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Observation of a fault tolerance threshold with concatenated codes

Grace M. Sommers1,*, Michael Foss-Feig2, David Hayes2, David A. Huse1, and Michael J. Gullans3

  • *Contact author: gsommers@princeton.edu
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Phys. Rev. Research 7, 043271 – Published 9 December, 2025

DOI: https://doi.org/10.1103/v477-jw8l

Abstract

We introduce a fault-tolerant protocol for code concatenation of a generalized Shor code using a butterfly network architecture with high-noise thresholds and low ancilla overhead to allow implementation on current devices. We develop a probability passing decoder using tensor networks that applies Bayesian updates to the marginal error probabilities after each layer of checks, achieving a state-preparation threshold of ec≈0.089 for erasure errors and ≈0.015 for unheralded noise. We implement our state-preparation protocol on ion-trap hardware with added noise to demonstrate the threshold behavior in a real quantum device. We further theoretically test the performance of our scheme as a quantum memory and for universal quantum computation through the preparation of low-noise magic states for state distillation and T-gate injection.

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Phys. Rev. Research 7, 043271– Published 9 December, 2025
  • Received 23 June 2025
  • Accepted 27 October 2025
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