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Decoding three-dimensional color codes with boundaries

Friederike Butt*, Lars Esser, and Markus Müller

  • *Contact author: f.butt@fz-juelich.de
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Phys. Rev. A 113, 042416 – Published 6 April, 2026

DOI: https://doi.org/10.1103/7cdb-px1d

Abstract

Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are promising candidates for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [S.-H. Lee et al., Quantum 9, 1609 (2025)], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of 2 higher than previously reported for this family of codes [N. Delfosse, Phys. Rev. A 89, 012317 (2014); S. Turner et al., arXiv:2003.11602]. We furthermore present qCodePlot3D, a python package for visualizing two-dimensional and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.

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Phys. Rev. A 113, 042416– Published 6 April, 2026
  • Received 15 December 2025
  • Revised 6 March 2026
  • Accepted 16 March 2026
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