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

Sergey Bravyi1, Giuseppe Carleo2, David Gosset3,4, and Yinchen Liu3,4

1IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, USA
2École Polytechnique Fédérale de Lausanne (EPFL), Institute of Physics, CH-1015 Lausanne, Switzerland
3Department of Combinatorics and Optimization and Institute for Quantum Computing, University of Waterloo
4Perimeter Institute for Theoretical Physics, Waterloo, Canada

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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.

We show how to map a quantum $k$-local Hamiltonian $H$ with the unique ground state $\psi$ to a continuous-time Markov Chain with the unique steady distribution describing the measurement of $\psi$ in the standard basis. The Markov chain is rapidly mixing whenever the spectral gap of $H$ is at least inverse polynomial in the system size. 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 show that for certain Hamiltonians our Markov Chain gives an efficient classical algorithm for sampling the ground state probability distribution.

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Cited by

[1] Massimo Bortone, Yannic Rath, and George H. Booth, "Impact of conditional modelling for a universal autoregressive quantum state", Quantum 8, 1245 (2024).

[2] Jiaqing Jiang, "Local Hamiltonian Problem with succinct ground state is MA-Complete", arXiv:2309.10155, (2023).

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