Obtaining the Quantum Fourier Transform from the Classical FFT with QR Decomposition

TL;DR

This paper demonstrates how to derive the Quantum Fourier Transform from the classical FFT using QR decomposition, illustrating a method to build quantum algorithms from classical counterparts.

Contribution

It provides a detailed, step-by-step conversion process from classical FFT to quantum Fourier Transform via QR decomposition, expanding on Coppersmith's work.

Findings

Quantum Fourier Transform can be systematically derived from classical FFT.

The QR decomposition approach generalizes previous methods.

The paper clarifies the relationship between classical and quantum Fourier transforms.

Abstract

We present the detailed process of converting the classical Fourier Transform algorithm into the quantum one by using QR decomposition. This provides an example of a technique for building quantum algorithms using classical ones. The Quantum Fourier Transform is one of the most important quantum subroutines known at present, used in most algorithms that have exponential speed up compared to the classical ones. We briefly review Fast Fourier Transform and then make explicit all the steps that led to the quantum formulation of the algorithm, generalizing Coppersmith's work.