diff --git a/codes/quantum/groups/stabilizer/lattice/2d_stabilizer.yml b/codes/quantum/groups/stabilizer/lattice/2d_stabilizer.yml index 9e483da5b..7f3930068 100644 --- a/codes/quantum/groups/stabilizer/lattice/2d_stabilizer.yml +++ b/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 changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-06-08' - user_id: VictorVAlbert diff --git a/codes/quantum/groups/stabilizer/lattice/translationally_invariant_stabilizer.yml b/codes/quantum/groups/stabilizer/lattice/translationally_invariant_stabilizer.yml index 76296917e..a5e57f549 100644 --- a/codes/quantum/groups/stabilizer/lattice/translationally_invariant_stabilizer.yml +++ b/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 changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-06-08' - user_id: VictorVAlbert diff --git a/codes/quantum/properties/block/topological/topological.yml b/codes/quantum/properties/block/topological/topological.yml index 21a174874..75b463d70 100644 --- a/codes/quantum/properties/block/topological/topological.yml +++ b/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 changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-06-08' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/small_distance/small/16/stab_16_4_4/stab_16_4_4.yml b/codes/quantum/qubits/small_distance/small/16/stab_16_4_4/stab_16_4_4.yml index a7df33832..7cc62429a 100644 --- a/codes/quantum/qubits/small_distance/small/16/stab_16_4_4/stab_16_4_4.yml +++ b/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 _meta: changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-26' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/small_distance/small/18/stab_18_4_4/stab_18_4_4.yml b/codes/quantum/qubits/small_distance/small/18/stab_18_4_4/stab_18_4_4.yml index 4e768b0df..9c6bd8c4b 100644 --- a/codes/quantum/qubits/small_distance/small/18/stab_18_4_4/stab_18_4_4.yml +++ b/codes/quantum/qubits/small_distance/small/18/stab_18_4_4/stab_18_4_4.yml @@ -74,6 +74,7 @@ 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 @@ -81,6 +82,8 @@ relations: # Begin Entry Meta Information _meta: changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-26' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/small_distance/small/6/stab_6_4_2/stab_6_4_2.yml b/codes/quantum/qubits/small_distance/small/6/stab_6_4_2/stab_6_4_2.yml index c0b38450a..720e40e5f 100644 --- a/codes/quantum/qubits/small_distance/small/6/stab_6_4_2/stab_6_4_2.yml +++ b/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 changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-20' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb108.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb108.yml index 5f2f9aed8..91a324bba 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb108.yml +++ b/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 _meta: # Change log - most recent first changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-26' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb288.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb288.yml index d53c4d86d..0dbf0b14d 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb288.yml +++ b/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 changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-26' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb72.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb72.yml index b6ed82da5..0755cb6cb 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb72.yml +++ b/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 _meta: # Change log - most recent first changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: AnthonyLeverrier date: '2026-07-22' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb90.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb90.yml index 8b10c09d3..213abcfb8 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/bb90.yml +++ b/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 changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-26' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/directional.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/directional.yml index cd17fb577..86ecfb3b7 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/directional.yml +++ b/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 _meta: # Change log - most recent first changelog: + - user_id: VictorVAlbert + date: '2026-10-13' + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-26' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/gross.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/gross.yml index 0dd6f9342..06eb4ae0d 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/gross.yml +++ b/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 diff --git a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/qcga.yml b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/qcga.yml index 880dc4c37..01e051713 100644 --- a/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/qcga.yml +++ b/codes/quantum/qubits/stabilizer/qldpc/balanced_product/lp/bb/qcga.yml @@ -18,19 +18,42 @@ description: | Codes can be classified by the weight of their checks, e.g., by BB\(w\) where \(w\) is the check weight. The qubit connectivity graph is not quite a 2D grid and is instead decomposable into two planar subgraphs of degree three; there exists an optimized layout minimizing Euclidean communication distance for check operators \cite{arxiv:2404.18809}. - There are \(n\) \(X\) and \(Z\) check operators, with each one of weight six. - Bivariate bicycle codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators. + There are \(n\) \(X\) and \(Z\) check operators, each of weight \(|A|+|B|\). + The weight-six BB codes of Ref. \cite{arxiv:2308.07915} are on par with the surface code in terms of threshold and admit a much higher ancilla-added encoding rate. + The price is check operators that are not geometrically local in a two-layer nearest-neighbor layout \cite{arxiv:2308.07915}. - See Refs. \cite{arxiv:2510.05211,arxiv:2510.06159} for examples of self-dual BB codes. Multivariate (e.g., trivariate) bicycle codes generalize BB codes to \hyperref[code:abelian_2bga]{Abelian 2BGA codes} with polynomials in more than two variables \cite{arxiv:2406.19151}; other variants and generalizations exist \cite{arxiv:2408.10001}. Versions with open boundary conditions are \hyperref[code:tile]{tile codes} \cite{arxiv:2504.09171}, obtained by pruning \cite{arxiv:2412.04181} or by condensing boundary anyons \cite{arxiv:2410.11942,arxiv:2504.08887}. There exist qudit BB codes that achieve \(kd^2 / n = 20\) \cite{arxiv:2602.20158}. + \subsection{BB codes as lattice codes on twisted tori} + + A BB code with polynomials \(A\) and \(B\) is the quotient by a torus of a 2D translation-invariant CSS code with two qubits per unit cell \cite{arxiv:2503.03827}. + In the \hyperref[topic:quantum-polynomial-mapping]{polynomial representation}, the stabilizer generators of that code are \((A,B|0,0)\) and \((0,0|\overline{B},\overline{A})\) and their translations \cite{arxiv:2503.03827}. + The bar denotes the antipode map \(x^iy^j\mapsto x^{-i}y^{-j}\). + The torus is specified by two lattice vectors \((0,\alpha)\) and \((\beta,\gamma)\), which impose \(y^{\alpha}=1\) and \(x^{\beta}y^{\gamma}=1\) on \(n=2\alpha\beta\) qubits \cite{arxiv:2503.03827}. + Every torus can be brought to this form by a unimodular change of basis, and the case \(\gamma\neq 0\) is called a \textit{twisted torus} \cite[Sec. II.B]{arxiv:2503.03827}. + The polynomials of the \hyperref[code:gross]{gross code} on the twisted torus with lattice vectors \((0,30)\) and \((6,6)\) yield a \([[360,12,d\leq 24]]\) code whose checks have range three \cite[Sec. III.B and Table IV]{arxiv:2503.03827}. + + A BB code satisfies the topological order condition iff \(A\) and \(B\) are coprime as Laurent polynomials \cite{arxiv:2503.03827,arxiv:2510.05211}. + Its \(m\)-type anyons are then the elements of the quotient