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  • Letter
  • Open Access

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

Zhiyuan Wang1,2,*, Michael Foss-Feig3, and Kaden R. A. Hazzard1,2

  • 1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
  • 2Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
  • 3Honeywell Quantum Solutions, 303 S. Technology Ct., Broomfield, Colorado 80021, USA

  • *zhiyuan.wang@rice.edu
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Phys. Rev. Research 3, L032047 – Published 16 August, 2021

DOI: https://doi.org/10.1103/PhysRevResearch.3.L032047

Abstract

Finite-size errors (FSEs), the discrepancies between an observable in a finite system and in the thermodynamic limit, are 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 |〈Ŝ(t)〉L−〈Ŝ(t)〉∞|≤C(2vt/L)cL−μ, with v, C, c, μ constants independent of L and t, which we compute explicitly. For periodic boundary conditions (PBCs), the constant c is twice as large as that for open boundary conditions (OBCs), suggesting that PBCs have smaller FSEs than OBCs 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.

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Phys. Rev. Research 3, L032047– Published 16 August, 2021
  • Received 24 September 2020
  • Revised 4 May 2021
  • Accepted 27 July 2021
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