Importance sampling for stochastic quantum simulations

TL;DR

This paper enhances stochastic quantum simulation methods by integrating importance sampling into the qDrift protocol, reducing simulation costs while maintaining accuracy, and providing rigorous bounds on bias and variance.

Contribution

It unifies importance sampling with qDrift, enabling more efficient quantum simulations with controlled bias and variance, and offers practical guidelines for implementation.

Findings

Reduced simulation cost for the same accuracy.

Rigorous bounds on bias and variance established.

Numerical validation on lattice nuclear effective field theory.

Abstract

Simulating many-body quantum systems is a promising task for quantum computers. However, the depth of most algorithms, such as product formulas, scales with the number of terms in the Hamiltonian, and can therefore be challenging to implement on near-term, as well as early fault-tolerant quantum devices. An efficient solution is given by the stochastic compilation protocol known as qDrift, which builds random product formulas by sampling from the Hamiltonian according to the coefficients. In this work, we unify the qDrift protocol with importance sampling, allowing us to sample from arbitrary probability distributions, while controlling both the bias, as well as the statistical fluctuations. We show that the simulation cost can be reduced while achieving the same accuracy, by considering the individual simulation cost during the sampling stage. Moreover, we incorporate recent work on composite channel and compute rigorous bounds on the bias and variance, showing how to choose the number of samples, experiments, and time steps for a given target accuracy. These results lead to a more efficient implementation of the qDrift protocol, both with and without the use of composite channels. Theoretical results are confirmed by numerical simulations performed on a lattice nuclear effective field theory.