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

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 $|⟨\hat{S}(t){⟩}_L-⟨\hat{S}(t){⟩}_{\infty }|\le C(2vt/L{)}^{cL-\mu }$, 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.

References in corpus (15)

• The density-matrix renormalization group in the age of matrix product states
• Probing many-body dynamics on a 51-atom quantum simulator
• Many body localization and thermalization in quantum statistical mechanics
• Quantum Simulation of Antiferromagnetic Spin Chains in an Optical Lattice
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• New Trends in Density Matrix Renormalization
• Nearly-linear light cones in long-range interacting quantum systems
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Exploring local quantum many-body relaxation by atoms in optical superlattices
• Scaling of diffusion constants in the spin-1/2 XX ladder
• Relaxation dynamics of a Fermi gas in an optical superlattice
• Relaxation of the collective magnetization of a dense 3D array of interacting dipolar S=3 atoms
• Temporal decay of Neel order in the one-dimensional Fermi-Hubbard model
• The ground state of a class of noncritical 1D quantum spin systems can be approximated efficiently
• Dynamical crossover in the transient quench dynamics of short-range transverse field Ising models

Cited by in corpus (5)

• Bounds in Nonequilibrium Quantum Dynamics
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• Locality of gapped ground states in systems with power-law decaying interactions
• Spectroscopic evidence for engineered hadron formation in repulsive fermionic $\text{SU}(N)$ Hubbard Models