On the Trotter Error in Many-body Quantum Dynamics with Coulomb Potentials

TL;DR

This paper establishes the optimal $1/4$-order convergence rate of Trotterization for simulating many-body quantum systems with Coulomb interactions, with explicit polynomial dependence on particle number, without regularization or discretization.

Contribution

It provides the first rigorous proof of the $1/4$-order convergence rate for Coulomb Hamiltonians, addressing unbounded operators directly and unifying the analysis of many-body quantum simulation.

Findings

Trotterization achieves a $1/4$-order convergence rate for Coulomb systems.

The convergence rate is proven to be optimal based on prior numerical evidence.

The analysis applies to all initial wavefunctions in the Hamiltonian's domain, without regularization.

Abstract

Efficient simulation of many-body quantum systems is central to advances in physics, chemistry, and quantum computing, with a key question being whether the simulation cost scales polynomially with the system size. In this work, we analyze many-body quantum systems with Coulomb interactions, which are fundamental to electronic and molecular systems. We prove that Trotterization for such unbounded Hamiltonians achieves a $1/4$-order convergence rate, with explicit polynomial dependence on the number of particles. The result holds for all initial wavefunctions in the domain of the Hamiltonian, and the $1/4$-order convergence rate is optimal, as previous work has numerically demonstrated that it can be saturated by a specific initial ground state. The main challenges arise from the many-body structure and the singular nature of the Coulomb potential. Our proof strategy differs from prior state-of-the-art Trotter analyses, addressing both difficulties in a unified framework. Our analysis treats the Coulomb potential as an unbounded operator without modification or regularization, and does not rely on spatial discretization, making it compatible with both first- and second-quantized circuit constructions.