Deriving CSS codes from a single zero-sum axiom

guobin ouyang

Published Oct 26, 2026• 0 citations

Abstract

We show that the algebraic skeleton of **CSS (Calderbank–Shor–Steane) quantum error-correcting codes** follows from a single axiom rather than being postulated. The axiom — *zero-sum*: the weights of a closed sequence of moves sum to zero — is the content of Z0 in the Zero programme, a discrete-physics derivation of spacetime and matter from one primitive. From it (together with local compensating moves) we obtain, in order: 1. a **cycle space** over $\mathbb F_2$, i.e. a classical linear code with parameters $[E,\ \beta_1=E-V+c,\ \mathrm{girth}]$; 2. the **commutation criterion** for vertex/face stabilisers, proved unconditionally when faces are closed words (Theorem A, §4); 3. the **closed form** for the number of logical qubits, $k=E-r_X-r_Z$ (Theorem B, §5); 4. the **distance** as an *essential girth*, and the observation that the naive $d=\mathrm{girth}$ is false (Theorem D, §6); 5. the **binary operator algebra on each edge**, including a *real* anticommutation $F_eZ_e=-Z_eF_e$ with no factor of $i$, and the tensor-product structure $(\mathbb C^2)^{\otimes E}$ as a consequence rather than an assumption (Theorems E and F, §7); 6. the **face-selection rule**: not determined by the four candidate primitives we tested (Theorem G, §8.1), but a necessary relation $\sum_f\partial f=0$ always holds, forcing $k=\beta_1-|F|+\mathrm{rel}$; when $K$ is a closed connected surface this collapses to $k=2g$, depending only on the genus (§8.2). The net result is a **reduction of borrowings**. Standard CSS constructions import three ingredients — a qubit register, Pauli anticommutation, and the tensor product. Here the first is *replaced* by a derived per-edge binary operator space (Theorem F), the second is *derived* (F2, or E1 in matrix form), and the third is *derived* (F4). That leaves exactly **one** genuine input — the choice of faces. We also report a structural negative result: the derived constraint is *weaker* than standard homology, so this route cannot **improve** on CSS codes (§9). ---

DOI: 10.5281/zenodo.23662387

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Calderbank-Shor-Steane codes on group-valued qudits

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quant-pharXiv:2602.19558Feb 24, 20260 citations

Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)

Aleks Kissinger

arXiv:2204.14038Apr 29, 202210 citations

A 67%-Rate CSS Code on the FCC Lattice: [[192,130,3]] from Weight-12 Stabilizers

Raghu Kulkarni

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Local Clifford equivalence to CSS codes: Polynomial-time classification and conversion

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Homological Quantum Error Correction with Torsion

S. Novák

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Toric codes all the way down: decoupling translation-invariant topological CSS codes in higher dimensions

Mingxin He, Yifei Wang, Hao Song +1 more

quant-pharXiv:2610.40592Oct 30, 20260 citations

Deriving CSS codes from a single zero-sum axiom

guobin ouyang

Published Oct 26, 2026• 0 citations

Publisher

Abstract

We show that the algebraic skeleton of CSS (Calderbank–Shor–Steane) quantum error-correcting codes follows from a single axiom rather than being postulated. The axiom — zero-sum: the weights of a closed sequence of moves sum to zero — is the content of Z0 in the Zero programme, a discrete-physics derivation of spacetime and matter from one primitive. From it (together with local compensating moves) we obtain, in order: 1. a cycle space over

\mathbb F_2

, i.e. a classical linear code with parameters

[E,\ \beta_1=E-V+c,\ \mathrm{girth}]

; 2. the commutation criterion for vertex/face stabilisers, proved unconditionally when faces are closed words (Theorem A, §4); 3. the closed form for the number of logical qubits,

k=E-r_X-r_Z

(Theorem B, §5); 4. the distance as an essential girth, and the observation that the naive

d=\mathrm{girth}

is false (Theorem D, §6); 5. the binary operator algebra on each edge, including a real anticommutation

F_eZ_e=-Z_eF_e

with no factor of

i

, and the tensor-product structure

(\mathbb C^2)^{\otimes E}

as a consequence rather than an assumption (Theorems E and F, §7); 6. the face-selection rule: not determined by the four candidate primitives we tested (Theorem G, §8.1), but a necessary relation

