Entanglement-Dependent Error Bounds for Hamiltonian Simulation

TL;DR

This paper links entanglement entropy to Hamiltonian simulation error bounds, showing that structured low-entanglement systems require significantly fewer resources for accurate quantum simulation.

Contribution

It provides the first entanglement-dependent error bounds for Trotter formulas, improving resource estimates for structured quantum systems and establishing fundamental entanglement-based separation results.

Findings

Error bounds scale with entanglement entropy $S_{max}$ instead of system size $n$.

One-dimensional area-law systems require $ ilde{ ext{O}}(n^2)$ fewer Trotter steps.

Volume-law entangled systems need $ ilde{ ext{O}}(n)$ more steps than area-law systems.

Abstract

We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy $S_\text{max}$ across all bipartitions, we prove that the first-order Trotter error scales as $\mathcal{O}(t^2 S_\text{max} \operatorname{polylog}(n)/r)$ rather than the worst-case $\mathcal{O}(t^2 n/r)$, where $n$ is the system size and $r$ is the number of Trotter steps. This yields improvements of $\tilde{\Omega}(n^2)$ for one-dimensional area-law systems and $\tilde{\Omega}(n^{3/2})$ for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves $2^{pS^*/2}$ for the $p$-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require $\tilde{\Omega}(n)$ more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.