[Paper Review] Accelerating variational quantum Monte Carlo using the variational quantum eigensolver

Ashley Montanaro, Stasja Stanisic|arXiv (Cornell University)|Jul 15, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes quantum-enhanced variational Monte Carlo (QEVMC), a hybrid quantum-classical method that uses noisy, near-term quantum computer samples—generated via the variational quantum eigensolver (VQE)—as initial distributions for variational Monte Carlo (VMC) simulations. The key contribution is that VQE-generated initial states accelerate convergence to the target quantum distribution and reduce energy compared to classical initial states, even when the VQE state itself is not exact, demonstrating practical speedups on noisy hardware.

ABSTRACT

Variational Monte Carlo (VMC) methods are used to sample classically from distributions corresponding to quantum states which have an efficient classical description. VMC methods are based on performing a number of steps of a Markov chain starting with samples from a simple initial distribution. Here we propose replacing this initial distribution with samples produced using a quantum computer, for example using the variational quantum eigensolver (VQE). We show that, based on the use of initial distributions generated by numerical simulations and by experiments on quantum hardware, convergence to the target distribution can be accelerated compared with classical samples; the energy can be reduced compared with the energy of the state produced by VQE; and VQE states produced by small quantum computers can be used to accelerate large instances of VMC. Quantum-enhanced VMC makes minimal requirements of the quantum computer and offers the prospect of accelerating classical methods using noisy samples from near-term quantum computers which are not yet able to accurately represent ground states of complex quantum systems.

Motivation & Objective

• To address slow convergence in classical variational Monte Carlo (VMC) due to poor initial distributions.
• To explore whether near-term quantum computers can accelerate classical quantum simulation methods like VMC.
• To demonstrate that noisy, imperfect quantum states from VQE can still enhance classical sampling efficiency.
• To validate that quantum-enhanced initial states improve MCMC convergence and energy estimation, even when the VQE state is not the true ground state.

Proposed method

• Replace the classical initial distribution in VMC with samples from a quantum computer, specifically from the variational quantum eigensolver (VQE).
• Use VQE to prepare a quantum state that approximates the ground state of a target Hamiltonian, even with noise and limited qubit count.
• Store the resulting quantum samples and use them as the starting distribution for a Markov chain Monte Carlo (MCMC) process in VMC.
• Apply standard Metropolis-Hastings MCMC sampling with the VQE-generated distribution as the initial state, without further quantum access after sampling.
• Evaluate performance via convergence speed to the target distribution and energy reduction compared to classical initial states.
• Test the method on two benchmark models: the Fermi-Hubbard model with Gutzwiller wavefunction and the transverse-field Ising model with neural network quantum states.

(a) $1\times 16$

Experimental results

Research questions

• RQ1 Can VQE-generated initial states accelerate convergence in variational Monte Carlo simulations?
• RQ2 Does using quantum samples as initial distributions lead to lower energy estimates than the original VQE state?
• RQ3 Can small-scale VQE on a noisy quantum computer accelerate VMC for larger systems beyond the quantum computer’s capacity?
• RQ4 Is the speedup due to better initial proximity to the target distribution or higher MCMC acceptance rates?

Key findings

• QEVMC accelerates convergence to the target distribution compared to classical initial distributions, with speedup factors observed in both the Fermi-Hubbard and transverse-field Ising models.
• VMC applied to VQE samples reduces the energy below that of the original VQE state, indicating improvement through classical sampling.
• Small quantum computers can generate VQE states that accelerate VMC for larger systems, demonstrating scalability beyond the quantum hardware’s direct simulation capacity.
• The speedup is primarily due to the VQE initial distribution being closer to the target distribution, not higher MCMC acceptance rates, as acceptance rates were slightly lower with VQE samples.
• QEVMC is resilient to noise: even with mixed states (e.g., 10% maximally mixed component), the method maintains performance close to ideal VQE states.
• The method enables practical use of near-term quantum hardware for accelerating classical quantum simulation, with minimal quantum access required after initial sampling.

(b) $1\times 24$