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

Abstract

Finite-size errors, the discrepancy between an observable in a finite-size 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 finite-size error is still missing. Here we derive rigorous upper bounds on the finite-size error of local observables in real time quantum dynamics simulations initiated from a product state and evolved under a general Hamiltonian in arbitrary spatial dimension. In locally interacting systems with a finite local Hilbert space, our bound implies | Ŝ(t) L − Ŝ(t) ∞| ≤ C(2vt/L) cL−µ , with v the Lieb-Robinson (LR) speed and C, c, µ 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 finite-size error than OBC at early times. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional quantum Ising and Fermi-Hubbard models.