given in (Grassl and Roetteler 2006).

</p>

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

<p>Forney et al. provided an example of a rate-1/3 quantum convolutional code by importing a particular classical quaternary convolutional code (Forney and Guha 2005). Grassl and Roetteler determined a noncatastrophic encoding circuit for Forney et al.'s rate-1/3 quantum convolutional code (Grassl and Roetteler 2006). The basic stabilizer and its first shift are as follows:

</p>

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</menclose>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdots {\begin{array}{|ccc|ccc|ccc|ccc|ccc|}I&I&I&X&X&X&X&Z&Y&I&I&I&I&I&I\\I&I&I&Z&Z&Z&Z&Y&X&I&I&I&I&I&I\\I&I&I&I&I&I&X&X&X&X&Z&Y&I&I&I\\I&I&I&I&I&I&Z&Z&Z&Z&Y&X&I&I&I\\\end{array}}\cdots }</annotation>

</semantics>

</math></span><img src="./9f06cf123f09b9f341afa23880275500d358cd7c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:65.563ex; height:13.843ex;" alt="{\displaystyle \cdots {\begin{array}{|ccc|ccc|ccc|ccc|ccc|}I&I&I&X&X&X&X&Z&Y&I&I&I&I&I&I\\I&I&I&Z&Z&Z&Z&Y&X&I&I&I&I&I&I\\I&I&I&I&I&I&X&X&X&X&Z&Y&I&I&I\\I&I&I&I&I&I&Z&Z&Z&Z&Y&X&I&I&I\\\end{array}}\cdots }" loading="lazy"></span></dd></dl>

<p>The code consists of all three-qubit shifts of the above generators. The vertical bars are a visual aid to illustrate the three-qubit shifts of the basic generators. The code can correct for an arbitrary single-qubit error in every other frame.

</p>

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

<p>Wilde and Brun have integrated the theory of <a href="Entanglement-assisted_stabilizer_code" class="mw-redirect" title="Entanglement-assisted stabilizer code">entanglement-assisted stabilizer codes</a> and quantum convolutional codes in a series of articles (Wilde and Brun 2007a, 2007b, 2008, 2009) to form a theory of entanglement-assisted quantum convolutional coding. This theory supposes that a sender and receiver share noiseless bipartite <a href="Quantum_entanglement" title="Quantum entanglement">entanglement</a> that they can exploit for protecting a stream of quantum information.

</p><p>(Wilde 2009), building on work of (Ollivier and Tillich 2004) and (Grassl and Roetteler 2006), also showed how to encode these codes with quantum shift register circuits, a natural extension of the theory of

classical <a href="Shift_register" title="Shift register">shift register</a> circuits.

</p>

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

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</style><cite id="CITEREFOllivierTillich2003" class="citation journal cs1">Ollivier, Harold; Tillich, Jean-Pierre (2003). "Description of a Quantum Convolutional Code". <i>Physical Review Letters</i>. <b>91</b> (17): 177902. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0304189">quant-ph/0304189</a></span>. <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/2003PhRvL..91q7902O">2003PhRvL..91q7902O</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.91.177902">10.1103/PhysRevLett.91.177902</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/14611378">14611378</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:17261900">17261900</a>.</cite></li>

<li><cite id="CITEREFOllivierTillich2004" class="citation journal cs1">Ollivier, H.; Tillich, J. -P. (2004). "Quantum convolutional codes: Fundamentals". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0401134">quant-ph/0401134</a></span>. <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/2004quant.ph..1134O">2004quant.ph..1134O</a>.</cite> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{cite journal}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (help)</span></li>

<li><cite id="CITEREFForney2005" class="citation book cs1"><a href="Dave_Forney" title="Dave Forney">Forney, G. David</a> (2005). "Simple rate-1/3 convolutional and tail-biting quantum error-correcting codes". <i>Proceedings. International Symposium on Information Theory, 2005. ISIT 2005</i>. pp. <span class="nowrap">1028–</span>1032. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0501099">quant-ph/0501099</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FISIT.2005.1523495">10.1109/ISIT.2005.1523495</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7803-9151-9</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14484674">14484674</a>.</cite></li>

<li><cite id="CITEREFDavid_ForneyGrasslGuha2007" class="citation journal cs1">David Forney, G. David; Grassl, Markus; Guha, Saikat (2007). "Convolutional and Tail-Biting Quantum Error-Correcting Codes". <i>IEEE Transactions on Information Theory</i>. <b>53</b> (3): <span class="nowrap">865–</span>880. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0511016">quant-ph/0511016</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2006.890698">10.1109/TIT.2006.890698</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:546490">546490</a>.</cite></li>

<li>M. Grassl and M. Roetteler, “Quantum convolutional codes: Encoders and structural properties,” in Forty-Fourth Annual Allerton Conference, 2006. Available at <a rel="nofollow" class="external free" href="http://www.csl.illinois.edu/allerton/archives/allerton06/PDFs/papers/0285.pdf">http://www.csl.illinois.edu/allerton/archives/allerton06/PDFs/papers/0285.pdf</a></li>

<li><cite id="CITEREFGrasslRotteler2006" class="citation book cs1">Grassl, Markus; Rotteler, Martin (2006). "Non-catastrophic Encoders and Encoder Inverses for Quantum Convolutional Codes". <i>2006 IEEE International Symposium on Information Theory</i>. pp. <span class="nowrap">1109–</span>1113. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0602129">quant-ph/0602129</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FISIT.2006.261956">10.1109/ISIT.2006.261956</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4244-0505-X</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1442">1442</a>.</cite></li>

