Quantum Gibbs Sampling in Infinite Dimensions: Generation, Mixing Times and Circuit Implementation

TL;DR

This paper introduces a rigorous, implementable framework for Gibbs sampling in infinite-dimensional quantum systems, addressing fundamental obstacles and establishing convergence results with practical implications.

Contribution

It extends dissipative Gibbs samplers to infinite dimensions using Dirichlet forms and spectral analysis, balancing implementability and convergence guarantees.

Findings

Constructed KMS-symmetric quantum Markov semigroups on separable Hilbert spaces.

Established quantitative convergence results, including fast thermalization regimes.

Identified Hamiltonians where implementability compromises convergence.

Abstract

We develop a rigorous and implementable framework for Gibbs sampling of infinite-dimensional quantum systems governed by unbounded Hamiltonians. Extending dissipative Gibbs samplers beyond finite dimensions raises fundamental obstacles, including ill-defined generators, the absence of spectral gaps on natural Banach spaces, and tensions between implementability and convergence guarantees. We overcome these issues by constructing KMS-symmetric quantum Markov semigroups on separable Hilbert spaces that are both well-posed and efficiently implementable on qubit hardware. Our generation theory is based on the abstract framework of Dirichlet forms, adapted here to the case of algebras of bounded operators over separable Hilbert spaces. Leveraging the spectral properties of our self-adjoint generators, we establish quantitative convergence results in trace distance, including regimes of fast thermalization. In contrast, we also identify Hamiltonians for which a naive choice of generators guaranteeing implementability generally comes at the cost of losing convergence of the associated evolutions, thereby establishing a strong trade-off between implementability and convergence. Our framework applies to a wide class of models, including Schr\"odinger operators, Gaussian systems, and Bose-Hubbard Hamiltonians, and provides a unified approach linking rigorous infinite-dimensional analysis with algorithmic Gibbs state preparation.