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arXiv:2502.19406

Single-shot and two-shot decoding with generalized bicycle codes

Hsiang-Ku Lin, Liu, Xingrui, Pak Kau Lim, Leonid P. Pryadko

Published Feb 26, 2025• 1 citation
arXivHTMLPDF

Abstract

Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA codes over abelian groups with one and two generators, with each block row of weight 3, relatively large dimensions, distances, and maximum syndrome distance $d_{\rm S}=3$. We simulated the performance of three such codes under phenomenological noise and standard circuit noise, using sliding window sequential decoding protocol covering $T\ge 1$ measurement rounds at a time, based on an in-house binary BP+OSD decoder. While true single-shot decoding ($T=1$) suffers from a significant loss of accuracy, already two-shot ($T=2$) decoding gives nearly the same logical error rates as multi-shot with much larger $T$. Comparison with the same codes but additional stabilizer generators dropped shows that redundancy significantly improves decoding accuracy for all $T\ge 1$.

DOI
10.48550/arxiv.2502.19406

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Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA codes over abelian groups with one and two generators, with each block row of weight 3, relatively large dimensions, distances, and maximum syndrome distance dS=3d_{\rm S}=3dS​=3. We simulated the performance of three such codes under phenomenological noise and standard circuit noise, using sliding window sequential decoding protocol covering T≥1T\ge 1T≥1 measurement rounds at a time, based on an in-house binary BP+OSD decoder. While true single-shot decoding (T=1T=1T=1) suffers from a significant loss of accuracy, already two-shot (T=2T=2T=2) decoding gives nearly the same logical error rates as multi-shot with much larger TTT. Comparison with the same codes but additional stabilizer generators dropped shows that redundancy significantly improves decoding accuracy for all T≥1T\ge 1T≥1.
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Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.
We consider quantum two-block group algebra (2BGA) codes, a family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle codes, except a cyclic group is replaced with an arbitrary finite group, generally non-Abelian. As special cases, 2BGA codes include a subset of square-matrix LP codes over Abelian groups, including quasicyclic codes, and all square-matrix hypergraph-product codes constructed from a pair of classical group codes. We establish criteria for permutation equivalence of 2BGA codes and give bounds for their parameters, both explicit and in relation to other quantum and classical codes. We also enumerate the optimal parameters of all inequivalent binary connected 2BGA codes with stabilizer generator weights $W\ensuremath{\le}8$, of length $n\ensuremath{\le}100$ for Abelian groups, and $n\ensuremath{\le}200$ for non-Abelian groups.
Quantum error correction (QEC) with single-shot decoding enables reduction of errors after every single round of noisy stabilizer measurement, easing the time-overhead requirements for fault tolerance. Notably, several classes of quantum low-density-parity-check (qLDPC) codes are known which facilitate single-shot decoding, potentially giving them an additional overhead advantage. However, the perceived advantage of single-shot decoding is limited because it can significantly degrade the effective code distance. This degradation may be compensated for by using a much larger code size to achieve the desired target logical error rate, at the cost of increasing the amount of syndrome information to be processed, as well as, increasing complexity of logical operations. Alternatively, in this work we study sliding-window decoding, which corrects errors from previous syndrome measurement rounds while leaving the most recent errors for future correction. We observe that sliding-window decoding significantly improves the logical memory lifetime and hence the effective distance compared to single-shot decoding on hypergraph-product codes and lifted-product codes. Remarkably, we find that this improvement may not cost a larger decoding complexity. Thus, the sliding-window strategy can be more desirable for fast and accurate decoding for fault-tolerant quantum computing with qLDPC codes.
Quantum error correction (QEC) codes are traditionally defined and searched for without specifying the manner in which its syndrome extraction circuits are executed using elementary gates and measurements. We show how morphing circuits introduced in Refs. [1-3] provide a way of optimising syndrome extraction circuits and codes directly in terms of connectivity, choice of two-qubit gate (ISWAP versus CNOT) and number of physical qubits. We discuss morphing circuits in code optimisation among Abelian two-block group algebra (2BGA) codes, handling boundaries for 2D codes, codes with single-shot properties, and improving performance in stability experiments against measurement and reset errors. We show that alternating syndrome extraction circuits - executed with alternating time-reversed rounds - can be viewed as a two-round morphing circuit whose fault-tolerant properties are computationally much easier to examine than non-alternating syndrome extraction circuits. Our methods find new codes and syndrome extraction circuits of practical interest, including Abelian 2BGA morphing circuits with better code parameters and connectivity than existing circuits. [1] Matt McEwen, Dave Bacon, and Craig Gidney. Relaxing hardware requirements for surface code circuits using time-dynamics. Quantum, 7:1172, 2023. [2] Craig Gidney and Cody Jones. New circuits and an open source decoder for the color code, 2023. [3] Mackenzie H. Shaw and Barbara M. Terhal. Lowering connectivity requirements for bivariate bicycle codes using morphing circuits.