Quantum matching pursuit: A quantum algorithm for sparse representations

TL;DR

This paper introduces a quantum algorithm for sparse signal representation that significantly reduces computational complexity, making real-time processing of high-dimensional signals more feasible with negligible error.

Contribution

It presents the first quantum matching pursuit algorithm that lowers complexity compared to classical methods assuming fault-tolerant quantum memory.

Findings

Reduces complexity of sparse representation computation by a polynomial factor.

Numerical experiments show negligible error in practical scenarios.

Establishes a foundation for quantum algorithms in signal processing applications.

Abstract

Representing signals with sparse vectors has a wide range of applications that range from image and video coding to shape representation and health monitoring. In many applications with real-time requirements, or that deal with high-dimensional signals, the computational complexity of the encoder that finds the sparse representation plays an important role. Quantum computing has recently shown promising speed-ups in many representation learning tasks. In this work, we propose a quantum version of the well-known matching pursuit algorithm. Assuming the availability of a fault-tolerant quantum random access memory, our quantum matching pursuit lowers the complexity of its classical counterpart of a polynomial factor, at the cost of some error in the computation of the inner products, enabling the computation of sparse representation of high-dimensional signals. Besides proving the computational complexity of our new algorithm, we provide numerical experiments that show that its error is negligible in practice. This work opens the path to further research on quantum algorithms for finding sparse representations, showing suitable quantum computing applications in signal processing.