Quantum algorithms for Gibbs sampling and hitting-time estimation

TL;DR

This work introduces two quantum algorithms for stochastic-process problems: Gibbs-state preparation and hitting-time estimation. The Gibbs algorithm achieves a running time of $\tilde{O}(\sqrt{N \beta/\mathcal{Z}})$ by approximating $e^{-eta H/2}$ as a finite linear combination of evolutions under a square-root Hamiltonian $\tilde{H}$ via a Hubbard–Stratonovich transformation, leveraging spectral-gap amplification and advanced Hamiltonian simulation. The hitting-time algorithm estimates the expected time to reach a marked configuration with complexity $\tilde{O}(1/(\varepsilon \Delta^{3/2}))$ by encoding $1/H$ as a linear combination of unitaries derived from $\tilde{H}$ and using amplitude estimation to obtain the result. Together, these results provide exponential improvement in $1/\varepsilon$ and quadratic improvement in $\beta$ for Gibbs-state sampling, and near-quadratic improvements in key parameters for hitting-time estimation, highlighting the potential of quantum techniques in statistical physics and randomized algorithms. The methods integrate Hubbard–Stratonovich decompositions, LCU, Hamiltonian simulation, spectral-gap amplification, and quantum metrology to achieve these speedups.

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.

Theorems & Definitions (12)

• Definition 1
• Theorem 1
• Lemma 1
• proof
• Corollary 1
• proof
• Theorem 2
• Corollary 2
• Lemma 2
• proof