A quantum Bluestein’s algorithm for arbitrary-size quantum Fourier transform

Abstract We propose a quantum analogue of Bluestein’s algorithm (QBA) that implements an algebraically exact N -point quantum Fourier transform (QFT) for arbitrary N . Our construction factors the N -dimensional QFT unitary U QFT N into three diagonal quadratic-phase gates and two standard radix-2 QFT subcircuits of size M = 2 m (with M ⩾ 2 N − 1 ). This achieves asymptotic gate complexity of O ( N ) and uses O ( log ⁡ N ) qubits. We validate the correctness of the algorithm through a concrete implementation in Qiskit, confirming that QBA produces the exact N -point discrete Fourier transform on arbitrary-length inputs. The proposed framework is especially relevant for biomedical time-series analysis and quantum-kernel methods, where irregular, non-power-of-two signal lengths are common and zero-padding may introduce distortions in the represented N -point spectrum.