Average-case Speedup for Product Formulas

TL;DR

This paper demonstrates that product formulas for quantum simulation perform significantly better on average than worst-case estimates suggest, especially for large, complex systems, by analyzing typical-case error scaling.

Contribution

It provides the first average-case analysis of Trotter error, showing improved gate complexity bounds for most input states and broad Hamiltonian classes.

Findings

Average-case Trotter error scales better than worst-case for most states.

Gate complexity improves for large, highly connected systems.

Results extend to fermionic Hamiltonians and SYK models.

Abstract

Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a local energy estimate. In this work, we give evidence that product formulas, in practice, may work much better than expected. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states, while the existing estimate is for the worst states. For general $k$-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from any orthogonal basis. The gate complexity significantly improves over the worst case for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms, with input states drawn from a fixed-particle number subspace, and with Gaussian coefficients (e.g., the SYK models). Technically, we employ a family of simple but versatile inequalities from non-commutative martingales called $\textit{uniform smoothness}$, which leads to $\textit{Hypercontractivity}$, namely $p$-norm estimates for $k$-local operators. This delivers concentration bounds via Markov's inequality. For optimality, we give analytic and numerical examples that simultaneously match our typical-case estimates and the existing worst-case estimates. Therefore, our improvement is due to asking a qualitatively different question, and our results open doors to the study of quantum algorithms in the average case.