Quantum Thermal State Preparation*

Preparing ground states and thermal states is essential for simulating quantum systems on quantum computers. Despite the hope for practical quantum advantage in quantum simulation, popular state preparation approaches have been challenged. Monte Carlo-style quantum Gibbs samplers have emerged as an alternative, but prior proposals have been unsatisfactory due to technical obstacles rooted in energy-time uncertainty.

We introduce simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating Nature-inspired quantum master equations (Lindbladians). Additionally we complete the first rigorous proof of finite-time thermalization for physically derived Lindbladians by developing a general analytic framework for nonasymptotic secular approximation and approximate detailed balance.

We also construct the first efficiently implementable and exactly detailed-balanced Lindbladian for arbitrary noncommutative Hamiltonians. To prepare the quantum Gibbs state, the resulting algorithm invokes Hamiltonian simulation for a time proportional to the mixing time and the inverse temperature β, up to polylogarithmic factors. In particular, the gate complexity reduces significantly for lattice Hamiltonians as the Lindblad operators are (quasi-)local (with radius about β) and only depend on local Hamiltonian patches. Meanwhile, purifying our Lindbladian yields a temperature-dependent family of frustration-free "parent Hamiltonians", prescribing an adiabatic path for the canonical purified Gibbs state (called the Thermal Field Double state in high-energy physics).

These favorable features suggest that our construction is the ideal quantum algorithmic counterpart of classical Markov chain Monte Carlo (MCMC) sampling; in particular it can be viewed as a noncommutative continuous-time counterpart of the classical Metropolis algorithm. Given the success of classical MCMC algorithms and the ubiquity of thermodynamics, we anticipate that quantum Gibbs sampling will become indispensable in quantum computing.