Quantum algorithms for the generalized eigenvalue problem

The generalized eigenvalue (GE) problems are of particular importance in various areas science engineering and machine learning. We present a variational quantum algorithm for finding the desired GE problem, $\mathcal{A}|\psi\rangle=\lambda\mathcal{B}|\psi\rangle$, by choosing suitable loss functions. Our approach imposes superposition trial state obtained eigenvectors with respect to weighting matrix $\mathcal{B}$ on Rayleigh-quotient. Furthermore, both values derivatives functions can be calculated near-term devices shallow circuit. Finally, we propose full eigensolver (FQGE) calculate minimal gradient descent algorithm. As demonstration principle, numerically implement our algorithms conduct 2-qubit simulation successfully find eigenvalues pencil $(\mathcal{A},\,\mathcal{B})$. experimental result indicates that FQGE is robust under Gaussian noise.