Quantum algorithms for Gibbs sampling and hitting-time estimation

Abstract

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 $\sqrt{N β/{\cal Z}}$ and polynomial in $\log(1/ε)$, where $N$ is the Hilbert space dimension, $β$ is the inverse temperature, ${\cal Z}$ is the partition function, and $ε$ is the desired precision of the output state. Our quantum algorithm exponentially improves the dependence on $1/ε$ and quadratically 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 $1/(εΔ^{3/2})$, where $ε$ is the absolute precision in the estimation and $Δ$ is a parameter determined by $P$, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the dependence on $1/ε$ and $1/Δ$ 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.