An efficient high dimensional quantum Schur transform

TL;DR

This paper introduces a new quantum algorithm for the high-dimensional Schur transform that is efficient in the number of tensor factors, the logarithm of the dimension, and the inverse precision, using symmetric group representation theory.

Contribution

It presents a novel quantum algorithm for the Schur transform that is polynomial in n, log d, and log epsilon, using a dual approach based on symmetric group representations.

Findings

Algorithm is polynomial in n, log d, and log epsilon.

Constructs quantum Fourier transform over permutation modules.

Potential applications in quantum information processing.

Abstract

The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an $n$ fold tensor product $V^{\otimes n}$ of a vector space $V$ of dimension $d$. Bacon, Chuang and Harrow \cite{BCH07} gave a quantum algorithm for this transform that is polynomial in $n$, $d$ and $\log\epsilon^{-1}$, where $\epsilon$ is the precision. In a footnote in Harrow's thesis \cite{H05}, a brief description of how to make the algorithm of \cite{BCH07} polynomial in $\log d$ is given using the unitary group representation theory (however, this has not been explained in detail anywhere. In this article, we present a quantum algorithm for the Schur transform that is polynomial in $n$, $\log d$ and $\log\epsilon^{-1}$ using a different approach. Specifically, we build this transform using the representation theory of the symmetric group and in this sense our technique can be considered a "dual" algorithm to \cite{BCH07}. A novel feature of our algorithm is that we construct the quantum Fourier transform over the so called \emph{permutation modules}, which could have other applications.