Quantum generalized least squares method in system identification

Abstract

The least squares method—usually used to estimate different parameters—also leads to solving the equation $A_{n \times n} \,\theta_{n \times 1} = y_{n \times 1}$ in system identification. The classical method (i.e., Cholesky decomposition) which is generally used to solve this problem has a computational complexity of $O\left( {n^{3} } \right)$. As we know, a greater number of samples yields better system modeling. With a larger input data, the computational complexity increases significantly. That is the case with Generalized Least Squares error identification (GLS method) as well. The proposed approach in this article to solve the aforementioned problem is to use quantum algorithms, which significantly reduce the computational complexity. Therefore, in this study, two methods, namely Classical Quantum GLS (C-QGLS) and Quantum GLS methods (QGLS), were proposed to ease the computational complexity. Unlike HHL, these two methods can estimate unbiased parameters despite the existence of color noise. Moreover, they can handle the issue of non-hermitian and ill-condition matrices. The complexity of C-QGLS and QGLS methods is $O\left( {{\text{poly}}\left( n \right)\,\log n} \right)$ and $O\left( {{\text{poly}}\,\log\, (n^{2} )} \right)$, respectively. The proposed methods showed significant superiority compared to the classical methods. Their limitations were fewer than the other Quantum Methods, and their performance was also acceptable.