When is randomization advantageous in quantum simulation?

TL;DR

This paper investigates when randomization improves quantum Hamiltonian simulation, showing it reduces gate counts significantly in certain regimes but becomes less advantageous at very high precision.

Contribution

It introduces a sparse-QSVT method with stochastic decompositions and analyzes error propagation, providing insights into when randomization is beneficial.

Findings

Randomized methods reduce gate counts by up to an order of magnitude for certain Hamiltonians.

Deterministic methods become more efficient as the target error decreases below approximately 10^{-3}.

Randomization benefits are most significant in moderate-precision regimes, especially for Hamiltonians with many terms and inhomogeneous coefficients.

Abstract

We study the regimes in which Hamiltonian simulation benefits from randomization. We introduce a sparse-QSVT construction based on composite stochastic decompositions, where dominant terms are treated deterministically and smaller contributions are sampled stochastically. Crucially, we analyze how stochastic and approximation errors propagate through block-encoding and QSVT procedures. To benchmark this approach, we construct ensembles of random Hamiltonians with controlled coefficient dispersion, locality, and number of terms, designed to favor randomization, and therefore providing an upper bound on its practical advantage. For Hamiltonians with many terms and highly inhomogeneous coefficient distributions, randomized methods reduce gate counts by up to an order of magnitude. However, this advantage is confined to moderate-precision regimes: as the target error decreases, deterministic methods become more efficient, with a crossover near $\varepsilon \sim 10^{-3}$. Although this regime partially overlaps with quantum chemistry Hamiltonians, realistic systems exhibit additional structure, such as commutation patterns, not captured by our model, which are expected to further favor deterministic approaches.