Unusual energy spectra of matrix product states

TL;DR

This paper investigates the unusual energy spectra of approximate matrix product states, revealing persistent high-energy tails that impact sampling methods, and proposes an effective variance extrapolation technique.

Contribution

It uncovers the persistent high-energy tails in the spectra of approximate matrix product states and neural network wavefunctions, and introduces a novel variance extrapolation method.

Findings

High-energy tails are common in approximate wavefunctions regardless of boundary or dimensionality.

Increasing bond dimension reduces but does not eliminate high-energy contributions.

The proposed variance extrapolation method effectively mitigates sampling fluctuations.

Abstract

In approximate ground states obtained from imaginary-time evolution, the spectrum of the state -- its decomposition into exact energy eigenstates -- falls off exponentially with the energy. Here we consider the energy spectra of approximate matrix product ground states, such as those obtained with the density matrix renormalization group. Despite the high accuracy of these states, contributions to the spectra are roughly constant out to surprisingly high energy, with an increase in the bond dimension reducing the amplitude but not the extent of these high-energy tails. The unusual spectra appear to be a general feature of compressed wavefunctions, independent of boundary or dimensionality, and are also observed in neural network wavefunctions. The unusual spectra can have a strong effect on sampling-based methods, yielding large fluctuations. The energy variance, which can be used to extrapolate observables to eliminate truncation error, is subject to these large fluctuations when sampled. Nevertheless, we devise a sampling-based variance approach which gives excellent and efficient extrapolations.