diff --git a/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml b/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml index 0bd0f68b7..071c6ab6b 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml +++ b/codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml @@ -41,6 +41,8 @@ relations: - code_id: qubit_css - code_id: 2bga detail: 'Bivariate bicycle codes are Abelian 2BGA codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).' + - code_id: abelian_lifted_product + detail: 'Bivariate bicycle codes are Abelian LP codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).' - code_id: qldpc cousins: - code_id: surface diff --git a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/lifted_product.yml b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/lifted_product.yml index f0f8838d4..0cbf8346f 100644 --- a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/lifted_product.yml +++ b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/lifted_product.yml @@ -20,7 +20,9 @@ description: | A code can be defined by \(LP(A,B)\), where \(A\) and \(B\) are a pair of matrices with elements from a \hyperref[topic:group-algebra]{group algebra}. Heuristically, the code is constructed as a hypergraph product code over the \hyperref[topic:group-algebra]{group algebra}, with each entry subsequently extended into a matrix. - More technically, a \textit{lifted product over} a ring \(R\) is a product of two chain complexes whose chains are free modules over \(R\). An interesting case is when \(R=\mathbb{F}_q [G]\), the \hyperref[topic:group-algebra]{group-\(G\) algebra} over the finite field \({\mathbb{F}}_q = GF(q)\); in this case, the product can be called a \(G\)-\textit{lifted product}. Just like its further generalization the balanced product, a lifted product code generalizes a hypergraph product code in that a reduction of symmetry is exploited to decrease the number of physical qubits required. + More technically, a \textit{lifted product over} a ring \(R\) is a product of two chain complexes whose chains are free modules over \(R\). + An interesting case is when \(R=\mathbb{F}_q [G]\), the \hyperref[topic:group-algebra]{group-\(G\) algebra} over the finite field \({\mathbb{F}}_q = GF(q)\); in this case, the product can be called a \(G\)-\textit{lifted product}. + Just like its further generalization the balanced product, a lifted product code generalizes a hypergraph product code in that a reduction of symmetry is exploited to decrease the number of physical qubits required. The key operation behind the \(G\)-lifted product is the \(G\)-\textit{lift}, a \hyperref[topic:group-algebra]{group-algebraic} version of the \hyperref[topic:lifting]{lifting} procedure of protograph LDPC codes. A combination of the lift and the usual hypergraph product yields lifted-product codes. diff --git a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/matrix/abelian_lifted_product.yml b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/matrix/abelian_lifted_product.yml index dd0e4d04b..afd7d970d 100644 --- a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/matrix/abelian_lifted_product.yml +++ b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/matrix/abelian_lifted_product.yml @@ -10,13 +10,12 @@ logical: galois name: 'Abelian LP code' introduced: '\cite{arxiv:1904.02703,arxiv:2012.04068}' -alternative_names: - - 'Quasi-cyclic LP code' - description: | An LP code for Abelian group \(G\). + The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) \cite[Sec. III.E]{arxiv:2012.04068}. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with constant rate and nearly constant distance. - The Abelian LP construction has been adapted to accommodate noise bias, yielding bias-tailored LP codes \cite{arxiv:2202.01702}. + + The Abelian LP construction has been adapted to accommodate noise bias, yielding \textit{bias-tailored LP codes} \cite{arxiv:2202.01702}. See Refs. \cite{arxiv:1904.02703,arxiv:2012.04068,arxiv:2111.07029,arxiv:2308.08648} for other explicit examples. diff --git a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/2bga.yml b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/2bga.yml index 939081591..b589d9855 100644 --- a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/2bga.yml +++ b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/2bga.yml @@ -21,6 +21,7 @@ description: | For a group of order \(\ell\), we get a 2BGA code of length \(n=2\ell\). A 2BGA code for an Abelian group is called an \textit{Abelian 2BGA code}. + A construction of such codes in terms of Kronecker products of circulant matrices was introduced in \cite{arxiv:1212.6703}. An \(\mathbb{F}_q\)-linear code isomorphic to a \(Z\) logical subspace of the 2BGA code LP\((a,b)\) can be most naturally defined as a linear space of pairs @@ -121,29 +122,31 @@ relations: detail: | Given \hyperref[topic:group-algebra]{group algebra} elements \(a,b\in \mathbb{F}_q[G]\) with weights \(W_a\) and \(W_b\) (i.e., number of non-zero terms in the expansion), the 2BGA code LP\((a,b)\) has stabilizer generators of uniform weight \(W_a+W_b\). - - code_id: quantum_quasi_cyclic - detail: | - Any Abelian 2BGA code - can be thought of as a multi-dimensional index-two quasi-cyclic code. - More precisely, any finite Abelian - group can be written as a direct product of several cyclic groups, - e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product - of \(D\) cyclic groups, which is equivalent to