Bethe states on a quantum computer: success probability and correlation functions

TL;DR

This paper analyzes a probabilistic quantum algorithm for preparing Bethe eigenstates of the Heisenberg chain, deriving success probabilities, studying their scaling, and demonstrating the computation of spin-spin correlations for short chains.

Contribution

It provides an exact formula for the success probability in terms of the Gaudin determinant and explores its behavior for large chain lengths.

Findings

Success probability decreases exponentially with chain length.

Feasibility of computing spin-spin correlations for short chains is demonstrated.

Derived an exact formula for success probability involving the Gaudin determinant.

Abstract

A probabilistic algorithm for preparing Bethe eigenstates of the spin-1/2 Heisenberg spin chain on a quantum computer has recently been found. We derive an exact formula for the success probability of this algorithm in terms of the Gaudin determinant, and we study its large-length limit. We demonstrate the feasibility of computing antiferromagnetic ground-state spin-spin correlation functions for short chains. However, the success probability decreases exponentially with the chain length, which precludes the computation of these correlation functions for chains of moderate length. Some conjectures for estimates of the Gaudin determinant are noted in an appendix.