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Subsystem many-hypercube codes: High-rate concatenated codes with low-weight syndrome measurements

Ryota Nakai1,* and Hayato Goto1,2

  • *Contact author: rnakai@riken.jp
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Phys. Rev. Applied 25, 014032 – Published 13 January, 2026

DOI: https://doi.org/10.1103/xbzn-vn37

Abstract

Quantum error-correcting codes (QECCs) require high encoding rates in addition to high thresholds unless a sufficiently large number of physical qubits are available. Many-hypercube (MHC) codes, defined as concatenations of the [[6,4,2]] quantum error-detecting code, have been proposed as high-performance and high-encoding-rate QECCs. However, concatenated codes have the disadvantage that the syndrome weight grows exponentially with respect to the concatenation level. To address this issue, here we propose subsystem quantum codes based on MHC codes. In particular, we study the smallest subsystem MHC codes, namely, subsystem codes derived from concatenated [[4,2,2]] error-detecting codes. The resulting codes have a constant syndrome-measurement weight of 4, while keeping high encoding rates. We build block-MAP and neural-network decoders and show that they demonstrate superior performance to the bounded-distance decoder.

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Phys. Rev. Applied 25, 014032– Published 13 January, 2026
  • Received 24 October 2025
  • Revised 18 November 2025
  • Accepted 10 December 2025
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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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