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Error resilience of fracton codes and near saturation of code-capacity threshold in three dimensions

Giovanni Canossa1,2, Lode Pollet1,2, Miguel A. Martin-Delgado3,4, Hao Song5,*, and Ke Liu6,7,1,2,†

  • 1Arnold Sommerfeld Center for Theoretical Physics, University of Munich, 80333 München, Germany
  • 2Munich Center for Quantum Science and Technology (MCQST), 80799 München, Germany
  • 3Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain
  • 4CCS-Center for Computational Simulation, Universidad Politócnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain
  • 5Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
  • 6Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China
  • 7Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Shanghai 201315, China

  • *Contact author: songhao@itp.ac.cn
  • †Contact author: ke.liu@ustc.edu.cn
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Phys. Rev. B 113, 104204 – Published 27 March, 2026

DOI: https://doi.org/10.1103/2y2z-hgfn

Abstract

Fracton codes have been intensively studied as novel topological states of matter, yet their fault-tolerant properties remain largely unexplored. Here, we investigate the optimal thresholds of self-dual fracton codes, in particular, the checkerboard code, against stochastic Pauli noise. By utilizing a statistical-mechanical mapping combined with large-scale parallel tempering Monte Carlo simulations, we calculate the optimal code-capacity threshold of the checkerboard code to be pth≃0.107(3). This value is the highest among known three-dimensional codes and nearly saturates the theoretical limit for topological codes. Our results further validate the generalized entropy relation for two mutually dual models, H(pth)+H(p̃th)≈1, and extend its applicability beyond standard topological codes. This verification indicates the Haah's code also possesses a code-capacity threshold near the theoretical limit pth≈0.11. These findings highlight fracton codes as highly resilient quantum memory and demonstrate the utility of duality techniques in analyzing intricate error-correcting codes.

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Phys. Rev. B 113, 104204– Published 27 March, 2026
  • Received 28 December 2025
  • Revised 24 February 2026
  • Accepted 27 February 2026
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