Adaptive variational quantum eigensolvers for highly excited states

TL;DR

This paper introduces an adaptive variational quantum algorithm, adaptive VQE-X, designed to approximate highly excited states in quantum many-body systems, demonstrating its effectiveness on spin chains and analyzing its scaling behavior.

Contribution

The paper proposes a novel adaptive variational algorithm, adaptive VQE-X, for approximating highly excited states, and benchmarks its performance against existing methods in spin chain models.

Findings

Adaptive VQE-X effectively approximates highly excited states in spin chains.

Including long-range two-body gates accelerates convergence.

An exponential number of parameters may be needed for accurate state approximation.

Abstract

Highly excited states of quantum many-body systems are central objects in the study of quantum dynamics and thermalization that challenge classical computational methods due to their volume-law entanglement content. In this work, we explore the potential of variational quantum algorithms to approximate such states. We propose an adaptive variational algorithm, adaptive VQE-X, that self-generates a variational ansatz for arbitrary eigenstates of a many-body Hamiltonian $H$ by attempting to minimize the energy variance with respect to $H$. We benchmark the method by applying it to an Ising spin chain with integrable and nonintegrable regimes, where we calculate various quantities of interest, including the total energy, magnetization density, and entanglement entropy. We also compare the performance of adaptive VQE-X to an adaptive variant of the folded-spectrum method. For both methods, we find a strong dependence of the algorithm's performance on the choice of operator pool used for the adaptive construction of the ansatz. In particular, an operator pool including long-range two-body gates accelerates the convergence of both algorithms in the nonintegrable regime. We also study the scaling of the number of variational parameters with system size, finding that an exponentially large number of parameters may be necessary to approximate individual highly excited states. Nevertheless, we argue that these methods lay a foundation for the use of quantum algorithms to study finite-energy-density properties of many-body systems.