Better bounds for low-energy product formulas

TL;DR

This paper improves the efficiency of quantum simulation using product formulas by deriving tighter error bounds for low-energy states, leading to potentially reduced runtime in practical quantum computations.

Contribution

It introduces low-energy state-specific error bounds for product formulas, enabling more efficient quantum simulations under certain conditions.

Findings

Tighter error bounds for low-energy states improve simulation accuracy.

Simulation efficiency is asymptotically enhanced for low-energy states.

Error bounds based on low-energy subspaces outperform traditional operator norm bounds.

Abstract

Product formulas are one of the main approaches for quantum simulation of the Hamiltonian dynamics of a quantum system. Their implementation cost is computed based on error bounds which are often pessimistic, resulting in overestimating the total runtime. In this work, we rigorously consider the error induced by product formulas when the state undergoing time evolution lies in the low-energy sector with respect to the Hamiltonian of the system. We show that in such a setting, the usual error bounds based on the operator norm of nested commutators can be replaced by those restricted to suitably chosen low-energy subspaces, yielding tighter error bounds. Furthermore, under some locality and positivity assumptions, we show that the simulation of generic product formulas acting on low-energy states can be done asymptotically more efficiently when compared with previous results.