\sum_f\partial f=0

always holds, forcing

k=\beta_1-|F|+\mathrm{rel}

; when

K

is a closed connected surface this collapses to

k=2g

, depending only on the genus (§8.2). The net result is a reduction of borrowings. Standard CSS constructions import three ingredients — a qubit register, Pauli anticommutation, and the tensor product. Here the first is replaced by a derived per-edge binary operator space (Theorem F), the second is derived (F2, or E1 in matrix form), and the third is derived (F4). That leaves exactly one genuine input — the choice of faces. We also report a structural negative result: the derived constraint is weaker than standard homology, so this route cannot improve on CSS codes (§9). ---

DOI: 10.5281/zenodo.23662387

Related papers

Calderbank-Shor-Steane codes on group-valued qudits

Ben T. McDonough, Jian-Hao Zhang, Victor V. Albert +1 more

Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting $X$ - and $Z$ -type checks. We introduce CSS-like codes on $G$ -valued qudits for any finite group $G$ that reduce to qubit CSS codes for $G = \mathbb{Z}_2$ yet generalize the Kitaev quantum double model for general groups. The $X$ -checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while $Z$ -checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when $G$ is non-Abelian and simple, every $G$ -covariant group-CSS code with suitably upper-bounded $Z$ -check weight and lower-bounded $Z$ -distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting $X$- and $Z$-type checks. We introduce CSS-like codes on $G$-valued qudits for any finite group $G$ that reduce to qubit CSS codes for $G = \mathbb{Z}_2$ yet generalize the Kitaev quantum double model for general groups. The $X$-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while $Z$-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when $G$ is non-Abelian and simple, every $G$-covariant group-CSS code with suitably upper-bounded $Z$-check weight and lower-bounded $Z$-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.

quant-pharXiv:2602.19558Feb 24, 20260 citations

Phase-free ZX diagrams are CSS codes (...or how to graphically grok the surface code)

Aleks Kissinger

In this paper, we demonstrate a direct correspondence between phase-free ZX diagrams, a graphical notation for representing and manipulating a certain class of linear maps on qubits, and Calderbank-Shor-Steane (CSS) codes, a large family of quantum error correcting codes constructed from classical codes, including for example the Steane code, surface codes, and colour codes. The stabilisers of a CSS code have an especially nice structure arising from a pair of orthogonal $\mathbb F_2$ -linear subspaces, or in the case of maximal CSS codes, a single subspace and its orthocomplement. On the other hand, phase-free ZX diagrams can always be efficiently reduced to a normal form given by the basis elements of an $\mathbb F_2$ -linear subspace. Here, we will show that these two ways of describing a quantum state by an $\mathbb F_2$ -linear subspace $S$ are in fact the same. Namely, the maximal CSS code generated by $S$ fixes the quantum state whose ZX normal form is also given by $S$ . This insight gives us an immediate translation from stabilisers of a maximal CSS code into a ZX diagram describing its associated state. We show that we can extend this translation to stabilisers and logical operators of any (possibly non-maximal) CSS code by "bending wires". To demonstrate the utility of this translation, we give a simple picture of the surface code and a fully graphical derivation of the action of physical lattice surgery operations on the space of logical qubits, completing the ZX presentation of lattice surgery initiated by de Beudrap and Horsman.In this paper, we demonstrate a direct correspondence between phase-free ZX diagrams, a graphical notation for representing and manipulating a certain class of linear maps on qubits, and Calderbank-Shor-Steane (CSS) codes, a large family of quantum error correcting codes constructed from classical codes, including for example the Steane code, surface codes, and colour codes. The stabilisers of a CSS code have an especially nice structure arising from a pair of orthogonal $\mathbb F_2$-linear subspaces, or in the case of maximal CSS codes, a single subspace and its orthocomplement. On the other hand, phase-free ZX diagrams can always be efficiently reduced to a normal form given by the basis elements of an $\mathbb F_2$-linear subspace. Here, we will show that these two ways of describing a quantum state by an $\mathbb F_2$-linear subspace $S$ are in fact the same. Namely, the maximal CSS code generated by $S$ fixes the quantum state whose ZX normal form is also given by $S$. This insight gives us an immediate translation from stabilisers of a maximal CSS code into a ZX diagram describing its associated state. We show that we can extend this translation to stabilisers and logical operators of any (possibly non-maximal) CSS code by "bending wires". To demonstrate the utility of this translation, we give a simple picture of the surface code and a fully graphical derivation of the action of physical lattice surgery operations on the space of logical qubits, completing the ZX presentation of lattice surgery initiated by de Beudrap and Horsman.

