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  • Open Access

Circuit-Based Characterization of Finite-Temperature Quantum Phases and Self-Correcting Quantum Memory

Ruochen Ma1,2,3,*, Vedika Khemani4,†, and Shengqi Sang4,‡

  • *Contact author: ruochenma@ucsb.edu
  • †Contact author: vkhemani@stanford.edu
  • ‡Contact author: sangsq@stanford.edu
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PRX Quantum 7, 033052 – Published 10 September, 2026

DOI: https://doi.org/10.1103/sh48-wmy4

Abstract

Quantum phases at zero temperature can be characterized as equivalence classes under local unitary transformations: two ground states within a gapped phase can be transformed into each other via a local unitary circuit. We generalize this circuit-based characterization of phases to systems at finite-temperature thermal equilibrium described by Gibbs states. We construct a channel circuit that approximately transforms one Gibbs state into another provided the two are connected by a path in parameter space along which a certain correlation-decay condition holds. For finite-dimensional systems of linear size L and approximation error ϵ, the locality of the circuit is Polylog(Poly(L)/ϵ). The correlation-decay condition, which we specify, is expected to be satisfied in the interior of many noncritical thermal phases, including those displaying discrete symmetry breaking and topological order. As an application, we show that any system in the same thermal phase as a zero-temperature topological code coherently preserves quantum information for a macroscopically long time, establishing self-correction as a universal property of thermal phases. As part of the proof, we provide explicit encoding and decoding channel circuits to encode information into, and decode it from, a system in thermal equilibrium.

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Popular Summary

Quantum phases of matter are usually studied through gapped many-body ground states. There, two systems are in the same phase if one can be transformed into the other by a local unitary circuit, without changing the long-distance properties that characterize the phase. Real materials and quantum devices, however, are at finite temperature and are described by thermal states. This raises a basic question: what does it mean, operationally, for thermal states to be in the same phase? We answer this question by extending the circuit-based viewpoint from ground states to thermal equilibrium. At zero temperature, an energy gap protects a phase. At finite temperature, we identify an analogous condition: a suitable correlation length remains finite. When this condition holds along a path between two thermal systems, we show that local quantum-channel circuits can convert one thermal state into the other in both directions. This correlation-decay condition is expected to hold in many noncritical thermal phases, including low-temperature phases with nontrivial order. This perspective also clarifies quantum memories at finite temperature. A self-correcting memory stores information passively: when coupled to a cold environment, it can retain quantum information for a time that grows exponentially with the size of the system. We show that this ability is shared by all systems in the same thermal phase as a zero-temperature topological code. Thus, self-correction is a universal property of a thermal phase, rather than a feature of a specific model.

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Outline

Information

PRX Quantum 7, 033052– Published 10 September, 2026
  • Received 19 February 2026
  • Revised 13 May 2026
  • Accepted 6 July 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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