Preparing ground states and thermal states is of key importance to simulating quantum systems on a quantum computer. Despite the hope for practical quantum advantage in quantum simulation, popular approaches like variational circuits or adiabatic algorithms appear to face serious difficulties. Monte-Carlo style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles related to energy-time uncertainty. We introduce [1] simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum Master Equations (Liouvillians) utilizing the operator Fourier transform. In addition, we construct the first provably accurate and efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems; this algorithm also benefits from a Szegedy-type quadratic improvement with respect to the mixing time. Our algorithms' cost has a favorable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Liouvillians. We contribute to the theory of thermalization by developing a general analytic framework that handles energy uncertainty through non-asymptotic secular approximation and approximate detailed balance, establishing our approximation guarantees and, as a byproduct yielding the first rigorous proof of finite-time thermalization for physically derived Liouvillians. Given the success of the classical Metropolis algorithm and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become an indispensable tool in quantum computing.

 

[1]: C-F Chen, MJ Kastoryano, FGSL Brandão, A Gilyen: https://arxiv.org/abs/2303.18224