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Generalized eigenvalue problems (GEPs) play an important role in the varietyof fields including engineering, machine learning and quantum chemistry.Especially, many problems in these fields can be reduced to finding the minimumor maximum eigenvalue of GEPs. One of the key problems to handle GEPs is thatthe memory usage and computational complexity explode as the size of the systemof interest grows. This paper aims at extending sequential quantum optimizersfor GEPs. Sequential quantum optimizers are a family of algorithms thatiteratively solve the analytical optimization of single-qubit gates in acoordinate descent manner. The contribution of this paper is as follows. First,we formulate the GEP as the minimization/maximization problem of the fractionalform of the expectations of two Hermitians. We then showed that the fractionalobjective function can be analytically minimized or maximized with respect to asingle-qubit gate by solving a GEP of a 4 $\times$ 4 matrix. Second, we showthat a system of linear equations (SLE) characterized by a positive-definiteHermitian can be formulated as a GEP and thus be attacked using the proposedmethod. Finally, we demonstrate two applications to important engineeringproblems formulated with the finite element method. Through the demonstration,we have the following bonus finding; a problem having a real-valued solutioncan be solved more effectively using quantum gates generating a complex-valuedstate vector, which demonstrates the effectiveness of the proposed method.

Variational quantum algorithm for generalized eigenvalue problems and its application to the finite element method