~

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

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.

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.
 
 

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

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
@@ -91,18 +97,19 @@ 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 \(b(x)\).
           The GB code ansatz is convenient for designing QLDPC codes and several extensions exist \cite{arxiv:2401.07583}.
       - 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: |