Estimating Bethe roots with VQE

TL;DR

This paper introduces a VQE-based method to estimate Bethe roots for the XXZ quantum spin chain, enabling the calculation of eigenvalues and eigenstates more efficiently, including complex roots, for small system sizes.

Contribution

It presents a novel VQE approach using Bethe states as trial states and Bethe roots as variational parameters, extending applicability to complex roots and excited states.

Findings

Successfully estimated Bethe roots for systems up to size 6.

Achieved accurate results for both ground and excited states.

Applicable to both open and closed XXZ chains.

Abstract

Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver (VQE) approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.