Bounding the finite-size error of quantum many-body dynamics simulations

TL;DR

This paper derives rigorous bounds on the finite-size errors in quantum many-body dynamics simulations, providing practical tools to assess the accuracy of finite system results in the thermodynamic limit.

Contribution

The authors present the first explicit, quantitative bounds on finite-size errors for local observables in real-time quantum dynamics, applicable to various boundary conditions and initial states.

Findings

Bound on FSE scales as (2vt/L)^{cL-μ} with explicit constants.

Periodic boundary conditions reduce FSE compared to open boundaries at early times.

FSE in ground state simulations decays exponentially with system size L.

Abstract

Finite-size error (FSE), the discrepancy between an observable in a finite system and in the thermodynamic limit, is ubiquitous in numerical simulations of quantum many body systems. Although a rough estimate of these errors can be obtained from a sequence of finite-size results, a strict, quantitative bound on the magnitude of FSE is still missing. Here we derive rigorous upper bounds on the FSE of local observables in real time quantum dynamics simulations initialized from a product state. In $d$-dimensional locally interacting systems with a finite local Hilbert space, our bound implies $ |\langle \hat{S}(t)\rangle_L-\langle \hat{S}(t)\rangle_\infty|\leq C(2v t/L)^{cL-\mu}$, with $v$, $C$, $c$, $\mu $ constants independent of $L$ and $t$, which we compute explicitly. For periodic boundary conditions (PBC), the constant $c$ is twice as large as that for open boundary conditions (OBC), suggesting that PBC have smaller FSE than OBC at early times. The bound can be generalized to a large class of correlated initial states as well. As a byproduct, we prove that the FSE of local observables in ground state simulations decays exponentially with $L$, under a suitable spectral gap condition. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.