Preparing Bethe Ansatz Eigenstates on a Quantum Computer

TL;DR

This paper introduces a quantum algorithm to prepare Bethe ansatz eigenstates of the XXZ spin chain, enabling direct measurement of physical properties that are analytically or numerically challenging.

Contribution

It presents a polynomial-time quantum algorithm for constructing Bethe eigenstates, utilizing amplitude amplification to improve success probability, with lower resource requirements than existing methods.

Findings

Algorithm efficiently prepares eigenstates for small systems.

Success probability can be boosted with amplitude amplification.

Resource requirements are lower than other quantum simulation methods.

Abstract

Several quantum many-body models in one dimension possess exact solutions via the Bethe ansatz method, which has been highly successful for understanding their behavior. Nevertheless, there remain physical properties of such models for which analytic results are unavailable, and which are also not well-described by approximate numerical methods. Preparing Bethe ansatz eigenstates directly on a quantum computer would allow straightforward extraction of these quantities via measurement. We present a quantum algorithm for preparing Bethe ansatz eigenstates of the spin-1/2 XXZ spin chain that correspond to real-valued solutions of the Bethe equations. The algorithm is polynomial in the number of T gates and circuit depth, with modest constant prefactors. Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplitude amplification to boost the success probability. The resource requirements for our approach are lower than other state-of-the-art quantum simulation algorithms for small error-corrected devices, and thus may offer an alternative and computationally less-demanding demonstration of quantum advantage for physically relevant problems.