<li>R. Johannesson and K. S. Zigangirov, <i>Fundamentals of Convolutional Coding</i>. Wiley-IEEE Press, 1999.</li>

<li><cite id="CITEREFWildeKroviBrun2010" class="citation book cs1">Wilde, Mark M.; Krovi, Hari; Brun, Todd A. (2010). "Convolutional entanglement distillation". <i>2010 IEEE International Symposium on Information Theory</i>. pp. <span class="nowrap">2657–</span>2661. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0708.3699">0708.3699</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FISIT.2010.5513666">10.1109/ISIT.2010.5513666</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4244-7892-7</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2409176">2409176</a>.</cite></li>

<li><cite id="CITEREFWildeBrun2010" class="citation journal cs1">Wilde, Mark M.; Brun, Todd A. (2010). "Entanglement-assisted quantum convolutional coding". <i>Physical Review A</i>. <b>81</b> (4): 042333. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0712.2223">0712.2223</a></span>. <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/2010PhRvA..81d2333W">2010PhRvA..81d2333W</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.81.042333">10.1103/PhysRevA.81.042333</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:8410654">8410654</a>.</cite></li>

<li><cite id="CITEREFWildeBrun2010" class="citation journal cs1">Wilde, Mark M.; Brun, Todd A. (2010). "Quantum convolutional coding with shared entanglement: General structure". <i>Quantum Information Processing</i>. <b>9</b> (5): <span class="nowrap">509–</span>540. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0807.3803">0807.3803</a></span>. <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/2010QuIP....9..509W">2010QuIP....9..509W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11128-010-0179-9">10.1007/s11128-010-0179-9</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:18185704">18185704</a>.</cite></li>

<li><cite id="CITEREFWilde2008" class="citation journal cs1">Wilde, Mark M. (2008). "Quantum Coding with Entanglement". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0806.4214">0806.4214</a></span>.</cite> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{cite journal}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (help)</span></li>

<li><cite id="CITEREFWildeBrun2009" class="citation journal cs1">Wilde, Mark M.; Brun, Todd A. (2009). "Extra shared entanglement reduces memory demand in quantum convolutional coding". <i>Physical Review A</i>. <b>79</b> (3): 032313. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0812.4449">0812.4449</a></span>. <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/2009PhRvA..79c2313W">2009PhRvA..79c2313W</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.79.032313">10.1103/PhysRevA.79.032313</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:67826844">67826844</a>.</cite></li>

<li><cite id="CITEREFWilde2009" class="citation journal cs1">Wilde, Mark M. (2009). "Quantum-shift-register circuits". <i>Physical Review A</i>. <b>79</b> (6): 062325. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0903.3894">0903.3894</a></span>. <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/2009PhRvA..79f2325W">2009PhRvA..79f2325W</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.79.062325">10.1103/PhysRevA.79.062325</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:56351003">56351003</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<div class="mw-heading mw-heading3"><h3 id="Publications">Publications</h3></div>

<ul><li><cite id="CITEREFHoushmandWilde2013" class="citation journal cs1">Houshmand, Monireh; Wilde, Mark M. (2013). "Recursive Quantum Convolutional Encoders are Catastrophic: A Simple Proof". <i>IEEE Transactions on Information Theory</i>. <b>59</b> (10): <span class="nowrap">6724–</span>6731. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1209.0082">1209.0082</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2013.2272932">10.1109/TIT.2013.2272932</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:15309497">15309497</a>.</cite></li>

<li><cite id="CITEREFLaiHsiehLu2016" class="citation journal cs1">Lai, Ching-Yi; Hsieh, Min-Hsiu; Lu, Hsiao-Feng (2016). "On the Mac <i>Williams</i> Identity for Classical and Quantum Convolutional Codes". <i>IEEE Transactions on Communications</i>. <b>64</b> (8): <span class="nowrap">3148–</span>3159. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1404.5012">1404.5012</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOMM.2016.2585641">10.1109/TCOMM.2016.2585641</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:7123143">7123143</a>.</cite></li>

<li><cite id="CITEREFPoulinTillichOllivier2007" class="citation journal cs1">Poulin, David; Tillich, Jean-Pierre; Ollivier, Harold (2007). "Quantum serial turbo-codes". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0712.2888">0712.2888</a></span>.</cite> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{cite journal}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (help)</span></li>

<li><cite id="CITEREFDjordjevic2012" class="citation book cs1">Djordjevic, Ivan (2012). <i>Quantum information processing and quantum error correction: an engineering approach</i>. Academic press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780123854919</bdi>.</cite></li>

<li><cite id="CITEREFBrun2013" class="citation book cs1">Brun, Todd A. (2013). Lidar, Daniel A.; Brun, Todd A. (eds.). <i>Quantum error correction</i>. Cambridge University Press. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1910.03672">1910.03672</a></span>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521897877</bdi>.</cite></li></ul>

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

<ul><li><a href="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<br>complexity theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

<ul><li>Codes

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

<li><a href="Stabilizer_code" title="Stabilizer code">stabilizer</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<br>implementations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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<br>programming</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="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><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Quantum information science</li>

<li><span class="noviewer" typeof="mw:File"><span title="Template"></span></span> Quantum mechanics topics</li></ul>

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