A rapidly mixing Markov chain from any gapped quantum many-body system

TL;DR

This paper establishes a connection between the spectral gap of local Hamiltonians in quantum many-body systems and the mixing time of a corresponding Markov chain, enabling efficient sampling of ground states.

Contribution

It introduces a method to construct rapidly mixing Markov chains from gapped quantum systems using the fixed-node Hamiltonian approach, extending previous sign-problem-free results.

Findings

The Markov chain mixes more rapidly than standard methods in numerical experiments.

The approach is efficient when ground state ratios are computable and the spectral gap is inverse polynomial.

Numerical sampling of the Haldane-Shastry Hamiltonian with up to 56 qubits demonstrates practical effectiveness.

Abstract

We consider the computational task of sampling a bit string $x$ from a distribution $\pi(x)=|\langle x|\psi\rangle|^2$, where $\psi$ is the unique ground state of a local Hamiltonian $H$. Our main result describes a direct link between the inverse spectral gap of $H$ and the mixing time of an associated continuous-time Markov Chain with steady state $\pi$. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes $\langle y|\psi\rangle/\langle x|\psi\rangle$ are efficiently computable, the spectral gap of $H$ is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in Quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed-node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain.