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  • Open Access

Abelian Multi-Cycle Codes for Single-Shot Error Correction

Hsiang-Ku Lin1, Pak Kau Lim1, Alexey A. Kovalev2, and Leonid P. Pryadko3,1,*

  • *Contact author: pryadko@google.com
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PRX Quantum 7, 020326 – Published 11 May, 2026

DOI: https://doi.org/10.1103/mj28-925w

Abstract

We construct a family of quantum low-density parity-check codes locally equivalent to higher-dimensional quantum hypergraph-product (QHP) codes. Similarly to QHP codes, the proposed codes have highly redundant sets of low-weight stabilizer generators, which improves decoding accuracy in a fault-tolerant regime and gives them single-shot properties. The advantage of the construction is that it gives shorter codes. We derive simple expressions for the dimension of the proposed codes in two important special cases, give bounds on the distances and explicitly construct some relatively short codes. Circuit simulations for codes locally equivalent to four-dimensional toric codes show a (pseudo)threshold close to 1.1%, better than for toric or surface codes with a similar noise model.

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Errors are a bane of quantum computation. They happen constantly, and special fault-tolerant hardware and software is required to operate in the presence of such errors. Our quantum error correcting codes, which we term abelian multi-cycle (AMC) codes, are designed to work in a fault-tolerant setting, in particular, in the presence of severe measurement errors. In the four-cycle case, our codes preserve the powerful features of quantum four-dimensional product codes such as self-correction, single-shot error correction and stability while requiring much shorter block lengths, making them significantly more practical for future quantum computers, as well as offering possibilities for near-term quantum hardware. In the simplest, two-cycle case, the construction boils down to generalized bicycle codes originally proposed by two of the authors back in 2013. Codes from this latter family, bivariate-bicycle (BB) codes, including the IBM “gross” codes, have recently come to the attention due to their high rates and distances (substantially improving over the surface codes), and excellent circuit performance.

In this work, we present a general algebraic construction, give simple expressions for the dimension (number of encoded qubits) and explicitly construct a number of codes based on four circulant matrices. We study the error-confinement profiles of three of the constructed codes, construct their circuit implementation and study their decoding performance with the help of a software package we made freely available. Our circuit simulations demonstrate a threshold close to 1.1%, exceeding those of surface and BB codes under comparable noise models. Together, these results represent both a theoretical advance in code design and a practical step toward more efficient fault-tolerant quantum error correction.

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PRX Quantum 7, 020326– Published 11 May, 2026
  • Received 26 August 2025
  • Revised 11 December 2025
  • Accepted 23 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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