A Low-Complexity Quantum Simulation Framework for Toeplitz-Structured Matrix and Its Application in Signal Processing

Toeplitz matrix reconstruction algorithms (TMRAs) are one of the central subroutines in array processing for wireless communication applications. The classical TMRAs have shown excellent accuracy in the spectral estimation for both uncorrelated and coherence sources in the recent era. However, TMRAs incorporate the classical eigenvalue decomposition technique for estimating the eigenvalues of the Toeplitz-structured covariance matrices that demand very high computational complexity for large arrays. We demonstrate a low-complexity quantum simulation framework exploiting the structured Hamiltonian of Toeplitz and circulant variants. In this framework, we consider two approaches for the estimation of the eigenvalue spectrum of a given Toeplitz-structured matrix: first, an analytical framework with Jordan form-based sparse-decomposition of a dense-Toeplitz matrix, and second, an approximation method for the conversion of a Toeplitz matrix into a circulant matrix embedding quantum subroutines. We have also compared the efficacy of the proposed algorithms with standard Hamiltonian simulation and quantum phase estimation techniques for different quantum time resolutions and gate complexities. We show quantum gate-complexity analysis for our proposed algorithms. Considering the large dimensions of the Toeplitz matrix, we have employed random matrix theory in deriving the error bounds for the estimated eigenvalues. The numerical results are obtained partly in a classical computer and in an IBM quantum simulator.