Approximate Quantum Fourier Transform with $O(n \log(n))$ T gates

TL;DR

This paper presents a method to implement an approximate Quantum Fourier Transform with a T-gate complexity of O(n log(n)), improving over the standard approach and enabling more efficient quantum algorithms.

Contribution

The authors introduce a novel approach using measurements, feedforward, and state reuse to reduce T-count complexity from O(n log^2(n)) to O(n log(n)).

Findings

Achieves asymptotic T-count of O(n log(n)) for approximate QFT.

Provides concrete quantum circuits demonstrating practical advantages.

Shows significant theoretical and practical improvements over previous methods.

Abstract

The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an $n$-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of $O(n \log^2(n))$. In this paper, we show how to obtain approximate QFT with the T-count of $O(n \log(n))$. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.