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  • Open Access

Low-Density Parity-Check Codes as Stable Phases of Quantum Matter

Chao Yin* and Andrew Lucas†

  • Department of Physics and Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

  • *Contact author: chao.yin@colorado.edu
  • †Contact author: andrew.j.lucas@colorado.edu
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PRX Quantum 6, 030329 – Published 19 August, 2025

DOI: https://doi.org/10.1103/361k-nj4b

Abstract

Phases of matter with robust ground-state degeneracy, such as the quantum toric code, are known to be capable of robust quantum information storage. Here, we address the converse question: given a quantum error-correcting code, when does it define a stable gapped quantum phase of matter, whose ground-state degeneracy is robust against perturbations in the thermodynamic limit? We prove that a low-density parity-check (LDPC) code defines such a phase, robust against all few-body perturbations, if its code distance grows at least logarithmically in the number of degrees of freedom, and it exhibits “check soundness.” Many constant-rate quantum LDPC expander codes have such properties, and define stable phases of matter with a constant zero-temperature entropy density, violating the third law of thermodynamics. Our results also show that quantum toric-code phases are robust to spatially nonlocal few-body perturbations. Similarly, phases of matter defined by classical codes are stable against symmetric perturbations. In the classical setting, we present improved locality bounds on the quasiadiabatic evolution operator between two nearby states in the same code phase.

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Popular Summary

The universal properties of quantum many-body ground states are typically understood through the lens of stable phases of matter, a framework that usually relies on the system being contained in finite spatial dimensions. Certain phases of matter in finite spatial dimensions, such as the toric code, are known to be absolutely stable to perturbations, a property that is related to the ability of the system to robustly store quantum information. Yet, many-body systems can also emerge in effectively infinite dimensions—for instance, in spin glasses or error-correcting codes. Do such systems represent stable phases of matter in the same sense? This work answers that question affirmatively. We prove that certain quantum low-density parity-check (LDPC) codes form absolutely stable phases of matter, robust against arbitrary local perturbations.

LDPC codes are foundational in classical error correction and are widely used in modern communication devices. Their quantum counterparts are now emerging as leading candidates for scalable quantum error correction, offering far lower overhead than traditional finite-dimensional codes like the toric code. We propose a sufficient condition for stability, grounded in coding theory, which generalizes the existing paradigm for the stability of the toric-code phase.

Intriguingly, these “infinite-dimensional” phases seem to defy the third law of thermodynamics: they have robust nonzero entropy density at zero temperature in the thermodynamic limit. Our work illustrates how error-correcting codes can open new perspectives on the foundations of statistical mechanics.

See Also

Low-Density Parity-Check Stabilizer Codes as Gapped Quantum Phases: Stability under Graph-Local Perturbations

Wojciech De Roeck, Vedika Khemani, Yaodong Li, Nicholas O’Dea, and Tibor Rakovszky
PRX Quantum 6, 030330 (2025)

Article Text

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Outline

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PRX Quantum 6, 030329– Published 19 August, 2025
  • Received 15 December 2024
  • Revised 14 April 2025
  • Accepted 2 July 2025
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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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