A Quantum Bluestein's Algorithm for Arbitrary-Size Quantum Fourier Transform

TL;DR

This paper introduces a quantum algorithm that efficiently computes the exact Fourier transform for any input size, overcoming limitations of traditional methods that require power-of-two dimensions.

Contribution

It presents a novel quantum Bluestein's algorithm that implements exact N-point QFT for arbitrary N with optimal gate complexity and qubit usage, avoiding larger Hilbert spaces.

Findings

Achieves asymptotic gate complexity O((log N)^2)

Uses O(log N) qubits, matching power-of-two QFT performance

Validated through Qiskit implementation and classical simulation

Abstract

We propose a quantum analogue of Bluestein's algorithm (QBA) that implements an exact $N$-point Quantum Fourier Transform (QFT) for arbitrary $N$. Our construction factors the $N$-dimensional QFT unitary into three diagonal quadratic-phase gates and two standard radix-2 QFT subcircuits of size $M = 2^m$ (with $M \ge 2N - 1$). This achieves asymptotic gate complexity $O((\log N)^2)$ and uses $O(\log N)$ qubits, matching the performance of a power-of-two QFT on $m$ qubits while avoiding the need to embed into a larger Hilbert space. We validate the correctness of the algorithm through a concrete implementation in Qiskit and classical simulation, confirming that QBA produces the exact $N$-point discrete Fourier transform on arbitrary-length inputs.