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

TL;DR

This paper demonstrates that a broad class of self-correcting quantum memories can rapidly reach thermal equilibrium within a logical sector using a quasi-local Gibbs sampler, providing insights into their dynamical properties and a fast state preparation method.

Contribution

It introduces a rapid convergence method for Gibbs states in self-correcting quantum memories, revealing their dynamical behavior and enabling efficient state preparation.

Findings

Quasi-local Gibbs sampler rapidly converges to low-temperature Gibbs state within a logical sector.

The approach applies to a broad class of self-correcting quantum memories on lattices.

Provides a polylogarithmic depth algorithm for rapid Gibbs state preparation of the 4D toric code.

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.