Bethe Ansatz, Quantum Circuits, and the F-basis

TL;DR

This paper connects the algebraic Bethe Ansatz with the F-basis, providing a systematic way to derive quantum circuits for Bethe states, and demonstrates new circuits for the inhomogeneous XXZ model.

Contribution

It shows that algebraic Bethe circuits can be derived via a basis change to the F-basis, linking two approaches and enabling new quantum circuit constructions.

Findings

Derived new quantum circuits for the inhomogeneous XXZ model.

Established the connection between algebraic Bethe circuits and the F-basis.

Highlighted the importance of the F-basis properties in circuit construction.

Abstract

The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as "algebraic Bethe circuits", have been developed to prepare Bethe states for the spin-1/2 XXZ model. These circuits represent a unitary formulation of the standard algebraic Bethe Ansatz, expressed using matrix-product states that act on both the spin chain and an auxiliary space. In this work, we systematize these previous results, and show that algebraic Bethe circuits can be derived by a change of basis in the auxiliary space. The new basis, identical to the "F-basis" known from the theory of quantum-integrable models, generates the linear superposition of plane waves that is characteristic of the coordinate Bethe Ansatz. We explain this connection, highlighting that certain properties of the F-basis (namely, the exchange symmetry of the spins) are crucial for the construction of algebraic Bethe circuits. We demonstrate our approach by presenting new quantum circuits for the inhomogeneous spin-1/2 XXZ model.