Highly-efficient quantum Fourier transformations for some nonabelian groups

TL;DR

This paper develops efficient quantum Fourier transform algorithms for specific nonabelian groups relevant to physics, significantly reducing quantum simulation costs for these groups.

Contribution

It introduces explicit quantum circuits and resource estimates for fast Fourier transforms on several nonabelian groups of interest in physics.

Findings

Reduces simulation costs by up to three orders of magnitude.

Provides explicit quantum circuit constructions.

Estimates resource scaling for fault-tolerant implementations.

Abstract

Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups of interest have seen less development. In particular, fast nonabelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the nonabelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics, $\mathbb{BT}$, $\mathbb{BO}$, $\Delta(27)$, $\Delta(54)$, and $\Sigma(36\times3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.