A hybrid classical-quantum algorithm for digital image processing

TL;DR

This paper introduces a hybrid classical-quantum algorithm for efficiently computing multi-dimensional Walsh-Hadamard transforms, enabling improved quantum image processing with lower complexity and optimized qubit usage.

Contribution

It presents a novel quantum approach for multidimensional Walsh-Hadamard transforms that reduces computational complexity and qubit requirements compared to existing methods.

Findings

Lower computational complexity ($O(N^d)$) compared to classical methods.

Efficient qubit usage requiring only $ ext{log}_2 N$ qubits.

Successful application to image filtering and noise removal tasks.

Abstract

A hybrid classical-quantum approach for evaluation of multi-dimensional Walsh-Hadamard transforms and its applications to quantum image processing are proposed. In this approach, multidimensional Walsh-Hadamard transforms are obtained using quantum Hadamard gates (along with state-preparation, shifting, scaling and measurement operations). The proposed approach for evaluation of multidimensional Walsh-Hadamard transform has a considerably lower computational complexity (involving $O(N^d)$ operations) in contrast to classical Fast Walsh-Hadamard transform (involving $O(N^d~\log_2 N^d)$ operations), where $d$ and $N$ denote the number of dimensions and degrees of freedom along each dimension. Unlike many other quantum image representation and quantum image processing frameworks, our proposed approach makes efficient use of qubits, where only $\log_2 N $ qubits are sufficient for sequential processing of an image of $ N \times N $ pixels. Selected applications of the proposed approach (for $ d=2 $) are demonstrated via computational examples relevant to basic image filtering and periodic banding noise removal and the results were found to be satisfactory.