Non-Uniform Quantum Fourier Transform

TL;DR

This paper develops a quantum algorithm for the Non-Uniform Discrete Fourier Transform (NUDFT), enabling efficient processing of irregularly sampled data with controlled accuracy and resource estimates, advancing quantum signal processing.

Contribution

It introduces the first explicit quantum algorithm for NUDFT based on low-rank matrix factorization and block encodings, filling a gap in quantum Fourier analysis for non-uniform sampling.

Findings

Quantum NUDFT algorithm with polylogarithmic complexity in precision

Resource estimates for quantum implementation under standard oracle assumptions

Framework applicable to irregularly sampled data in quantum computing

Abstract

The Discrete Fourier Transform (DFT) is central to the analysis of uniformly sampled signals, yet many practical applications involve non-uniform sampling, requiring the Non-Uniform Discrete Fourier Transform (NUDFT). While quantum algorithms for the standard DFT are well established, a corresponding framework for the non-uniform case remains underdeveloped. This work introduces a quantum algorithm for the Non-Uniform Quantum Fourier Transform (NUQFT) based on a low-rank factorization of the NUDFT matrix. The factorization is translated into an explicit quantum construction using block encodings, Quantum Signal Processing, and the Linear Combination of Unitaries framework, yielding an $\epsilon$-accurate block encoding of the NUDFT matrix with controlled approximation error from both classical truncation and quantum implementation. Under standard oracle access assumptions for non-uniform sampling points, we derive explicit, non-asymptotic gate-level resource estimates. The resulting complexity scales polylogarithmically with target precision, quadratically with the number of qubits through the quantum Fourier transform, and logarithmically with a geometry-dependent conditioning parameter induced by the non-uniform grid. This establishes a concrete and resource-efficient quantum analogue of the NUDFT and provides a foundation for quantum algorithms on irregularly sampled data.