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arXiv:2510.05211 · npj Quantum Information

Self-dual bivariate bicycle codes with transversal Clifford gates

Zijian Liang, Yu-An Chen

Published Jul 15, 2026• 0 citations
arXivHTMLPDF

Abstract

Bivariate bicycle codes are promising candidates for high-threshold, low-overhead fault-tolerant quantum memories. Meanwhile, color codes are the most prominent self-dual CSS codes, supporting transversal Clifford gates that have been demonstrated experimentally. In this work, we combine these advantages and introduce a broad family of self-dual bivariate bicycle codes. These codes achieve higher encoding rates than surface and color codes while admitting transversal CNOT, Hadamard, and S gates. In particular, we enumerate weight-8 self-dual bivariate bicycle codes with up to n ⩽ 200 physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality. Representative examples include codes with parameters [[ n , k , d ]]: [[16, 4, 4]], [[40, 6, 6]], [[56, 6, 8]], [[64, 8, 8]], [[120, 8, 12]], [[152, 6, 16]], and [[160, 8, 16]].

DOI
10.1038/s41534-026-01303-1

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Bivariate bicycle codes are promising candidates for high-threshold, low-overhead fault-tolerant quantum memories. Meanwhile, color codes are the most prominent self-dual CSS codes, supporting transversal Clifford gates that have been demonstrated experimentally. In this work, we combine these advantages and introduce a broad family of self-dual bivariate bicycle codes. These codes achieve higher encoding rates than surface and color codes while admitting transversal CNOT, Hadamard, and S gates. In particular, we enumerate weight-8 self-dual bivariate bicycle codes with up to n ⩽ 200 physical qubits, realized on twisted tori that enhance code distance and improve stabilizer locality. Representative examples include codes with parameters [[ n , k , d ]]: [[16, 4, 4]], [[40, 6, 6]], [[56, 6, 8]], [[64, 8, 8]], [[120, 8, 12]], [[152, 6, 16]], and [[160, 8, 16]].
Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.
High-rate bivariate bicycle (BB) codes are promising low-overhead quantum memories, but their stabilizer checks typically have weight 666 or higher, making syndrome extraction challenging. We introduce subsystem bivariate bicycle (SBB) codes, a translation-invariant CSS subsystem construction that realizes BB-code logical structure using local weight-444 gauge measurements. Their stabilizer syndromes are inferred by multiplying the corresponding gauge outcomes. We further show that nonlocal stabilizers in translation-invariant CSS subsystem codes can be detected using a determinantal-ideal criterion based on the gauge-operator commutation matrix. When this criterion excludes nonlocal stabilizers, a finite-depth Clifford circuit decouples gauge qubits and identifies the protected subsystem with a corresponding BB stabilizer code. An SBB code is topological, meaning that it has no nontrivial local logical operators, if and only if the corresponding BB code is topological. A finite search yields low-overhead examples including [[27,6,3]][[27,6,3]][[27,6,3]], [[75,10,5]][[75,10,5]][[75,10,5]], and [[108,12,6]][[108,12,6]][[108,12,6]]; the latter encodes six times more logical qubits than a subsystem surface code at the same block length and distance. These results show how gauge degrees of freedom can make high-rate BB logical structure compatible with local weight-444 syndrome extraction.
We compare and study different Bivariate-Bicycle (BB) encodings and node choices for distributed quantum operations such as transversal non-local CNOTs. We observe that while some encodings have more physical qubits requiring more ebits for a distributed computation, reducing ebit consumption alone might not be the best criterion for selecting an encoding for the logical qubits, if the goal is to reduce the logical error rate of the distributed computation; for example, one should also consider encodings with a larger distance, which using more physical qubits can provide. We consider distributed, or non-local, CNOTs and computation of the global gate (GCZ) over distributed logical qubits as examples. The choice of encoding enables particular concurrent operations; e.g., we show that a transversal physical GCZ on a self-dual [[120,8,12]]BB[[120,8,12]] BB[[120,8,12]]BB code can realize eight concurrent logical GCZ operations after accounting for the logical permutation induced by transversal Hadamard.
Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates. As examples, we construct[[98,6,12]][[{98,6,12}]][[98,6,12]]and[[162,8,12]][[{162, 8, 12}]][[162,8,12]]BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block group algebra codes, which might be of independent interest.
We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a [[140,6,14]][[140,6,14]][[140,6,14]] code with kd2/n=8.40kd^2/n = 8.40kd2/n=8.40. In the code-capacity setting, the [[140,6,14]][[140,6,14]][[140,6,14]] code achieves a pseudothreshold of 8.0%8.0\%8.0% and kd2/n=8.40kd^2/n = 8.40kd2/n=8.40, exceeding the best multivariate bicycle code of Voss et al.\ (7.9%7.9\%7.9%, kd2/n=2.67kd^2/n = 2.67kd2/n=2.67). With circuit-level depolarizing noise, pseudothresholds reach 0.59%0.59\%0.59% for [[140,6,14]][[140,6,14]][[140,6,14]] and 0.53%0.53\%0.53% for [[84,6,10]][[84,6,10]][[84,6,10]]. On the SI1000 superconducting noise model, the [[140,6,14]][[140,6,14]][[140,6,14]] code achieves a per-round per-observable rate of 5.6×10−55.6 \times 10^{-5}5.6×10−5 at p=0.20%p = 0.20\%p=0.20%. We additionally present two self-dual codes with weight-8 stabilizers: [[54,14,5]][[54,14,5]][[54,14,5]] (kd2/n=6.48kd^2/n = 6.48kd2/n=6.48) and [[128,20,8]][[128,20,8]][[128,20,8]] (kd2/n=10.0kd^2/n = 10.0kd2/n=10.0). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.