Non-Uniform Quantum Fourier Transform

TL;DR

This work introduces a quantum algorithm for the Non-Uniform Quantum Fourier Transform (NUQFT) based on a low-rank factorization of the NUDFT matrix, establishing a concrete and resource-efficient quantum analogue of the NUDFT and providing a foundation for quantum algorithms on irregularly sampled data.

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 $ε$-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.

Figures (5)

• Figure 1: Quantum oracle for multiplication and addition implementing $\lvert x \rangle \lvert y \rangle \lvert z \rangle \mapsto \lvert x \rangle \lvert y \rangle \lvert xy+z \rangle$.
• Figure 2: Quantum circuit $U_{\text{LCU}}$ implementing the LCU algorithm.
• Figure 3: Quantum circuit implementing $U_{\vec{v}_r}$ for $r \geq 1$ in the special case $n = 5$.
• Figure 4: Quantum circuit implementing $O^{(m)}_{\vec{s}}$.
• Figure 5: Quantum circuit implementing the operator $U_{\vec{u}_r} = \sum_{q=0}^{K-1} \alpha'_{qr} U_{qr}$. We have abbreviated $\vec{t}-\vec{s}/N$ by $\vec{y}$.

Theorems & Definitions (36)

• Definition 2
• Lemma 3: Lemma 53 in gilyen2019qsvt
• Lemma 4
• Proposition 5: Theorem 3 in gilyen2019qsvt
• Example 6
• Proposition 7
• proof
• Lemma 8
• proof
• Proposition 9