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  • Open Access

Biased-noise thresholds of zero-rate holographic codes with tensor-network decoding

J. Fan1,2,*, M. Steinberg1,2,*,†, A. Jahn3, C. Cao4, and S. Feld1,2

  • *These authors contributed equally to this work.
  • †Contact author: matt.steinberg3@gmail.com
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Phys. Rev. A 114, 012447 – Published 20 July, 2026

DOI: https://doi.org/10.1103/rcxg-x3hy

Abstract

A crucial insight for practical quantum error correction is that different types of errors, such as single-qubit Pauli operators, typically occur with different probabilities. Finding an optimal quantum code under such biased noise is a challenging problem, related to the (generally unknown) maximum capacity of the corresponding noisy channel. A benchmark for this capacity is given by the hashing bound, which describes the performance of random stabilizer codes and leads to the matter of identifying codes that come close to the bound while also being efficiently decodable. In this work, we perform a comprehensive analysis of asymptotically zero-rate holographic codes under biased noise. We show that many representatives from such models of this code class fulfill both the channel optimality and efficient decoding guarantees for tensor-network codes. In fact, all tested holographic codes are found to reach the hashing bound in some bias regime, while several built from the [[5,1,2]] surface code and [[6,1,3]] code exceed state-of-the-art code performance in the 2-Pauli noise regime. Furthermore, we consider Clifford deformations which allow all considered codes to reach the hashing bound for 1-Pauli noise as well. Our results establish that holographic codes, which were previously shown to possess efficient tensor-network decoders, also exhibit competitive thresholds under biased noise.

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Outline

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Phys. Rev. A 114, 012447– Published 20 July, 2026
  • Received 23 August 2024
  • Revised 27 April 2026
  • Accepted 1 July 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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