A quantum algorithm for the quantum Schur-Weyl transform

TL;DR

This paper presents a versatile quantum algorithm that generalizes classical combinatorial algorithms, efficiently computing the quantum Schur-Weyl transform for any quantum parameter value, unifying several known algorithms within a single framework.

Contribution

It introduces a quantum algorithm that smoothly interpolates between classical RSK and dual RSK algorithms through a quantum parameter q, extending the Bacon-Chuang-Harrow algorithm.

Findings

Efficient quantum algorithm for all q in [0, ∞]

Unifies classical RSK and dual RSK algorithms

Provides a quantum framework for classical combinatorial transforms

Abstract

We construct an efficient quantum algorithm to compute the quantum Schur-Weyl transform for any value of the quantum parameter $q \in [0,\infty]$. Our algorithm is a $q$-deformation of the Bacon-Chuang-Harrow algorithm, in the sense that it has the same structure and is identically equal when $q=1$. When $q=0$, our algorithm is the unitary realization of the Robinson-Schensted-Knuth (or RSK) algorithm, while when $q=\infty$ it is the dual RSK algorithm together with phase signs. Thus, we interpret a well-motivated quantum algorithm as a generalization of a well-known classical algorithm.