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

TL;DR

This paper extends quantum optimization algorithms to efficiently solve generalized eigenvalue problems, demonstrating applications in engineering via the finite element method and revealing that complex-valued quantum states can enhance solution effectiveness.

Contribution

The paper formulates GEPs as fractional expectation minimization/maximization problems and develops a quantum optimizer that analytically solves single-qubit subproblems, applying it to engineering problems.

Findings

The fractional expectation function can be minimized/maximized analytically with respect to a single-qubit gate.

The method can solve systems of linear equations formulated as GEPs.

Using complex-valued quantum states improves solution efficiency for real-valued problems.

Abstract

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