six_semion, qcga on twisted tori, lattice entries (arXiv:2503.03827, 2312.11170, 2510.05211)

codes/quantum/groups/stabilizer/lattice/2d_stabilizer.yml

@@ -16,7 +16,12 @@ description: |
   Any 2D topological order requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on 2D lattices of qubits or qutrits \cite{arxiv:quant-ph/0308021,arxiv:1102.0770,arxiv:1803.02213}.
 
   Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial code via a local constant-depth Clifford circuit \cite{arxiv:1812.11193}.
-  There are algorithms which determine the fusion and braiding rules \cite{arxiv:2312.11170} as well as boundaries and twist defects \cite{arxiv:2410.11942} of a 2D translationally invariant modular-qudit stabilizer code for any qudit dimension.
+  An algorithm checks the topological order condition of a 2D translationally invariant modular-qudit stabilizer code of any qudit dimension and extracts its anyon string operators, fusion rules, topological spins, and braiding \cite{arxiv:2312.11170}.
+  Another algorithm determines boundaries and twist defects \cite{arxiv:2410.11942}.
+
+  On a finite torus, an anyon whose translation period does not divide the torus lengths is absent \cite{arxiv:2503.03827}.
+  The ground-state degeneracy can then be smaller than the number of anyon types of the underlying topological order \cite{arxiv:2211.00299,arxiv:2503.03827,arxiv:2503.04699}.
+  See the \hyperref[code:qcga]{BB code} entry for the logical dimension of BB codes on twisted tori.
 
 features:
   encoders:
@@ -49,6 +54,8 @@ relations:
 _meta:
   # Change log - most recent first
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+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-06-08'
     - user_id: VictorVAlbert

codes/quantum/groups/stabilizer/lattice/translationally_invariant_stabilizer.yml

@@ -37,6 +37,12 @@ description: |
   For open boundary conditions, this mapping extends the mapping used in quantum convolutional codes to multiple spatial dimensions.
   \end{defterm}
 
+  In this representation, the syndrome of a Pauli operator is the vector of its symplectic products with the stabilizer generators, which defines the \textit{excitation map} \cite{arxiv:1204.1063,arxiv:2312.11170}.
+  A code satisfies the \textit{topological order condition} if the kernel of the excitation map is the stabilizer module \cite{arxiv:1204.1063,arxiv:2312.11170}.
+  In other words, every local operator commuting with all stabilizers must be a product of stabilizers \cite{arxiv:1204.1063}.
+  This condition stems from the \hyperref[topic:tqo]{TQO-1 condition} \cite{arxiv:2312.11170}.
+  For qubit codes with as many generators as qubits per unit cell, the number of logical qubits on a torus is the dimension of the cokernel of the excitation map \cite{arxiv:1204.1063}.
+
   \subsection{Bosonic lattice stabilizer codes}
 
   Bosonic lattice stabilizer codes can contain discrete or continuous subgroups and can admit logical qudit and/or oscillator logical subspaces.
@@ -111,6 +117,8 @@ relations:
 _meta:
   # Change log - most recent first
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+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-06-08'
     - user_id: VictorVAlbert

codes/quantum/properties/block/topological/topological.yml

@@ -113,6 +113,7 @@ features:
       K=\sum_{a\in A}\left(d_{a}/D\right)^{\chi(\Sigma^{2})}~,
     \end{align}
     and a generalization of the formula to the non-orientable case can be found in Ref. \cite{arxiv:1612.07792}.
+    For a torus, \(K=|A|\) is the number of anyon types \cite{doi:10.1007/bf01217730,arxiv:2503.03827}.
   threshold:
     - 'Topological-code families can be used to obtain a fault-tolerance threshold, although the numerical threshold value and overhead depend strongly on the syndrome-extraction and decoding protocol \cite[Ch. 10]{preset:GottesmanBook}.'
   encoders:
@@ -151,6 +152,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-06-08'
     - user_id: VictorVAlbert

codes/quantum/qubits/small_distance/small/16/stab_16_4_4/stab_16_4_4.yml

@@ -95,7 +95,7 @@ relations:
       detail: 'The \([[16,4,4]]\) code is the 4.8.8 color code on the smallest torus, whose two pairs of opposite boundaries are glued straight and with a half-period shift, respectively.'
     - code_id: self_dual_css
     - code_id: qcga
-      detail: 'The \([[16,4,4]]\) code is among the weight-eight self-dual bivariate bicycle (BB) codes on twisted tori enumerated in \cite{arxiv:2510.05211}.'
+      detail: 'The \([[16,4,4]]\) code is the weight-eight self-dual bivariate bicycle (BB) code with \(A=1+x+y+y^{-1}\) and \(B=\overline{A}\) on the twisted torus with lattice vectors \((0,4)\) and \((2,2)\) \cite[Table I]{arxiv:2510.05211}.'
     - code_id: qubit_concatenated
       detail: 'Using the concatenation convention of the Zoo, the inner code is the \([[8,4,2]]\) \hyperref[topic:symplectic-doubling]{symplectic double} of the \([[4,2,2]]\) code. The \([[16,4,4]]\) twisted color code is its block concatenation with outer \([[4,2,2]]\) blocks along a \(ZX\) duality \cite{arxiv:2510.18753}.'
     - code_id: small_distance_qubit_stabilizer
@@ -111,6 +111,8 @@ relations:
 # Begin Entry Meta Information
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   changelog:
+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/small_distance/small/18/stab_18_4_4/stab_18_4_4.yml

