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Spectral Properties and Coding Transitions of Haar-Random Quantum Codes

Grace M. Sommers1,*, J. Alexander Jacoby1,2, Zack Weinstein3,4, David A. Huse1, and Sarang Gopalakrishnan5

  • *Contact author: gsommers@princeton.edu
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PRX Quantum 7, 020328 – Published 13 May, 2026

DOI: https://doi.org/10.1103/d4wh-hqcp

Abstract

A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random stabilizer codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher detection threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.

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Popular Summary

Quantum computers show profound promise in diverse applications, such as optimization, cryptography, materials simulation, and even climate science. However, current qudits (the quantum states upon which logical operations are performed) rapidly deteriorate as interactions with their surroundings introduce errors. Quantum error-correcting codes make these quantum states robust by redundantly encoding them in physical systems. In the large-system limit, some codes display nonzero error-rate thresholds, below which the encoded state is nominally recoverable with perfect reliability (though the procedure may be onerous in practice). This threshold can be interpreted as a so-called mixed-state phase transition, which is information-theoretic in character.

In this interdisciplinary study, we bring to bear tools from many-body physics, statistical physics, and classical coding theory to examine this threshold in random codes. We show that they exhibit a transition at the so-called “hashing bound,” where the entropy of possible errors is equal to the maximum achievable entropy of the physical system on which quantum states are encoded. Additionally, we show that if the noisy state is probabilistically rejected, information from an accepted state is information-theoretically recoverable up to a higher threshold.

Our work suggests several avenues for future investigation, such as (i) generalizing the perturbation theory method to more structured codes and noise models, (ii) incorporating the effect of locality into the beyond-hashing phase diagram, and (iii) further exploring the duality features of quantum codes.

Article Text

Supplemental Material

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PRX Quantum 7, 020328– Published 13 May, 2026
  • Received 1 November 2025
  • Revised 18 February 2026
  • Accepted 3 April 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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