A Faster Quantum Fourier Transform

TL;DR

This paper introduces a significantly faster quantum Fourier transform algorithm that reduces gate complexity for both approximate and exact versions by leveraging a recursive qubit partitioning approach.

Contribution

It presents a novel recursive formulation of the QFT that achieves asymptotic improvements in gate complexity for both approximate and exact implementations.

Findings

Approximate QFT implemented with $ ext{O}(n( ext{log log } n)^2)$ gates.

Exact QFT achieved with $ ext{O}(n( ext{log } n)^2)$ gates.

Reduction in ancilla qubits needed for both approximate and exact QFT.

Abstract

We present an asymptotically improved algorithm for implementing the Quantum Fourier Transform (QFT) in both the exact and approximate settings. Historically, the approximate QFT has been implemented in $\Theta(n \log n)$ gates, and the exact in $\Theta(n^2)$ gates. In this work, we show that these costs can be reduced by leveraging a novel formulation of the QFT that recurses on two partitions of the qubits. Specifically, our approach yields an $\Theta(n(\log \log n)^2)$ algorithm for the approximate QFT using $\Theta(\log n)$ ancillas, and an $\Theta(n(\log n)^2)$ algorithm for the exact QFT requiring $\Theta(n)$ ancillas.