ring \(\mathbb{F}_2[x^{\pm1},y^{\pm1}]/\langle A,B\rangle\) \cite{arxiv:2503.03827}. + The number of logical qubits on any torus is at most \(k_{\max}=2\dim\mathbb{F}_2[x^{\pm1},y^{\pm1}]/\langle A,B\rangle\), twice the number of independent \(m\)-type anyons \cite[Thm. 1]{arxiv:2503.03827}. + On the torus with lattice vectors \((0,\alpha)\) and \((\beta,\gamma)\), the number of logical qubits is \cite[Thm. 3]{arxiv:2503.03827} + \begin{align} + k=2\dim\frac{\mathbb{F}_2[x^{\pm1},y^{\pm1}]}{\langle A,B,y^{\alpha}-1,x^{\beta}y^{\gamma}-1\rangle}~. + \end{align} + The maximum \(k=k_{\max}\) is attained iff \(y^{\alpha}-1\) and \(x^{\beta}y^{\gamma}-1\) lie in the ideal \(\langle A,B\rangle\) \cite[Corr. 2.2]{arxiv:2503.03827}. + Otherwise, \(k0\) has distance two regardless of check weight \cite[Thm. 1]{arxiv:2606.02418}.' notes: - 'A database of bivariate bicycle codes is available in QECDB \cite{preset:QECDB}.' - 'A catalogue of BB codes found by LLM-guided search is available at \href{https://github.com/qiskit-community/qcode-discovery}{qcode-discovery} \cite{arxiv:2606.02418}.' + - 'See Ref. \cite[Tables I-IV]{arxiv:2503.03827} for weight-six BB codes of this form on twisted tori with \(n\leq 400\), and Ref. \cite[Tables I and II]{arxiv:2510.05211} for weight-eight self-dual BB codes with \(n\leq 200\).' + - 'See Ref. \cite{arxiv:2510.06159} for further self-dual BB codes.' features: transversal_gates: @@ -39,7 +62,7 @@ features: general_gates: - 'Certain bivariate bicycle codes admit a cup product structure and can thus have logical gates in the \term{Clifford hierarchy} implemented by constant-depth Clifford circuits \cite{arxiv:2410.16250}.' - rate: 'When ancilla qubit overhead is included, the encoding rate surpasses that of the surface code. + rate: 'When ancilla qubit overhead is included, the encoding rate of the weight-six codes of Ref. \cite{arxiv:2308.07915} surpasses that of the surface code. A general \([[n,k,d]]\) bivariate bicycle code requires \(n\) ancilla qubits for encoding, meaning that its \textit{ancilla-added encoding rate} is \(k/2n\). BB codes have zero design rate, so all of their logical qubits come from dependent stabilizer generators \cite{arxiv:2609.30069}.' @@ -75,19 +98,25 @@ relations: detail: 'BB codes are PBB codes with \(C=D=0\) \cite{arxiv:2606.02418}.' - code_id: mirror detail: 'BB codes are mirror codes up to qubit permutations and Hadamard gates \cite[Prop. 3.21]{arxiv:2603.05496}.' + - code_id: 2d_stabilizer + detail: 'A BB code with polynomials \(A\) and \(B\) is the quotient by a torus \(\mathbb{Z}^2/\Lambda\) of a 2D translation-invariant CSS code with two qubits per unit cell \cite{arxiv:2503.03827,arxiv:2510.05211}. The generators of that code are \((A,B|0,0)\) and \((0,0|\overline{B},\overline{A})\), with range given by the degrees of the Laurent representatives of \(A\) and \(B\) \cite{arxiv:2503.03827}. Holding the polynomials fixed while enlarging the torus, including twisted tori, yields a 2D lattice stabilizer code family with checks of bounded range \cite{arxiv:2410.11942,arxiv:2503.03827}.' cousins: - code_id: triangular_color - detail: 'Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes \cite[Exam. 1]{arxiv:2412.04181}.' + detail: 'Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes \cite[Exam. 1]{arxiv:2412.04181}. The 6.6.6 color code on a torus is the self-dual BB code with \(A=1+x+y\) \cite[Exam. 2]{arxiv:2510.05211}.' + - code_id: self_dual_css + detail: 'BB codes with \(B=\overline{A}\), where the bar denotes the antipode map \(x^iy^j\mapsto x^{-i}y^{-j}\), have \(H_X=H_Z\) and are self-dual CSS codes \cite{arxiv:2510.05211}. Weight-six ones, \(A=1+x^ay^b+x^cy^d\), have \(k_{\max}=4|ad-bc|\) and reduce to \(|ad-bc|\) decoupled copies of the 6.6.6 color code \cite[Thm. 1 and Exam. 2]{arxiv:2510.05211}. Weight-eight ones are doubly even, and \(A=1+x+y+y^{-1}\) yields the \([[16,4,4]]\) and \([[64,8,8]]\) codes \cite[Table I]{arxiv:2510.05211}.' - code_id: