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arXiv:2512.15349 · Physica Scripta

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

Nan-Hong, Kuo, Renata Wong

quant-phSubmitted Dec 17, 2025• Announced Dec 18, 2025• 0 citations
v1 · Dec 17, 2025v2 · Dec 23, 2025
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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.

Journal reference
Phys. Scr. 101 385104, 2026
DOI
10.1088/1402-4896/aea647
Comments
9 pages, 4 figures

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