Quantum algorithms for Gibbs sampling and hitting-time estimation

We present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in p Nβ/Z and polynomial in log(1/epsilon), where N is the Hilbert space dimension, β is the inverse temperature, Z is the partition function, and epsilon is the desired precision of the output state. Our quantum algorithm exponentially improves the complexity dependence on 1/epsilon and polynomially improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix P, it runs in time almost linear in 1/(epsilon ∆3/2 ), where epsilon is the absolute precision in the estimation and ∆ is a parameter determined by P, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the complexity dependence on 1/epsilon and 1/∆ of the analog classical algorithm for hitting-time estimation. Both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.