Matrix product state approximations to quantum states of low energy variance

TL;DR

This paper demonstrates how tensor network algorithms can efficiently approximate low-energy quantum states with small energy variance in one-dimensional systems, revealing states with narrow spectral support and moderate entanglement.

Contribution

It introduces a method to produce matrix product states with decreasing energy variance as bond dimension grows, and proves the existence of low-variance states with moderate entanglement.

Findings

Variance as small as 1/log N achievable with polynomial bond dimension

States with narrow spectral support can have moderate entanglement entropy

Provides a simplified proof and improved bounds for the Berry-Esseen theorem in spin systems

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.