Bounding the finite-size error for simulations of quantum many-body systems

Finite size errors are ubiquitous in numerical simulations of quantum many body systems, and an estimation of these errors is crucial to the assessment of the reliability of their results. In this talk I present rigorous upper bounds on finite size error of local observables measured in gapped ground states of locally-interacting systems, as well as in real time quench dynamics simulations initiated from a product state. The key step of our method relies on the well-known Lieb-Robinson (LR) bound, which is a direct consequence of locality. We show that the error bounds are practically useful, enabled by the recent tightening of the LR bounds [1]. For example, in a ground state simulation of the transverse field Ising model with J=1, h=2 and L=25 sites, the relative error for a center site observable is less than 2% and decays exponentially with system size. In a quench dynamics simulation of the same model, the relative error bound remains to be within 1% up to the time at which the system equilibrates, and decays superexponentially with system size at fixed time.

References:
[1] Z. Wang and K. R. A. Hazzard, arXiv:1908.03997 (2019)