Convergence monitoring of quantum Gibbs samplers

Preparing Gibbs states is a fundamental task in quantum simulation, with applications ranging from many-body physics to Bayesian inference. While quantum Gibbs samplers (QGSs) have emerged as efficient tools for this task, determining when these algorithms have successfully reached thermal equilibrium remains a significant practical challenge. Unlike classical Markov chain Monte Carlo (MCMC), where practitioners rely on empirical diagnostics like trace plots, direct measurement of a quantum system to check for convergence is costly because it disturbs the state and requires restarting the simulation.

This paper introduces a convergence-monitoring criterion that avoids the overhead of auxiliary measurements. The approach exploits the weak-measurement record already produced by the Lindbladian simulation of QGSs. In these samplers, the system interacts with an environment via jump operators that exchange energy. At thermal equilibrium, the system satisfies detailed balance, which implies that the net energy flow vanishes. The authors demonstrate that this physical balance manifests as a specific symmetry in the distribution of quasi-frequencies (the energy-exchange events) recorded during the simulation.

Because the equilibrium symmetry of these quasi-frequencies is determined by the algorithm's design parameters—specifically the inverse temperature and the frequency resolution—rather than the system's Hamiltonian, the authors can define a precise target for the frequency distribution. The proposed stopping rule monitors the mean and the third centered moment of the recorded quasi-frequencies. It stops the simulation when the batch-means confidence ellipsoid of these statistics is contained within a predefined tolerance region. This procedure is computationally efficient, as it processes data already generated by the sampler, and it is compatible with both standard and qubit-efficient quantum Gibbs sampling architectures.