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

Nan-Hong Kuo, Renata Wong

TL;DR

The paper addresses the limitation that the standard Quantum Fourier Transform (QFT) is naturally efficient only for sizes that are powers of two, which complicates exact transforms for arbitrary N. It introduces Quantum Bluestein's Algorithm (QBA), which reduces an N-point QFT to a convolution using three diagonal quadratic-phase gates and two radix-2 QFT subcircuits on a workspace of size $M=2^m\ge 2N-1$, enabling exact computation with $O((\log N)^2)$ gates on $O(\log N)$ qubits. The approach preserves the asymptotic resource scaling of the standard QFT while removing the size restriction, and validation via Qiskit simulations demonstrates exact N-point DFT outputs for arbitrary-length inputs. The work provides a practical, open-source framework for implementing exact, scalable QFTs of arbitrary sizes, with potential impact on quantum signal processing and kernel methods in quantum machine learning.

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.

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

TL;DR

The paper addresses the limitation that the standard Quantum Fourier Transform (QFT) is naturally efficient only for sizes that are powers of two, which complicates exact transforms for arbitrary N. It introduces Quantum Bluestein's Algorithm (QBA), which reduces an N-point QFT to a convolution using three diagonal quadratic-phase gates and two radix-2 QFT subcircuits on a workspace of size , enabling exact computation with gates on qubits. The approach preserves the asymptotic resource scaling of the standard QFT while removing the size restriction, and validation via Qiskit simulations demonstrates exact N-point DFT outputs for arbitrary-length inputs. The work provides a practical, open-source framework for implementing exact, scalable QFTs of arbitrary sizes, with potential impact on quantum signal processing and kernel methods in quantum machine learning.

Abstract

We propose a quantum analogue of Bluestein's algorithm (QBA) that implements an exact -point Quantum Fourier Transform (QFT) for arbitrary . Our construction factors the -dimensional QFT unitary into three diagonal quadratic-phase gates and two standard radix-2 QFT subcircuits of size (with ). This achieves asymptotic gate complexity and uses qubits, matching the performance of a power-of-two QFT on 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 -point discrete Fourier transform on arbitrary-length inputs.

Paper Structure

This paper contains 29 sections, 39 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Quantum Bluestein's Algorithm
  4. Quantum circuit construction
  5. 1. Input chirp
  6. 2. Forward QFT
  7. 3. Convolution in the Fourier basis
  8. 4. Inverse QFT
  9. 5. Output de-chirp
  10. Complexity analysis
  11. Proof of correctness
  12. Simulation using Qiskit
  13. Conclusion
  14. Code availability
  15. Worked examples
  16. ...and 14 more sections

Figures (5)

  • Figure 1: High-level circuit diagram for the Quantum Bluestein's Algorithm (QBA). The algorithm computes the arbitrary-size QFT by sequentially applying an input chirp, a standard power-of-two QFT, a Fourier-domain convolution operator, an inverse QFT, and an output de-chirp.
  • Figure 2: Quantum circuit for $N=3$ generated using QBA.
  • Figure 3: Histogram of probabilities for $N=3$ generated using QBA.
  • Figure 4: Quantum circuit for $N=6$ generated using QBA.
  • Figure 5: Histogram of probabilities for $N=6$ generated using QBA.

Theorems & Definitions (1)

  • proof