Bounding the finite-size error of quantum many-body dynamics simulations
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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 ∣∣⟨ˆS(t)⟩L−⟨ˆS(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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Wang, Zhiyuan, Foss-Feig, Michael and Hazzard, Kaden R.A.. "Bounding the finite-size error of quantum many-body dynamics simulations." Physical Review Research, 3, no. 3 (2021) American Physical Society: https://doi.org/10.1103/PhysRevResearch.3.L032047.