Quantum wave packet transforms with compact frequency support

TL;DR

This work develops quantum circuit implementations for wave packets with compact frequency support, focusing on Gabor atoms and wavelets in the frequency domain. It introduces a suite of building blocks, including $Q_{\, ext{G}(M)}$, $V_{\, ext{G}}$, and $T_{\, ext{W}}$, to realize sharp and blended frequency windows across Gabor atoms and Shannon/Meyer wavelets, and analyzes their ancilla usage and computational complexity. The proposed transforms, $U_{\, ext{GS}}$, $U_{\, ext{GB}}$, $U_{\, ext{WS}}$, and $U_{\, ext{WB}}$, leverage both exact constructions and ε-approximate schemes via QSVT, enabling efficient quantum processing of wave packets with compact frequency support. The results open pathways to extending these methods to other wave packets and higher-dimensional settings, with explicit circuit constructions, complexity analyses, and rigorous proofs of key equalities.

Abstract

Different kinds of wave packet transforms are widely used for extracting multi-scale structures in signal processing tasks. This paper introduces the quantum circuit implementation of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. Our approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers. The resulting implementation is different from the existing quantum algorithms for spatially compactly supported wavelets and can be readily extended to quantum transforms of other wave packets with compact frequency support.

Figures (20)

• Figure 1: (a) Illustration for Gabor atom tiling. Each pair of tiles symmetric with respect to the $x$ axis represents the essential support of a basis function $\uppsi_{j,p}$. The red, orange, yellow, and green tiles represent $\uppsi_{4,0}$, $\uppsi_{3,0}$, $\uppsi_{2,0}$, and $\uppsi_{1,0}$ respectively. All tiles with the same $\omega$ are at the same level $j$. For instance, the blue tile represents $\uppsi_{3,3}$. (b) Illustration for wavelet tiling. Each pair of tiles symmetric with respect to the $x$ axis represents the essential support of a basis function $\uppsi_{j,p}$. This diagram shows the truncation up to $n=4$. The red, orange, yellow, green, and blue tiles represent $\uppsi_{1,0}$, $\uppsi_{2,0}$, $\uppsi_{3,0}$, $\uppsi_{4,0}$, and $\uppsi_{2,3}$ respectively. The purple tiles represent the scaling function.
• Figure 2: (a) CNOT gate. (b) $\ket{0}$-controlled-NOT gate. (c) SWAP gate. (d) An example of a multi-qubit control gate $\ket{11}\!\bra{11}\otimes U+(I_4-\ket{11}\!\bra{11})\otimes I$. The two filled nodes indicate that $U$ is applied to the third qubit if and only if both the first two qubits are at state $\ket{1}$.
• Figure 3: The process of sharp Gabor atoms transform. Reshuffle the indices in the frequency domain, and then perform an inverse Fourier transform on each of the $2B$-size blocks.
• Figure 4: The circuit of $U_{\mathrm{GS}}$ when $N=64$ and $B=4$.
• Figure 5: Illustration of reallocation when $N = 8B$. Each basis function $\widehat{\psi_{2Bj+p}}$ contains two bumps supported at $[(j-\frac{1}{2})B,(j+\frac{3}{2})B)$ and $[-(j+\frac{3}{2})B, -(j-\frac{1}{2})B)$. The arrows indicate how the small tails are transposed such that all the relevant $\hat{f}(k)$'s are placed in $[jB,(j+1)B)\cup[-(j+1)B, -jB)$ with correct proportion. All these reallocations are unitary manipulations. This figure exclusively illustrates magnitude transpositions, omitting the change of phase. In particular, the self-transpositions are, in fact, phase rotations.