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

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.

💡 Research Summary

The paper presents a systematic method for converting the classical Fast Fourier Transform (FFT) into its quantum counterpart, the Quantum Fourier Transform (QFT), by employing QR decomposition. The authors begin by reviewing the structure of the classical FFT, emphasizing its recursive divide‑and‑conquer nature. For an input vector of size (N = 2^{n}), the FFT splits the data into even and odd indexed subsets, applies a smaller DFT of size (N/2) to each, and then recombines them using “twiddle” factors (W_N^k = e^{-2\pi i k/N}). This process can be expressed in matrix form as a product of a permutation matrix (P), a diagonal twiddle matrix (D), and a block‑diagonal matrix containing two copies of the smaller Fourier matrix (F_{N/2}).

The core contribution is the observation that the overall Fourier matrix (F_N) can be factorized via QR decomposition into a unitary matrix (Q_N) and an upper‑triangular matrix (R_N). The authors show that (Q_N) corresponds exactly to a sequence of quantum gates that are already familiar from standard QFT circuits: Hadamard gates applied to individual qubits and controlled‑phase rotations (CR gates) where a higher‑order qubit controls a lower‑order qubit with a phase (e^{2\pi i k/2^{m}}). The upper‑triangular factor (R_N) contains only global phase factors on the diagonal, which have no observable effect on quantum measurements and can therefore be ignored in a physical implementation.

By recursively applying QR decomposition to the block‑diagonal component (F_{N/2}\otimes I_2), the authors derive a product representation \