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Quantum block codes are useful in quantum computing and in quantum communications. The encoding circuit for a large block code typically has a high complexity although those for modern codes do have lower complexity.

Quantum convolutional coding theory offers a different paradigm for coding quantum information. The convolutional structure is useful for a quantum communication scenario where a sender possesses a stream of qubits 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.

Quantum convolutional stabilizer codes borrow heavily from the structure of their classical counterparts. 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.

Definition

A quantum convolutional stabilizer code acts on a Hilbert space ${\displaystyle {\mathcal {H}},}$ which is a countably infinite tensor product of two-dimensional qubit Hilbert spaces indexed over integers ≥ 0 ${\displaystyle \left\{{\mathcal {H}}_{i}\right\}_{i\in \mathbb {Z} ^{+}}}$:

${\displaystyle {\mathcal {H}}={\displaystyle \bigotimes \limits _{i=0}^{\infty }}\ {\mathcal {H}}_{i}.}$

A sequence ${\displaystyle \mathbf {A} }$ of Pauli matrices ${\displaystyle \left\{A_{i}\right\}_{i\in \mathbb {Z} ^{+}}}$, where

${\displaystyle \mathbf {A} ={\displaystyle \bigotimes \limits _{i=0}^{\infty }}\ A_{i},}$

can act on states in ${\displaystyle {\mathcal {H}}}$. Let ${\displaystyle \Pi ^{\mathbb {Z} ^{+}}}$ denote the set of all Pauli sequences. The support supp${\displaystyle \left(\mathbf {A} \right)}$ of a Pauli sequence ${\displaystyle \mathbf {A} }$ is the set of indices of the entries in ${\displaystyle \mathbf {A} }$ that are not equal to the identity. The weight of a sequence ${\displaystyle \mathbf {A} }$ is the size ${\displaystyle \left\vert {\text{supp}}\left(\mathbf {A} \right)\right\vert }$ of its support. The delay del${\displaystyle \left(\mathbf {A} \right)}$ of a sequence ${\displaystyle \mathbf {A} }$ is the smallest index for an entry not equal to the identity. The degree deg${\displaystyle \left(\mathbf {A} \right)}$ of a sequence ${\displaystyle \mathbf {A} }$ is the largest index for an entry not equal to the identity. E.g., the following Pauli sequence

${\displaystyle {\begin{array}{cccccccc}I&X&I&Y&Z&I&I&\cdots \end{array}},}$

has support ${\displaystyle \left\{1,3,4\right\}}$, weight three, delay one, and degree four. A sequence has finite support if its weight is finite. Let ${\displaystyle F(\Pi ^{\mathbb {Z} ^{+}})}$ denote the set of Pauli sequences with finite support. The following definition for a quantum convolutional code utilizes the set ${\displaystyle F(\Pi ^{\mathbb {Z} ^{+}})}$ in its description.

A rate ${\displaystyle k/n}$-convolutional stabilizer code with ${\displaystyle 0\leq k\leq n}$ is a commuting set ${\displaystyle {\mathcal {G}}}$ of all ${\displaystyle n}$-qubit shifts of a basic generator set ${\displaystyle {\mathcal {G}}_{0}}$. The basic generator set ${\displaystyle {\mathcal {G}}_{0}}$ has ${\displaystyle n-k}$ Pauli sequences of finite support:

${\displaystyle {\mathcal {G}}_{0}=\left\{\mathbf {G} _{i}\in F(\Pi ^{\mathbb {Z} ^{+}}):1\leq i\leq n-k\right\}.}$

The constraint length ${\displaystyle \nu }$ of the code is the maximum degree of the generators in ${\displaystyle {\mathcal {G}}_{0}}$. A frame of the code consists of ${\displaystyle n}$ qubits.

A quantum convolutional code admits an equivalent definition in terms of the delay transform or ${\displaystyle D}$-transform. The ${\displaystyle D}$-transform captures shifts of the basic generator set ${\displaystyle {\mathcal {G}}_{0}}$. Let us define the ${\displaystyle n}$-qubit delay operator ${\displaystyle D}$ acting on any Pauli sequence ${\displaystyle \mathbf {A} \in \Pi ^{\mathbb {Z} ^{+}}}$ as follows:

${\displaystyle D\left(\mathbf {A} \right)=I^{\otimes n}\otimes \mathbf {A.} }$

We can write ${\displaystyle j}$ repeated applications of ${\displaystyle D}$ as a power of ${\displaystyle D}$:

${\displaystyle D^{j}\left(\mathbf {A} \right)=I^{\otimes jn}\otimes \mathbf {A.} }$

Let ${\displaystyle D^{}}$Zdroj:https://en.wikipedia.org?pojem=Quantum_convolutional_code
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