Quantum Algorithms for Wave Packet Transforms with Compact Frequency Support

Core Concepts

This paper introduces quantum circuit implementations of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. The approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers.

Abstract

The paper presents quantum algorithms for implementing wave packet transforms with compact frequency support, such as Gabor atoms and wavelets.
For sharp Gabor atoms, the implementation involves reshuffling the Fourier transform of the input signal and then performing an inverse Fourier transform for each sharp frequency window.
For blended Gabor atoms, the approach first reallocates parts of the Fourier transform to the proper locations, which is a unitary process, and then applies the circuit of the sharp Gabor atoms.
For wavelets with compact frequency support, such as Shannon wavelets and Meyer wavelets, the paper shows that they can be realized with no more than three ancilla qubits, in contrast to wavelets with compact spatial support which typically require more ancilla qubits.
The paper also discusses the implementation of the required diagonal matrices using bit manipulations or quantum singular value transformation (QSVT), depending on the complexity of the frequency window functions.

Stats

The paper does not provide any specific numerical data or metrics to support the key arguments.

Quotes

"Previous studies on quantum wavelet transforms have mostly utilized finite-size filters in the spatial domain, which are only applicable to spatially compactly supported wavelets. As far as we know, no existing algorithm has explored their implementations on quantum computers."
"We show that these wavelets may be realized with no more than three ancilla qubits, thanks to their favorable forms in the frequency domain. In contrast, for the wavelets with compact spatial support, the number of ancilla qubits typically grows with the order of the wavelet."

Key Insights Distilled From

Quantum Wave Packet Transforms with compact frequency support

by Hongkang Ni,... at arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00929.pdf

Deeper Inquiries

How can the quantum algorithms presented in this paper be extended to other types of wave packet transforms beyond Gabor atoms and wavelets

The quantum algorithms discussed in the paper for Gabor atoms and wavelets can be extended to other types of wave packet transforms by following a similar approach of operating in the frequency space and manipulating signals tailored for quantum computation. For different types of wave packets with compact frequency support, such as curvelets or wave atoms, the key lies in designing the quantum circuits to accommodate the specific characteristics of these wave packets. By adapting the reallocation and reshuffling techniques used for Gabor atoms and wavelets, it is possible to implement quantum transforms for a broad class of wave packets with compact frequency support. This extension would involve defining the basis functions in the frequency domain, constructing the corresponding quantum circuits, and optimizing the algorithms for efficient quantum computation.

What are the potential limitations or challenges in applying these quantum wave packet transform algorithms to real-world signal processing tasks

When applying quantum wave packet transform algorithms to real-world signal processing tasks, several potential limitations and challenges may arise. One challenge is the scalability of these algorithms to handle large-scale signal processing tasks efficiently. As the size of the input data increases, the computational resources required for quantum wave packet transforms also grow, posing constraints on the practical implementation of these algorithms. Additionally, the accuracy and precision of the quantum algorithms in representing complex signals need to be carefully considered, as any errors or inaccuracies in the quantum computations can impact the quality of the signal processing results. Moreover, the integration of quantum wave packet transforms into existing signal processing pipelines and frameworks may require significant modifications and adaptations to ensure seamless compatibility and interoperability with classical processing techniques.

What are the connections between the quantum wave packet transform techniques and other quantum algorithms for encoding differential operators or enabling quantum sampling

The quantum wave packet transform techniques presented in the paper have connections to other quantum algorithms for encoding differential operators and enabling quantum sampling. In the context of encoding differential operators, quantum wave packet transforms can be utilized to efficiently represent and manipulate the differential operators associated with wave packets in quantum computations. By leveraging the sparsity and locality properties of wave packet bases, quantum algorithms can encode differential operators in a compressed and efficient manner, facilitating the solution of differential equations and related tasks on quantum computers. Furthermore, the quantum wave packet transform techniques can be integrated with quantum sampling methods to enhance the efficiency of sampling complex signals and data structures in quantum algorithms. By combining these techniques, quantum algorithms can achieve improved performance in various signal processing and data analysis applications.