@@ -74,13 +74,16 @@ relations:
       detail: |
         The \([[18,4,4]]\) color code is simultaneously the \hyperref[code:qcga]{bivariate bicycle code} with \(\ell=m=3\), \(A=x+1+y^2\) and \(B=y+1+x^2\) \cite{arxiv:2505.09684}, i.e., an Abelian 2BGA code over \(\mathbb{Z}_3\times\mathbb{Z}_3\).
         It is equivalently the \(\ell=m=3\) member of the bivariate bicycle family \(A=1+x+xy\), \(B=1+y+xy\), which yields 6.6.6 color codes whenever \(\ell\) and \(m\) are multiples of three \cite[Exam. 1]{arxiv:2412.04181}.
+        It is also the weight-six self-dual BB code with \(A=1+x+y\) on that torus \cite[Exam. 2]{arxiv:2510.05211}.
     - code_id: self_dual_css
     - code_id: small_distance_qubit_stabilizer
 
 
 # Begin Entry Meta Information
 _meta:
   changelog:
+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/small_distance/small/6/stab_6_4_2/stab_6_4_2.yml

@@ -61,6 +61,8 @@ relations:
       detail: 'The \([[6,4,2]]\) error-detecting code is a color code defined on a single hexagon of the 6.6.6 or 4.6.12 tilings. The \([[6,4,2]]\) code can be concatenated with the surface code to yield the 6.6.6 color code \cite[Appx. A]{arxiv:1604.04062}.'
     - code_id: 4612_color
       detail: 'The \([[6,4,2]]\) error-detecting code is a color code defined on a single hexagon of the 6.6.6 or 4.6.12 tilings.'
+    - code_id: qcga
+      detail: 'The \([[6,4,2]]\) error-detecting code is the weight-six self-dual BB code with the 6.6.6 color-code polynomial \(A=1+x+y\) on the twisted torus with lattice vectors \((3,0)\) and \((1,1)\) \cite[Exam. 2]{arxiv:2510.05211}.'
   cousins:
     - code_id: qubit_concatenated
       detail: 'Concatenations of this code with itself yield the level-\(r\) \([[6^r,4^r,2^r]]\) many-hypercube code \cite{arxiv:2403.16054}.
@@ -71,6 +73,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-09-20'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb108.yml

@@ -30,14 +30,15 @@ features:
 relations:
   parents:
     - code_id: qcga
-    - code_id: 2d_stabilizer
-      detail: 'The three-quarters-gross code belongs to the \((3,3)\)-BB family of 2D lattice stabilizer codes obtained by holding the check pattern fixed while enlarging the torus \cite{arxiv:2308.07915,arxiv:2410.11942}.'
+      detail: 'The three-quarters-gross code is the BB code with \(A=1+x+x^{-1}y^3\) and \(B=1+y+x^3y^{-1}\) on the torus with lattice vectors \((0,9)\) and \((6,0)\) \cite[Table I]{arxiv:2503.03827}.'
 
 
 # Begin Entry Meta Information
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   # Change log - most recent first
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       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb288.yml

@@ -27,14 +27,14 @@ features:
 relations:
   parents:
     - code_id: qcga
-    - code_id: 2d_stabilizer
-      detail: 'The double-gross code belongs to a family of 2D lattice stabilizer codes obtained by holding its Laurent-polynomial check pattern fixed while enlarging the torus \cite{arxiv:2308.07915,arxiv:2412.04181}.'
 
 
 # Begin Entry Meta Information
 _meta:
   # Change log - most recent first
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       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb72.yml

@@ -30,14 +30,15 @@ features:
 relations:
   parents:
     - code_id: qcga
-    - code_id: 2d_stabilizer
-      detail: 'The half-gross code belongs to the \((3,3)\)-BB family of 2D lattice stabilizer codes obtained by holding the check pattern fixed while enlarging the torus \cite{arxiv:2308.07915,arxiv:2410.11942}.'
+      detail: 'The half-gross code places the checks of the gross code on the \(6\times 6\) torus \cite[Table 3]{arxiv:2308.07915}.'
 
