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

• URL: http://arxiv.org/abs/2009.12032v2
• Date: Tue, 4 May 2021 21:27:41 GMT
• Title: Bounding the finite-size error of quantum many-body dynamics simulations
• Authors: Zhiyuan Wang, Michael Foss-Feig, and Kaden R. A. Hazzard
• Abstract summary: We derive rigorous upper bounds on the Finite-size error (FSE) of local observables in real time quantum dynamics simulations from a product state. 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.
• Score: 6.657101721138396
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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