Rapid mixing for Gibbs samplers for quantum spin systems with all-to-all interactions

Abstract

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 $1$, 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.