Rapid Mixing of Quantum Gibbs Samplers for Weakly-Interacting Quantum Systems
Štěpán Šmíd, Richard Meister, Mario Berta, Roberto Bondesan
TL;DR
This work proves rapid mixing for algorithmic Gibbs samplers across weakly interacting quantum spin, fermionic, and bosonic systems, using tailor-made oscillator-norm techniques to bound convergence to the Gibbs state $\sigma_\beta = e^{-\beta H}/Z$. It provides explicit, system-size-dependent mixing bounds and perturbation thresholds, yielding end-to-end quantum Gibbs-state preparation with $\tilde{O}(n^2)$ Hamiltonian-simulation time and extending rapid mixing to non-commuting qudit models and bosonic Lindbladians at arbitrary temperature. The results offer the first efficient mixing bounds for broad classes of non-commuting and bosonic systems and establish explicit maximal interaction strengths $\lambda_{\max}$ under which rapid mixing persists, improving robustness to noise and potentially enabling practical quantum advantages in thermal-state preparation. The methodology centers on a problem-specific oscillator-norm framework that can be adapted to other Lindbladians, providing a versatile route to rigorous end-to-end complexity analyses of quantum simulation tasks.
Abstract
Dissipative quantum algorithms for state preparation in many-body systems are increasingly recognised as promising candidates for achieving large quantum advantages in application-relevant tasks. Recent advances in algorithmic, detailed-balance Lindbladians enable the efficient simulation of open-system dynamics converging towards desired target states. However, the overall complexity of such schemes is governed by system-size dependent mixing times. In this work, we analyse algorithmic Lindbladians for Gibbs state preparation and prove that they exhibit rapid mixing, i.e., convergence in time poly-logarithmic in the system size. We first establish this for non-interacting spin systems, free fermions, and free bosons, and then show that these rapid mixing results are stable under perturbations, covering weakly interacting qudits and perturbed non-hopping fermions. Our results constitute the first efficient mixing bounds for non-commuting qudit models and bosonic systems at arbitrary temperatures. Compared to prior spectral-gap-based results for fermions, we achieve exponentially faster mixing, further featuring explicit constants on the maximal allowed interaction strength. This not only improves the overall polynomial runtime for quantum Gibbs state preparation, but also enhances robustness against noise. Our analysis relies on oscillator norm techniques from mathematical physics, where we introduce tailored variants adapted to specific Lindbladians $\unicode{x2014}$ an innovation that we expect to significantly broaden the scope of these methods.