A Practical Quantum Algorithm for the Schur Transform

TL;DR

This paper presents an efficient, practical quantum algorithm for the Schur transform, improving on previous methods by optimizing gate complexity and resource usage for qubits and qudits, with applications in quantum computing.

Contribution

The paper simplifies and extends the existing Schur transform algorithm, providing a practical construction with optimized gate complexity and resource requirements for qubits and qudits.

Findings

Decomposes the Schur transform into O(n^4 log(n/ε)) Clifford+T gates for qubits.

Uses exactly 2⌊log₂(n)⌋-1 ancillary qubits.

Extends the algorithm to qudits with complexity depending on dimension d.

Abstract

We describe an efficient quantum algorithm for the quantum Schur transform. The Schur transform is an operation on a quantum computer that maps the standard computational basis to a basis composed of irreducible representations of the unitary and symmetric groups. We simplify and extend the algorithm of Bacon, Chuang, and Harrow, and provide a new practical construction as well as sharp theoretical and practical analyses. Our algorithm decomposes the Schur transform on $n$ qubits into $O\left(n^4\log\left(\frac{n}{\epsilon}\right)\right)$ operators in the Clifford+T fault-tolerant gate set and uses exactly $2\lfloor\log_2(n)\rfloor-1$ ancillary qubits. We extend our qubit algorithm to decompose the Schur transform on $n$ qudits of dimension $d$ into $O\left(n^{d^2+2}\log^p\left(\frac{n^{d^2+1}}{\epsilon}\right)\right)$ primitive operators from any universal gate set, for $p\approx3.97$.