Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Di Fang, Xiaoxu Wu, Avy Soffer

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.

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

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 -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 -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.

Paper Structure

This paper contains 13 sections, 15 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Main Results and Proof Idea Overview
  3. Problem Setup and Main Results
  4. Challenges and Proof Strategies
  5. One-body Intuition
  6. $N$-body Solution Norm Estimate
  7. $N$-body Trotter Error Estimate
  8. Trotter Error Estimates
  9. Proof of \ref{['thm Tt']} and \ref{['thm Tt-Hpsi-ver']}
  10. Conclusion and Remarks
  11. Auxiliary Estimates for the One-Body Problem
  12. Auxiliary Estimates for the $N$-Body Problem
  13. Proof of the estimate \ref{['Qineq']}

Key Result

Theorem 1

Let $H = A + B$ be the $N$-body Hamiltonian with Coulomb interactions given by N-SEeq:N-V_defcon: c0, where $A = -\Delta$ denotes the kinetic part and $B = V(x)$ the Coulomb interaction potential. Then for any initial state $\psi_0 \in H^2$, the long-time Trotter error for a total evolution time $T> where $t = T/L$ is the short-time Trotter step size, and $\tilde{C}>0$ is a universal constant depe

Theorems & Definitions (29)

  • Theorem 1: Long-time Trotter Error
  • Theorem 2
  • Lemma 3
  • Theorem 4: One-body Short-time Trotter Error
  • Lemma 5: Energy estimate -- Step 1
  • Lemma 6: Commutator estimate -- Step 2
  • Theorem 7
  • Lemma 8
  • proof : Proof of \ref{['thmN']}
  • proof : Proof of \ref{['Key lem: N-body est']}
  • ...and 19 more