a representation - \begin{align} - G=\langle x_1,\ldots,x_D|x_j^{m_j}=1, x_jx_ix_j^{-1}x_i^{-1}=1, \forall 1\le i,j\le D\rangle. - \end{align} - Respectively, an element of the \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is - a finite field, can be written as a \(D\)-variate polynomial in - \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of - order \(m_j\) not exceeding \(m_j-1\). - An equivalent - construction in terms of Kronecker products of circulant matrices was - introduced in \cite{arxiv:1212.6703}. - Related higher-dimensional quasi-cyclic and convolutional quantum codes have been constructed in - \cite{arxiv:2305.00137}. - code_id: group +# - code_id: quantum_quasi_cyclic +# detail: | +# Any Abelian 2BGA code +# can be thought of as a multi-dimensional index-two quasi-cyclic code. +# More precisely, any finite Abelian +# group can be written as a direct product of several cyclic groups, +# e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product +# of \(D\) cyclic groups, which is equivalent to a representation +# \begin{align} +# G=\langle x_1,\ldots,x_D|x_j^{m_j}=1, x_jx_ix_j^{-1}x_i^{-1}=1, \forall 1\le i,j\le D\rangle. +# \end{align} +# Respectively, an element of the \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is +# a finite field, can be written as a \(D\)-variate polynomial in +# \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of +# order \(m_j\) not exceeding \(m_j-1\). +# An equivalent +# construction in terms of Kronecker products of circulant matrices was +# introduced in \cite{arxiv:1212.6703}. +# Related higher-dimensional quasi-cyclic and convolutional quantum codes have been constructed in +# \cite{arxiv:2305.00137}. + + # Begin Entry Meta Information _meta: # Change log - most recent first diff --git a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/generalized_bicycle.yml b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/generalized_bicycle.yml index 7f131bcd1..7a1cef38b 100644 --- a/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/generalized_bicycle.yml +++ b/codes/quantum/qudits_galois/stabilizer/qldpc/lp/scalar/generalized_bicycle.yml @@ -12,8 +12,14 @@ short_name: 'GB' introduced: '\cite{arxiv:1212.6703,manual:{M. B. Hastings, LR codes, private communication, 2014.},arxiv:2203.17216}' # Hastings ref from 1812.02101 +alternative_names: + - 'Quasi-cyclic LP code' +# https://arxiv.org/pdf/2012.04068 + description: | A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz \cite{arxiv:quant-ph/0304161} from a pair of equivalent index-two quasi-cyclic linear codes. + Equivalently, the codes can constructed via the lifted-product construction for \(G\) being a cyclic group \cite[Sec. III.E]{arxiv:2012.04068}. + Various instances of qubit GB codes are constructed in Ref. \cite{arxiv:2203.17216} (for \(k=2\)) and in Refs. \cite{arxiv:2306.16400,arxiv:2311.16980}. The stabilizer generator matrix of a \([[ n=2\ell,k,d]]_q\) GB\((a,b)\) code, constructed from polynomials \(a(x)\) and \(b(x)\), can be refined to the form @@ -91,10 +97,11 @@ relations: The 2BGA code LP\((a,b)\) is then just a generalized bicycle code GB\([a(x),b(x)]\) constructed from the polynomials \(a(x)\) and \(b(x)\) corresponding to \(a,b\in \mathbb{F}_q[\mathbb{Z}_{\ell}]\). - code_id: abelian_lifted_product detail: | - A code GB\((a,b)\) with circulants of size \(\ell\) is a special case of a lifted-product code LP\((A,B)\) code over the Abelian \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) associated with the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle\), with \(1\times 1\) matrices \(A=a(x)\), \(B=b(x)\) given by the corresponding polynomials. + A code GB\((a,b)\) with circulants of size \(\ell\) is a special case of a lifted-product code LP\((A,B)\) code over the Abelian \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) associated with a cyclic group, with \(1\times 1\) matrices \(A=a(x)\), \(B=b(x)\) given by the corresponding polynomials. + Quasi-cyclic LP codes, i.e., LP codes constructed from cyclic groups, are equivalent to GB codes \cite[Sec. III.E]{arxiv:2012.04068}. cousins: - code_id: sc_qldpc - detail: 'Qubit GB stabilizer generator matrices is equivalent to a 1D SC-QLDPC code, see \cite[Remark 7]{arxiv:2305.00137}.' + detail: 'Qubit GB codes can be categorized as 1D SC-QLDPC codes, see \cite[Remark 7]{arxiv:2305.00137}.' - code_id: qldpc detail: | Stabilizer generators of the code GB\((a,b)\) have weights given by the sum of weights of polynomials \(a(x)\) and \(b(x)\). @@ -102,7 +109,7 @@ relations: - code_id: single_shot detail: 'A qubit GB code \([[n,k,d]]_2\) has \(k\) non-trivial relations between the syndrome bits, which is expected to help with operation in a fault-tolerant regime (in the presence of syndrome measurement errors). See Ref. \cite{arxiv:2306.16400} for many examples of such codes.' - code_id: hypergraph_product - detail: 'An arbitrary GB code of length \(2\ell\) is equivalent \cite{arxiv:2203.17216} to a rotated quantum hypergraph-product code with periodicity vectors \(\vec{L}_{1}\) and \(\vec{L}_{2}\) such that \(\lvert{\vec{L}_{1}\times\vec{L}_{2}\rvert=\ell}\).' + detail: 'An arbitrary qubit GB code of length \(2\ell\) is equivalent \cite{arxiv:2203.17216} to a rotated quantum hypergraph-product code with periodicity vectors \(\vec{L}_{1}\) and \(\vec{L}_{2}\) such that \(\lvert{\vec{L}_{1}\times\vec{L}_{2}\rvert=\ell}\).' # - code_id: quantum_quasi_cyclic # detail: |