The Quantum Schur Transform: I. Efficient Qudit Circuits

TL;DR

This paper introduces an efficient quantum circuit family for implementing the Schur transform on multiple qudits, enabling more practical quantum information processing by leveraging representation theory and subgroup bases.

Contribution

It provides the first polynomial-time quantum circuit construction for the Schur transform, surpassing the limitations of the quantum Fourier transform paradigm.

Findings

Circuit complexity is polynomial in n, d, and log(1/epsilon)

Uses subgroup adapted basis and Wigner-Eckart theorem for construction

Enables efficient quantum protocols in information theory

Abstract

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of epsilon, then our circuit construction uses a number of gates which is polynomial in n, d and log(1/epsilon). The important insights we use to perform this construction are the selection of the appropriate subgroup adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and is an important new efficient quantum circuit family which goes significantly beyond the standard paradigm of the quantum Fourier transform.