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 (73)

• 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
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Quantum many-body systems out of equilibrium
• Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas
• Exploring the many-body localization transition in two dimensions
• Engineered 2D Ising interactions on a trapped-ion quantum simulator with hundreds of spins
• Quantum Simulation of Antiferromagnetic Spin Chains in an Optical Lattice
• Realizing a lattice spin model with polar molecules
• Non-local propagation of correlations in long-range interacting quantum systems
• Experimental realization of a long-range antiferromagnet in the Hubbard model with ultracold atoms
• Realizing quantum Ising models in tunable two-dimensional arrays of single Rydberg atoms
• Quantum phase transitions
• Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry
• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Dynamics of quantum phase transition: exact solution in quantum Ising model
• Universal digital quantum simulation with trapped ions
• Spectral Gap and Exponential Decay of Correlations
• Quantum Simulators: Architectures and Opportunities
• Toward the first quantum simulation with quantum speedup
• Fermi-Hubbard physics with atoms in an optical lattice
• Thermalization and prethermalization in isolated quantum systems: a theoretical overview
• Digitized adiabatic quantum computing with a superconducting circuit
• Spin and Charge Resolved Quantum Gas Microscopy of Antiferromagnetic Order in Hubbard Chains
• Studying Two Dimensional Systems With the Density Matrix Renormalization Group
• Many-body interferometry of a Rydberg-dressed spin lattice
• New Trends in Density Matrix Renormalization
• Site-resolved measurement of the spin-correlation function in the Hubbard model
• Extended slow dynamical regime close to the many-body localization transition
• The Ergodic Side of the Many-Body Localization Transition
• Bad metallic transport in a cold atom Fermi-Hubbard system
• Non-equilibrium quantum magnetism in a dipolar lattice gas
• Probing Slow Relaxation and Many-Body Localization in Two-Dimensional Quasi-Periodic Systems
• Nearly-linear light cones in long-range interacting quantum systems
• Many-body dynamics of dipolar molecules in an optical lattice
• Probing quench dynamics across a quantum phase transition into a 2D Ising antiferromagnet
• Dynamical typicality of quantum expectation values
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Pushing the limits of the eigenstate thermalization hypothesis towards mesoscopic quantum systems
• Observing the space- and time-dependent growth of correlations in dynamically tuned synthetic Ising antiferromagnets
• Exploring local quantum many-body relaxation by atoms in optical superlattices
• Locality of temperature
• Spin-current autocorrelations from single pure-state propagation
• Persistence of locality in systems with power-law interactions
• Finite speed of quantum scrambling with long range interactions
• Observation of a Transition Between Dynamical Phases in a Quantum Degenerate Fermi Gas
• Strictly linear light cones in long-range interacting systems of arbitrary dimensions
• Emergence of multi-body interactions in few-atom sites of a fermionic lattice clock
• Probing the Bose-Glass--Superfluid Transition using Quantum Quenches of Disorder
• Regression relation for pure quantum states and its implications for efficient computing
• Hierarchy of linear light cones with long-range interactions
• Simulating adiabatic evolution of gapped spin systems
• Relaxation of an isolated dipolar-interacting Rydberg quantum spin system
• Many-body localization, symmetry, and topology
• Nonequilibrium dynamics in the one-dimensional Fermi-Hubbard model: A comparison of the nonequilibrium Green functions approach and the density matrix renormalization group method
• Locality and digital quantum simulation of power-law interactions
• Direct observation of ultrafast many-body electron dynamics in an ultracold Rydberg gas
• Extending rotational coherence of interacting polar molecules in a spin-decoupled magic trap
• Simulating Bosonic Baths with Error Bars
• Tightening the Lieb-Robinson Bound in Locally-Interacting Systems
• Lightcone renormalization and quantum quenches in one-dimensional Hubbard models
• Scaling of diffusion constants in the spin-1/2 XX ladder
• Non-thermal fixed points: Universal dynamics far from equilibrium
• Relaxation dynamics of a Fermi gas in an optical superlattice
• Dynamical error bounds for continuum discretisation via Gauss quadrature rules, -- a Lieb-Robinson bound approach
• Operator growth bounds from graph theory
• Size-Driven Quantum Phase Transitions
• Relaxation of the collective magnetization of a dense 3D array of interacting dipolar S=3 atoms
• Optimization of finite-size errors in finite-temperature calculations of unordered phases
• 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
• Bound states and photon emission in non-Hermitian nanophotonics
• Detecting quantum phase transitions in the quasi-stationary regime of Ising chains
• 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