Quantum convolutional code

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 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 theoryoffers a different paradigm for coding quantum information. The convolutionalstructure is useful for a quantum communication scenario where a senderpossesses a stream of qubits to send to a receiver. The encoding circuit for aquantum convolutional code has a much lower complexity than an encodingcircuit needed for a large block code. It also has a repetitive pattern sothat the same physical devices or the same routines can manipulate the streamof quantum information. Quantum convolutional stabilizer codes borrowheavily from the structure of their classical counterparts.Quantum convolutional codes are similar because some of the qubits feed backinto a repeated encoding unitary and give the code a memory structure likethat of a classical convolutional code. The quantum codes feature onlineencoding and decoding of qubits. This feature gives quantum convolutionalcodes both their low encoding and decoding complexity and their ability tocorrect a larger set of errors than a block code with similar parameters. A quantum convolutional stabilizer code acts on a Hilbert space H , {displaystyle {mathcal {H}},} which is a countably infinite tensor product of two-dimensional qubit Hilbert spaces indexed over integers ≥ 0 { H i } i ∈ Z + {displaystyle left{{mathcal {H}}_{i} ight}_{iin mathbb {Z} ^{+}}} : A sequence A {displaystyle mathbf {A} } of Pauli matrices { A i } i ∈ Z + {displaystyle left{A_{i} ight}_{iin mathbb {Z} ^{+}}} , where can act on states in H {displaystyle {mathcal {H}}} . Let Π Z + {displaystyle Pi ^{mathbb {Z} ^{+}}} denote the setof all Pauli sequences. The support supp ( A ) {displaystyle left(mathbf {A} ight)} of aPauli sequence A {displaystyle mathbf {A} } is the set of indices of the entries in A {displaystyle mathbf {A} } that are not equal to the identity. The weight of a sequence A {displaystyle mathbf {A} } is the size | supp ( A ) | {displaystyle leftvert { ext{supp}}left(mathbf {A} ight) ightvert } of its support. The delay del ( A ) {displaystyle left(mathbf {A} ight)} of asequence A {displaystyle mathbf {A} } is the smallest index for an entry not equal to theidentity. The degree deg ( A ) {displaystyle left(mathbf {A} ight)} of a sequence A {displaystyle mathbf {A} } is the largest index for an entry not equal to the identity.E.g., the following Pauli sequence has support { 1 , 3 , 4 } {displaystyle left{1,3,4 ight}} , weight three, delay one, and degreefour. A sequence has finite support if its weight is finite. Let F ( Π Z + ) {displaystyle F(Pi ^{mathbb {Z} ^{+}})} denote the set of Pauli sequences with finitesupport. The following definition for a quantum convolutional code utilizesthe set F ( Π Z + ) {displaystyle F(Pi ^{mathbb {Z} ^{+}})} in its description. A rate k / n {displaystyle k/n} -convolutional stabilizer code with 0 ≤ k ≤ n {displaystyle 0leq kleq n} is a commuting set G {displaystyle {mathcal {G}}} of all n {displaystyle n} -qubit shifts of a basicgenerator set G 0 {displaystyle {mathcal {G}}_{0}} . The basic generator set G 0 {displaystyle {mathcal {G}}_{0}} has n − k {displaystyle n-k} Pauli sequences of finite support: The constraint length ν {displaystyle u } of the code is the maximum degree of thegenerators in G 0 {displaystyle {mathcal {G}}_{0}} . A frame of the code consists of n {displaystyle n} qubits.