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.

24 pages, 3 figures, Accepted in Quantum

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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