Quantum Gibbs sampling through the detectability lemma

TL;DR

This paper introduces improved quantum Gibbs state preparation methods using the detectability lemma, avoiding Lindbladian simulation and achieving significant speedups in spectral gap dependence.

Contribution

It develops new Gibbs state preparation techniques that bypass Lindbladian simulation and combines the detectability lemma with quantum singular value transformation for efficiency gains.

Findings

Reduces Gibbs state preparation cost by a factor of O(M) for local Lindbladians.

Achieves quadratic speedup in spectral gap dependence for ground state projection.

Improves dependence on Lindbladian spectral gap for local commuting Hamiltonians.

Abstract

Gibbs state preparation is an important subroutine in quantum computing. In this work we use the detectability lemma to improve Gibbs state preparation. Specifically, we design new Gibbs state preparation methods that do not rely on simulating Lindbladian evolution, thus avoiding the overhead from it. For local Lindbladians consisting of $M$ terms, this approach reduces the cost by a factor of $O(M)$. We also combine the detectability lemma operator and quantum singular value transformation to implement ground state projection operators of frustration-free Hamiltonians, resulting in a quadratic speedup in the spectral gap dependence. Applying this method to Lindbladians for the Gibbs state of local commuting Hamiltonians, we achieve quadratically better dependence on the Lindbladian spectral gap.