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Circuit-based characterization of finite-temperature quantum phases and self-correcting quantum memory

Ruochen Ma, Vedika Khemani, Shengqi Sang

TL;DR

The paper develops a circuit-based framework to classify finite-temperature quantum phases by connecting Gibbs states through local channels under a stability condition called stable clustering. It proves the existence of Polylog-range locally reversible channels mapping Gibbs states along stable paths, and shows a continuous-time Lindbladian version for commuting Hamiltonians. As an application, it demonstrates that systems in the same thermal phase as a zero-temperature topological code support self-correcting quantum memory with encoding/decoding channels and memory times scaling as $\exp(L^{\mathcal{O}(1)})$. The work connects clustering, approximately Markovian structure, and recoverability of Gibbs states, offering a state-centric perspective on thermal phases with potential extensions to symmetry and higher-dimensional codes.

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 ${\rm polylog}({\rm poly}(L)/ε)$. 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.

Circuit-based characterization of finite-temperature quantum phases and self-correcting quantum memory

TL;DR

The paper develops a circuit-based framework to classify finite-temperature quantum phases by connecting Gibbs states through local channels under a stability condition called stable clustering. It proves the existence of Polylog-range locally reversible channels mapping Gibbs states along stable paths, and shows a continuous-time Lindbladian version for commuting Hamiltonians. As an application, it demonstrates that systems in the same thermal phase as a zero-temperature topological code support self-correcting quantum memory with encoding/decoding channels and memory times scaling as . The work connects clustering, approximately Markovian structure, and recoverability of Gibbs states, offering a state-centric perspective on thermal phases with potential extensions to symmetry and higher-dimensional codes.

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 and approximation error , the locality of the circuit is . 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.

Paper Structure

This paper contains 19 sections, 8 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setup and Definitions
  3. Constructing channel circuits connecting Gibbs states
  4. Local Interaction Variation
  5. Global Interaction Variation
  6. Zero-temperature limit
  7. Thermal phases of self-correcting quantum memories
  8. Continuous-time construction for commuting Hamiltonians
  9. Summary and outlook
  10. Proof of Lemma 1
  11. Proof of Lemma 3
  12. Symmetric local interaction variation: example of Ising symmetry
  13. Derivation of Eq.\ref{['eq:totalerror']}
  14. Proof of local reversibility
  15. Gapped Ground States
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\rho_{{\boldsymbol\beta}}$ and $\rho_{{\boldsymbol\beta}'}$ be two Gibbs states connected by a path in ${\boldsymbol\beta}$-space along which stable clustering is satisfied. Then there exists a $2\epsilon_{\mathcal{C}}$-locally reversible channel circuit $\mathcal{C}$ with range ${\rm rng\,}{\m

Figures (3)

  • Figure 1: Annulus tripartition for defining clustering of correlations, illustrated in 2D. $A$ is a small region, $B$ is a region surrounding $A$, and $C$ denotes the rest of the system.
  • Figure 3: Geometry for local interaction variation.
  • Figure 4: Illustration of global interaction variation. Two gates ${\mathcal{M}}^{(i)}$ and ${\mathcal{M}}^{(j)}$, used for varying interactions on $A_{x_i}$ and $A_{x_j}$ respectively, can be implemented in parallel if $x_i$ and $x_j$ are at least $2r_b$ apart.

Theorems & Definitions (11)

  • Definition 2: Stable clustering
  • Definition 3: Local reversibility (LR)
  • Theorem 1
  • Lemma 1: Clustering implies LPPL. Theorem 25 in Ref. 2025LPPL, modified
  • Lemma 2: Gibbs states are locally Markovian. Ref.2025CR
  • Lemma 3: Markov property implies recoverability. Ref. 2015JRWW, modified
  • Theorem 2
  • Definition 4: Local indistinguishability (LI) for mixed states 2025localreversibility
  • Lemma 4
  • Theorem 3: Self-correction is a property of thermal phases
  • ...and 1 more