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

Low-overhead pieceable fault-tolerant construction of logical controlled-phase circuit for degenerate quantum code

Chen Lin, Guowu Yang, Jinzhao Wu

We propose a non-transversal but pieceable fault-tolerant conversion circuit that is used to convert encoded information between five-qubit code and seven-qubit CSS code. Since a syndrome extraction circuit requiring fewer ancillary qubit resources would facilitate the realization of large-scale quantum computations, we further adapt a flag-assisted fault-tolerant syndrome measurement scheme to reduce the cost of ancillary preparation. Numerical simulations are also performed to further analyze the performance of our conversion method.We propose a non-transversal but pieceable fault-tolerant conversion circuit that is used to convert encoded information between five-qubit code and seven-qubit CSS code. Since a syndrome extraction circuit requiring fewer ancillary qubit resources would facilitate the realization of large-scale quantum computations, we further adapt a flag-assisted fault-tolerant syndrome measurement scheme to reduce the cost of ancillary preparation. Numerical simulations are also performed to further analyze the performance of our conversion method.

arXiv:2105.07133IEEE International Conference on Artificial Intelligence and Computer Applications (ICAICA)Jun 1, 20200 citations

Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly

K. Sai Mineesh Reddy, Navin Kashyap

This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$ -rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$ ; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$ , respectively.This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.

quant-pharXiv:2601.08568Jan 14, 20260 citations

Structure of CSS and CSS-T quantum codes

Elena Berardini, A. Caminata, Alberto Ravagnani

arXiv:2310.16504Designs Codes and CryptographyMay 24, 202412 citations

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.

This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal

Z

-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal

T

realizes the logical transversal

S^{\dagger}

; (b) show that the condition

C_2 \ast C_1 \subseteq C_1^{\perp}

is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal

T

implements the logical identity and the logical transversal

T

, respectively.