arXiv:2204.14038Apr 29, 202210 citations

A 67%-Rate CSS Code on the FCC Lattice: [[192,130,3]] from Weight-12 Stabilizers

Raghu Kulkarni

We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination $K=12$ ); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verification confirms CSS validity ( $H_{X}H_{Z}^{T}=0$ over GF(2)) and reveals $k=2L^{3}+2$ logical qubits: $k=130$ at $L=4$ and $k=434$ at $L=6$ , yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance $d=3$ is proven exactly by exhaustive elimination of all weight- $\le 2$ candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at $L=4$ and [[648, 434, 3]] at $L=6$ . This rate is 24x higher than the cubic 3D toric code (2.8% at $d=4$ ), though at a lower distance ( $d=3$ vs. $d=4$ ); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has $3L^{3}$ edges but only $L^{3}-2$ independent stabilizer constraints, leaving $k=2L^{3}+2$ logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at $p=0.001$ (and 63x at $p=0.0005$ ), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination $K=12$); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verification confirms CSS validity ($H_{X}H_{Z}^{T}=0$ over GF(2)) and reveals $k=2L^{3}+2$ logical qubits: $k=130$ at $L=4$ and $k=434$ at $L=6$, yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance $d=3$ is proven exactly by exhaustive elimination of all weight-$\le 2$ candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at $L=4$ and [[648, 434, 3]] at $L=6$. This rate is 24x higher than the cubic 3D toric code (2.8% at $d=4$), though at a lower distance ($d=3$ vs. $d=4$); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has $3L^{3}$ edges but only $L^{3}-2$ independent stabilizer constraints, leaving $k=2L^{3}+2$ logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at $p=0.001$ (and 63x at $p=0.0005$), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.

quant-pharXiv:2603.20294Mar 20, 20260 citations

Local Clifford equivalence to CSS codes: Polynomial-time classification and conversion

Mingyuan Wang, Stefan Krastanov

Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure $X$ and pure $Z$ stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For $m$ possibly redundant generator rows on $n$ qubits, the algorithm uses $O(mn^2+n^4)$ binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure $X$ and pure $Z$ stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For $m$ possibly redundant generator rows on $n$ qubits, the algorithm uses $O(mn^2+n^4)$ binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.

quant-pharXiv:2610.28737Oct 22, 20260 citations

Homological Quantum Error Correction with Torsion

S. Novák

Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.Homological quantum error correction uses tools of algebraic topology and homological algebra to derive Calderbank-Shor-Steane quantum error correcting codes from cellulations of topological spaces. This work is an exploration of the relevant topics, a journey from classical error correction, through homology theory, to CSS codes acting on qudit systems. Qudit codes have torsion in their logical spaces. This is interesting to study because it gives us extra logical qudits, of possibly different dimension. Apart from examples and comments on the topic, we prove an original result, the Structure Theorem for the Qudit Logical Space, an application of the Universal Coefficient Theorem from homological algebra, which gives us information about the logical space when torsion is involved, and that improves on a previous result in the literature. Furthermore, this work introduces our own abstracted and restricted version of the general notion of a cell complex, suited exactly to our needs.

arXiv:2405.03559May 6, 20240 citations

Toric codes all the way down: decoupling translation-invariant topological CSS codes in higher dimensions

Mingxin He, Yifei Wang, Hao Song +1 more

Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant $p$ -ary CSS codes, where $p$ is prime ( $p=2$ for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and $m$ nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of $p^{m-1}$ in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.Topological Calderbank-Shor-Steane (CSS) codes in more than two spatial dimensions offer features useful for fault-tolerant quantum computation, such as transversal non-Clifford gates and single-shot decoding, but a systematic understanding of them is still lacking. One approach is to decouple a code: map it to a stack of higher-form toric codes by a finite-depth local CSS-preserving Clifford circuit, possibly after adding ancillas. For translation-invariant $p$-ary CSS codes, where $p$ is prime ($p=2$ for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and $m$ nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of $p^{m-1}$ in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.

