Bottlenecks in quantum channels and finite temperature phases of matter

TL;DR

This paper extends the classical bottleneck theorem to quantum channels, providing bounds on relaxation times and insights into quantum phases and Gibbs sampling in low-temperature many-body systems.

Contribution

It introduces a quantum bottleneck ratio concept, linking low-weight regions to long relaxation times, and applies this to analyze quantum phases and Gibbs sampler performance.

Findings

Quantum bottleneck ratio bounds relaxation times.

Low-temperature quantum systems have exponentially large mixing times.

Certain quantum models with energy barriers are proven to mix slowly.

Abstract

We prove an analogue of the "bottleneck theorem", well-known for classical Markov chains, for Markovian quantum channels. In particular, we show that if two regions (subspaces) of Hilbert space are separated by a region that has very low weight in the channel's steady state, then states initialized on one side of this barrier will take a long time to relax, putting a lower bound on the mixing time in terms of an appropriately defined "quantum bottleneck ratio". Importantly, this bottleneck ratio involves not only the probabilities of the relevant subspaces, but also the size of off-diagonal matrix elements between them. For low-temperature quantum many-body systems, we use the bottleneck theorem to bound the performance of any quasi-local Gibbs sampler. This leads to a new perspective on thermally stable quantum phases in terms of a decomposition of the Gibbs state into multiple components separated by bottlenecks. As a concrete application, we show rigorously that weakly perturbed commuting projector models with extensive energy barriers (including certain classical and quantum expander codes) have exponentially large mixing times.