A Complexity-Efficient Quantum Architecture and Simulation for Eigen Spectrum Estimation of Vandermonde System in a Large Antenna Array

Vandermonde matrix finds a wide range of applications in the digital signal processing domain including antenna array processing. Eigenspectrum estimation remains a challenge in such a system when the matrix dimension increases. This precise problem can be well addressed in a quantum simulation framework based on quantum Hamiltonian simulation (QHS). However, the structural exploitation of the matrix is never considered, which can give benefits in terms of quantum gate complexity. In this work, we exploit the special structural property of a dense Vandermonde matrix by decomposing it into several sparse matrices and then use them for a complexity-efficient eigenvalue spectrum estimation in a quantum framework. In this context, we propose an iterative quantum simulation framework for a Vandermonde-structured Hamiltonian called quantum Vandermonde simulation (QVS). The proposed approach requires fewer quantum gate operations compared to the standard QHS of a dense Vandermonde matrix. In this work, we have considered a large antenna array system, that exploits a delay Vandermonde matrix (DVM) to create a multi-beam beamformer using the proposed low complex quantum architecture. Extensive theoretical and numerical analysis in terms of complexity and other quality parameters have been conducted to show the benefit of the proposed method.