The Bethe Ansatz as a Quantum Circuit

TL;DR

This paper provides an analytical derivation of quantum circuits from the Bethe ansatz, revealing new insights into the wavefunction structure and the equivalence of different Bethe ansatz formulations.

Contribution

It offers the first analytical expression for quantum gates derived from the Bethe ansatz and introduces diagrammatic rules for constructing Bethe wavefunctions as Matrix Product States.

Findings

Derived explicit quantum circuit gates from Bethe ansatz

Introduced diagrammatic rules for Bethe wavefunction construction

Unified coordinate and algebraic Bethe ansatz perspectives

Abstract

The Bethe ansatz represents an analytical method enabling the exact solution of numerous models in condensed matter physics and statistical mechanics. When a global symmetry is present, the trial wavefunctions of the Bethe ansatz consist of plane wave superpositions. Previously, it has been shown that the Bethe ansatz can be recast as a deterministic quantum circuit. An analytical derivation of the quantum gates that form the circuit was lacking however. Here we present a comprehensive study of the transformation that brings the Bethe ansatz into a quantum circuit, which leads us to determine the analytical expression of the circuit gates. As a crucial step of the derivation, we present a simple set of diagrammatic rules that define a novel Matrix Product State network building Bethe wavefunctions. Remarkably, this provides a new perspective on the equivalence between the coordinate and algebraic versions of the Bethe ansatz.