topological_abelian - detail: 'BB codes have been investigated in terms of their anyons and topological order \cite{arxiv:2503.04699}.' + detail: 'BB codes have been investigated in terms of their anyons and topological order \cite{arxiv:2503.04699,arxiv:2503.03827}. A BB code satisfies the topological order condition iff its polynomials are coprime \cite{arxiv:2503.03827}. Its \(m\)-type anyons are then the elements of \(\mathbb{F}_2[x^{\pm1},y^{\pm1}]/\langle A,B\rangle\), and the number of logical qubits on any torus is at most twice the dimension of this ring \cite[Thm. 1]{arxiv:2503.03827}.' - code_id: generalized_bicycle - detail: 'GB codes (BB codes) are 2BGA codes over the cyclic group \(\mathbb{Z}_{\ell}\) (Abelian group \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\)). The two codes are the same when \(r\) and \(s\) are relatively prime due to the isomorphism \(\mathbb{Z}_{r} \times \mathbb{Z}_{s} \cong \mathbb{Z}_{\ell = rs}\).' + detail: 'GB codes (BB codes) are 2BGA codes over the cyclic group \(\mathbb{Z}_{\ell}\) (Abelian group \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\)). The two codes are the same when \(r\) and \(s\) are relatively prime due to the isomorphism \(\mathbb{Z}_{r} \times \mathbb{Z}_{s} \cong \mathbb{Z}_{\ell = rs}\). BB codes on twisted tori with lattice vectors \((0,\ell)\) and \((1,\gamma)\) reduce to GB codes upon substituting \(x=y^{-\gamma}\) \cite[Sec. III.C]{arxiv:2503.03827}.' # Begin Entry Meta Information _meta: # Change log - most recent first changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-27' - user_id: VictorVAlbert diff --git a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml index c7918efca..76e188d76 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml @@ -92,6 +92,8 @@ relations: detail: 'The \(D\)-dimensional twisted toric code reduces to the toric code for \(D=2\) and a square lattice.' - code_id: cyclic_hgp detail: 'The toric code can be obtained from a hypergraph product of two repetition codes \cite[Exam. 6]{arxiv:1202.0928}. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family \cite[Exam. 5]{arxiv:1202.0928}.' + - code_id: qcga + detail: 'The toric code on any torus \(\mathbb{Z}^2/\Lambda\), including twisted tori, is the BB code with \(A=1+x\) and \(B=1+y\) \cite[Exam. 1 and Appx. B]{arxiv:2503.03827}.' cousins: - code_id: string_net detail: 'The toric code is the Turaev-Viro/Levin-Wen string-net code for the \(\mathbb{Z}_2\) input category; equivalently, the construction of Ref. \cite{arxiv:1002.2816} on a genus-one handlebody yields the toric code.' @@ -113,6 +115,8 @@ relations: # Begin Entry Meta Information _meta: changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-23' - user_id: VictorVAlbert diff --git a/codes/quantum/qudits/stabilizer/topological/double_semion.yml b/codes/quantum/qudits/stabilizer/topological/double_semion.yml index 9d0e4af10..efc9d2e04 100644 --- a/codes/quantum/qudits/stabilizer/topological/double_semion.yml +++ b/codes/quantum/qudits/stabilizer/topological/double_semion.yml @@ -21,6 +21,8 @@ description: | This stabilizer code family is inequivalent to a CSS code via a constant-depth Clifford circuit \cite[Thm. 1.1]{arxiv:1506.08883}. Similarly, the double semion model has a sign problem \cite{arxiv:1506.08883,arxiv:2005.05343} that cannot be eliminated via such a circuit. However, the sign problem can be eliminated via a non-local circuit \cite{arxiv:2509.03708}. + The \(\mathbb{Z}_4\) CSS code obtained from the double-semion code by \hyperref[topic:symplectic-doubling]{symplectic doubling} realizes two copies of the \(\mathbb{Z}_2\) toric code rather than double-semion order \cite{arxiv:2311.18003}\cite[Sec. VI.B]{arxiv:2312.11170}. + A \textit{shifted} variant condenses pairs of \(e^2 m^2\) anyons separated by \(l\) lattice sites instead of neighboring ones and realizes the same anyon theory, with \(l=0\) recovering the original code \cite[Sec. III.C]{arxiv:2312.11170}. relations: @@ -37,6 +39,8 @@ relations: _meta: # Change log - most recent first changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-06-08' - user_id: nathanan diff --git a/codes/quantum/qudits/stabilizer/topological/six_semion.yml b/codes/quantum/qudits/stabilizer/topological/six_semion.yml new file mode 100644 index 000000000..5478b832f --- /dev/null +++ b/codes/quantum/qudits/stabilizer/topological/six_semion.yml @@ -0,0 +1,40 @@ +####################################################### +## This is a code entry in the error correction zoo. ## +## https://github.com/errorcorrectionzoo ## +####################################################### + +code_id: six_semion +physical: qudits +logical: qudits + +name: 'Six-semion stabilizer code' +introduced: '\cite{arxiv:2112.11394}' + +description: | + A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) and two qudits per edge of the square lattice that realizes the six-semion topological order. + The code is obtained from two copies of the \(\mathbb{Z}_4\) toric code by \hyperref[topic:code-switching]{condensing} the bosons \(e_1^2 m_1^2\) and \(e_1^2 e_2^2 m_2^2\) \cite{arxiv:2112.11394,arxiv:2312.11170}. + Its stabilizer generators are four terms from the stabilizer group of the two toric codes together with four condensation terms \cite[Sec. VI.C]{arxiv:2312.11170}. + + The six-semion anyon theory is the Abelian TQD of \(G=\mathbb{Z}_2\times\mathbb{Z}_2\) with both Type-I cocycles and the Type-II cocycle \cite{arxiv:2112.11394}. + Its 16 anyons obey \(\mathbb{Z}_4\times\mathbb{Z}_4\) fusion rules and consist of four bosons, six semions, and six anti-semions \cite{arxiv:2112.11394}. + Two basis anyons of order four have topological spin \(-i\) and mutual braiding phase \(i\) \cite[Table V]{arxiv:2312.11170}. + +relations: + parents: + - code_id: tqd_abelian_stabilizer + detail: 'The six-semion code realizes the Abelian TQD of \(G=\mathbb{Z}_2\times\mathbb{Z}_2\) with both Type-I cocycles and the Type-II cocycle, and is obtained by condensing bosons in two \(\mathbb{Z}_4\) toric codes \cite{arxiv:2112.11394}.' + cousins: + - code_id: qudit_surface + detail: 'The six-semion code is obtained from two copies of the \(\mathbb{Z}_4\) surface code by condensing the bosons \(e_1^2 m_1^2\) and \(e_1^2 e_2^2 m_2^2\) \cite{arxiv:2112.11394,arxiv:2312.11170}.' + - code_id: double_semion + detail: 'The double-semion and six-semion codes are obtained by condensing bosons in one and two copies of the \(\mathbb{Z}_4\) toric code, respectively \cite{arxiv:2112.11394}.' + - code_id: three_fermion + detail: 'The six-semion and three-fermion anyon theories are the \(r=2\) and \(r=1\) members of the family \(F_{2^r}\) of Abelian anyon theories with fusion group \(\mathbb{Z}_{2^r}\times\mathbb{Z}_{2^r}\) \cite{arxiv:2112.11394,doi:10.1088/1751-8121/abc6c0}.' + + +# Begin Entry Meta Information +_meta: + # Change log - most recent first + changelog: + - user_id: VictorVAlbert + date: '2026-10-11' diff --git a/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/scalar/generalized_bicycle.yml b/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/scalar/generalized_bicycle.yml index 435e9e8e8..ce2ae3f6c 100644 --- a/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/scalar/generalized_bicycle.yml +++ b/codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/scalar/generalized_bicycle.yml @@ -94,6 +94,9 @@ features: - 'Depolarizing noise: \(15\%\) for a family of 6-limited \([[2^{m+1}-2,2m]]\) GB codes with BP-OSD decoder \cite[Appx. C]{arxiv:1904.02703}.' +notes: + - 'See Ref. \cite[Tables V-VIII]{arxiv:2503.03827} for weight-six qubit GB codes with \(n\leq 400\) induced from \hyperref[code:qcga]{BB codes} on twisted \(1\times n/2\) tori.' + relations: parents: - code_id: abelian_2bga @@ -125,6 +128,8 @@ relations: _meta: # Change log - most recent first changelog: + - user_id: VictorVAlbert + date: '2026-10-11' - user_id: VictorVAlbert date: '2026-09-27' - user_id: VictorVAlbert