quant-pharXiv:2610.40592Oct 30, 20260 citations

\mathbb F_2

, i.e. a classical linear code with parameters

[E,\ \beta_1=E-V+c,\ \mathrm{girth}]

; 2. the commutation criterion for vertex/face stabilisers, proved unconditionally when faces are closed words (Theorem A, §4); 3. the closed form for the number of logical qubits,

k=E-r_X-r_Z

(Theorem B, §5); 4. the distance as an essential girth, and the observation that the naive

d=\mathrm{girth}

is false (Theorem D, §6); 5. the binary operator algebra on each edge, including a real anticommutation

F_eZ_e=-Z_eF_e

with no factor of

i

, and the tensor-product structure

(\mathbb C^2)^{\otimes E}

\sum_f\partial f=0

always holds, forcing

k=\beta_1-|F|+\mathrm{rel}

; when

K

is a closed connected surface this collapses to

k=2g

Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting

X

- and

Z

-type checks. We introduce CSS-like codes on

G

-valued qudits for any finite group

G

that reduce to qubit CSS codes for

G = \mathbb{Z}_2

yet generalize the Kitaev quantum double model for general groups. The

X

-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while

Z

-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when

G

is non-Abelian and simple, every

G

-covariant group-CSS code with suitably upper-bounded

Z

-check weight and lower-bounded

Z

-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.

\mathbb F_2

-linear subspaces, or in the case of maximal CSS codes, a single subspace and its orthocomplement. On the other hand, phase-free ZX diagrams can always be efficiently reduced to a normal form given by the basis elements of an

\mathbb F_2

-linear subspace. Here, we will show that these two ways of describing a quantum state by an

\mathbb F_2

-linear subspace

S

are in fact the same. Namely, the maximal CSS code generated by

S

fixes the quantum state whose ZX normal form is also given by

S

. This insight gives us an immediate translation from stabilisers of a maximal CSS code into a ZX diagram describing its associated state. We show that we can extend this translation to stabilisers and logical operators of any (possibly non-maximal) CSS code by "bending wires". To demonstrate the utility of this translation, we give a simple picture of the surface code and a fully graphical derivation of the action of physical lattice surgery operations on the space of logical qubits, completing the ZX presentation of lattice surgery initiated by de Beudrap and Horsman.

We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination

K=12

); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verification confirms CSS validity (

H_{X}H_{Z}^{T}=0

over GF(2)) and reveals

k=2L^{3}+2

logical qubits:

k=130

L=4

and

k=434

L=6

, yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance

d=3

is proven exactly by exhaustive elimination of all weight-

\le 2

candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at

L=4

and [[648, 434, 3]] at

L=6

. This rate is 24x higher than the cubic 3D toric code (2.8% at

d=4

), though at a lower distance (

d=3

vs.

d=4

); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has

3L^{3}

edges but only

L^{3}-2

independent stabilizer constraints, leaving

k=2L^{3}+2

logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at

p=0.001

(and 63x at

p=0.0005

), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.

Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure

X

and pure

Z

stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For

m

possibly redundant generator rows on

n

qubits, the algorithm uses

O(mn^2+n^4)

binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.

p

-ary CSS codes, where

p

is prime (

p=2

for qubits), we require the circuit to commute with the translations of a superlattice. In two dimensions, such a decoupling exists once these translations act trivially on every anyon type. In higher dimensions, excitations are sorted by their dimension into superselection layers, which can remain glued together even when translations act trivially on each layer. We show that this gluing is the only remaining obstruction: a code can be decoupled on a superlattice exactly when, after coarse-graining to it, translations act trivially on every layer and the gluing vanishes. Moreover, for a code with finitely many excitation types in each layer and

m

nonzero layers, once translations act trivially on every layer, further coarse-graining by a factor of

p^{m-1}

in every direction removes the aforementioned gluing. Physically, the same criterion determines when the code encodes the maximal number of logical qudits on a finite torus. Finally, we give an explicit algorithm that constructs the decoupling as a circuit of SUM (qudit CNOT) and SWAP gates.