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arXiv:2507.10976

Rapid mixing for Gibbs states within a logical sector: a dynamical view of self-correcting quantum memories

Thiago Bergamaschi, Reza Gheissari, Yunchao Liu

Published Jul 15, 2025• 0 citations
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Abstract

Self-correcting quantum memories store logical quantum information for exponential time in thermal equilibrium at low temperatures. By definition, these systems are slow mixing. This raises the question of how the memory state, which we refer to as the Gibbs state within a logical sector, is created in the first place. In this paper, we show that for a broad class of self-correcting quantum memories on lattices with parity check redundancies, a quasi-local quantum Gibbs sampler rapidly converges to the corresponding low-temperature Gibbs state within a logical sector when initialized from a ground state. This illustrates a dynamical view of self-correcting quantum memories, where the "syndrome sector" rapidly converges to thermal equilibrium, while the "logical sector" remains metastable. As a key application, when initialized from a random ground state, this gives a rapid Gibbs state preparation algorithm for the 4D toric code in $\mathrm{polylog}(n)$ depth. The main technical ingredients behind our approach are new, low-temperature decay-of-correlation properties for these metastable states.

DOI
10.48550/arxiv.2507.10976

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Self-correcting quantum memories store logical quantum information for exponential time in thermal equilibrium at low temperatures. By definition, these systems are slow mixing. This raises the question of how the memory state, which we refer to as the Gibbs state within a logical sector, is created in the first place. In this paper, we show that for a broad class of self-correcting quantum memories on lattices with parity check redundancies, a quasi-local quantum Gibbs sampler rapidly converges to the corresponding low-temperature Gibbs state within a logical sector when initialized from a ground state. This illustrates a dynamical view of self-correcting quantum memories, where the "syndrome sector" rapidly converges to thermal equilibrium, while the "logical sector" remains metastable. As a key application, when initialized from a random ground state, this gives a rapid Gibbs state preparation algorithm for the 4D toric code in polylog(n)\mathrm{polylog}(n)polylog(n) depth. The main technical ingredients behind our approach are new, low-temperature decay-of-correlation properties for these metastable states.
Passive error correction protects logical information forever (in the thermodynamic limit) by updating the system based only on local information and few-body interactions. A paradigmatic example is the classical two-dimensional Ising model: a Metropolis-style Gibbs sampler retains the sign of the initial magnetization (a logical bit) for thermodynamically long times in the low-temperature phase. Known models of passive quantum error correction similarly exhibit thermodynamic phase transitions to a low-temperature phase wherein logical qubits are protected by thermally stable topological order. Here, in contrast, we show that certain families of constant-rate classical and quantum low-density parity check codes have no thermodynamic phase transitions at nonzero temperature, but nonetheless exhibit ergodicity-breaking dynamical transitions: below a critical nonzero temperature, the mixing time of local Gibbs sampling diverges in the thermodynamic limit. Slow Gibbs sampling of such codes enables fault-tolerant passive quantum error correction using finite-depth circuits. This strategy is well suited to measurement-free quantum error correction, and may present a desirable experimental alternative to conventional quantum error correction based on syndrome measurements and active feedback. It has been commonly assumed that self-correcting quantum memories are only possible in systems with finite-temperature phase transitions to topological order. Here the authors show a complete breakdown of this expectation in quantum low-density parity-check codes.
Understanding when quantum many-body systems rapidly converge to thermal equilibrium is a central problem in quantum statistical mechanics. While spatial locality provides powerful tools for systems with an underlying geometry, fully connected models require estimates that accommodate interactions between every pair of spins. We study Gibbs-state preparation for two-local quantum Hamiltonians on arbitrary interaction graphs, including fully connected systems, via Lindblad dynamics that satisfies Kubo--Martin--Schwinger (KMS) detailed balance. Our approach is based on a quantum analogue of the classical Dobrushin condition, which controls the influence of local perturbations on the rest of the system. Using the quantum Wasserstein distance of order 111, we quantify how these perturbations propagate under the dynamics and derive contraction estimates in a high-temperature regime controlled by the maximum total interaction strength per site. These estimates require no assumptions on spatial geometry or graph growth and yield mixing times logarithmic in the system size and the inverse error tolerance, uniformly over all initial states.
We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in O(log⁡(n/ε))\mathcal{O}\left(\log (n/\varepsilon)\right)O(log(n/ε)) time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order 111. We further provide O(polylog(n/ε))\mathcal{O}(\text{polylog}(n/\varepsilon))O(polylog(n/ε)) depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.
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.
We propose a polynomial-time algorithm for preparing the Gibbs state of the two-dimensional toric code Hamiltonian at any temperature, starting from any initial condition, significantly improving upon prior estimates that suggested exponential scaling with inverse temperature. Our approach combines the Lindblad dynamics using a local Davies generator with simple global jump operators to enable efficient transitions between logical sectors. Our proof also shows that the Lindblad dynamics with a digitally implemented low-temperature local Davies generator is able to efficiently drive the quantum state towards the ground state manifold.