Algebraic Bethe Circuits

TL;DR

This paper develops a method to implement the Algebraic Bethe Ansatz on quantum computers by converting non-unitary matrices into unitaries, enabling efficient eigenstate preparation and verification of the Yang-Baxter equation.

Contribution

It introduces a novel unitary formulation of the ABA using QR decomposition, allowing direct quantum implementation and state preparation for integrable models.

Findings

Successfully prepared eigenstates of XX and XXZ models on IBM quantum computers.

Verified a new unitary form of the Yang-Baxter equation on quantum hardware.

Achieved efficient eigenstate preparation with resources matching previous methods.

Abstract

The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary $R$ matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-$\frac{1}{2}$ XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on $4$ sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.