 
 # Begin Entry Meta Information
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   # Change log - most recent first
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     - user_id: AnthonyLeverrier
       date: '2026-07-22'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb90.yml

@@ -26,8 +26,6 @@ features:
 relations:
   parents:
     - code_id: qcga
-    - code_id: 2d_stabilizer
-      detail: 'The \([[90,8,10]]\) BB6 code belongs to a family of 2D lattice stabilizer codes obtained by holding its Laurent-polynomial check pattern fixed while enlarging the torus \cite{arxiv:2308.07915,arxiv:2412.04181}.'
   cousins:
     - code_id: balanced_product
       detail: 'The \([[90,8,10]]\) BB code can be formulated as a balanced product of two cyclic codes \cite{arxiv:2411.03302}.'
@@ -37,6 +35,8 @@ relations:
 _meta:
   # Change log - most recent first
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+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/directional.yml

@@ -54,18 +54,22 @@ features:
 relations:
   parents:
     - code_id: qcga
-      detail: 'Directional codes form a subfamily of BB codes. Writing the support of an \(X\)-type stabilizer generator on horizontal and vertical edges as bivariate polynomials \(f\) and \(g\), the parity-check matrices are \(H_X=[B(f)\mid B(g)]\) and \(H_Z=[B(\overline{g})\mid B(\overline{f})]\), where \(B\) denotes the binary representation of the quotient ring and the bar denotes the reciprocal polynomial \cite{arxiv:2606.19482}. Every stabilizer generator then has weight \(w\), and every data qubit participates in exactly \(w\) checks, so the Tanner graph is \((w,w)\)-regular \cite{arxiv:2606.19482}.'
-    - code_id: 2d_stabilizer
-      detail: 'Directional codes are 2D lattice stabilizer codes whose generators are supported on translates of a fixed bounded string of the square lattice \cite{arxiv:2507.19430}.'
+      detail: 'Directional codes form a subfamily of BB codes. Writing the support of an \(X\)-type stabilizer generator on horizontal and vertical edges as bivariate polynomials \(f\) and \(g\), the parity-check matrices are \(H_X=[B(f)\mid B(g)]\) and \(H_Z=[B(\overline{g})\mid B(\overline{f})]\), where \(B\) denotes the binary representation of the quotient ring and the bar denotes the reciprocal polynomial \cite{arxiv:2606.19482}. Every stabilizer generator then has weight \(w\), and every data qubit participates in exactly \(w\) checks, so the Tanner graph is \((w,w)\)-regular \cite{arxiv:2606.19482}. Directional codes are 2D lattice stabilizer codes whose generators are supported on translates of a fixed bounded string of the square lattice \cite{arxiv:2507.19430}.'
   cousins:
     - code_id: toric
-      detail: 'The unrotated toric code is the \(NE^2N\) directional code on a square torus, obtained with uniform scheduling and retaining full circuit-level distance \cite{arxiv:2507.19430}. The rotated toric code arises from the same word on a diamond-shaped twisted torus for even distance.'
+      detail: 'The unrotated toric code is the \(NE^2N\) directional code on a square torus, obtained with uniform scheduling and retaining full circuit-level distance \cite{arxiv:2507.19430}.'
+    - code_id: rotated_surface
+      detail: 'The rotated toric code is the \(NE^2N\) directional code on a diamond-shaped twisted torus for even distance \cite{arxiv:2507.19430}.'
 
 
 # Begin Entry Meta Information
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+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: VictorVAlbert
       date: '2026-09-26'
     - user_id: VictorVAlbert

codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/gross.yml

@@ -62,8 +62,7 @@ realizations:
 relations:
   parents:
     - code_id: qcga
-    - code_id: 2d_stabilizer
-      detail: 'The gross code belongs to the \((3,3)\)-BB family of 2D lattice stabilizer codes obtained by holding the check pattern fixed while enlarging the torus \cite{arxiv:2410.11942}.'
+      detail: 'The gross code is the BB code with \(A=1+x+x^{-1}y^3\) and \(B=1+y+x^3y^{-1}\) on the torus with lattice vectors \((0,12)\) and \((6,0)\) \cite[Exam. 3 and Table II]{arxiv:2503.03827}.'
   cousins:
     - code_id: surface
       detail: 'The gross code requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The gross code is equivalent to 8 copies of the surface code via a constant-depth Clifford circuit, and is an element of a larger family of 2D stabilizer codes \cite{arxiv:2410.11942}. An architecture combining the surface and gross codes was proposed in \cite{arxiv:2411.03202}.'
@@ -75,6 +74,8 @@ _meta:
   changelog:
     - user_id: VictorVAlbert
       date: '2026-10-12'
+    - user_id: VictorVAlbert
+      date: '2026-10-11'
     - user_id: AnthonyLeverrier
       date: '2026-07-22'
     - user_id: VictorVAlbert