A Multigraph Approach for Performing the Quantum Schur Transform

TL;DR

This paper introduces a multigraph-based method inspired by representation theory to improve the efficiency of the Quantum Schur Transform, enabling straightforward basis changes for any system size and dimension.

Contribution

It develops a new multigraph called the Schur-Weyl-Young graph and a branching rule, simplifying the quantum basis transformation process.

Findings

Improved formula for transition amplitudes between tableaux for d=2

Introduction of the Schur-Weyl-Young graph integrating Weyl and Young tableaux

A new branching rule enabling basis change for any n and d

Abstract

We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case $d=2$, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any $n$ and $d$.