Matrix product state approximations to quantum states of low energy variance

Abstract

We show how to efficiently simulate pure quantum states in one dimensional systems that have both finite energy density and vanishingly small energy fluctuations. We do so by studying the performance of a tensor network algorithm that produces matrix product states whose energy variance decreases as the bond dimension increases. Our results imply that variances as small as $\propto 1/\log N$ can be achieved with polynomial bond dimension. With this, we prove that there exist states with a very narrow support in the bulk of the spectrum that still have moderate entanglement entropy, in contrast with typical eigenstates that display a volume law. Our main technical tool is the Berry-Esseen theorem for spin systems, a strengthening of the central limit theorem for the energy distribution of product states. We also give a simpler proof of that theorem, together with slight improvements in the error scaling, which should be of independent interest.

arXiv comment24 pages, 3 figures, Accepted in Quantum

References in corpus (26)

• Chaos and Quantum Thermalization
• Area laws for the entanglement entropy - a review
• Entanglement in quantum critical phenomena
• Average Entropy of a Subsystem
• Quantum many-body scars
• Criticality, the area law, and the computational power of PEPS
• Matrix product operator representations
• The computational complexity of PEPS
• Weak Ergodicity Breaking and Quantum Many-Body Scars in Spin-1 XY Magnets
• Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Algorithms for quantum simulation at finite energies
• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Entanglement-ergodic quantum systems equilibrate exponentially well
• Revivals imply quantum many-body scars
• Optimal Quantum Thermometry with Coarse-grained Measurements
• Classical simulation of short-time quantum dynamics
• Gaussian quantum fluctuations in interacting many particle systems
• Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems
• Canonical Universality
• Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems
• How much entanglement is needed to reduce the energy variance?
• Probing finite-temperature observables in quantum simulators of spin systems with short-time dynamics
• Classical algorithms for many-body quantum systems at finite energies
• Approximating the long time average of the density operator: Diagonal ensemble
• Connecting probability distributions of different operators and generalization of the Chernoff-Hoeffding inequality

Cited by in corpus (7)

• Energy-filtered quantum states and the emergence of non-local correlations
• Typical thermalization of low-entanglement states
• Out-of-equilibrium Eigenstate Thermalization Hypothesis
• Preparing low-variance states using a distributed quantum algorithm
• High-temperature partition functions and classical simulatability of long-range quantum systems
• Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians
• Imaginary Time Spectral Transforms for Excited State Preparation