Non-Hermitian generalized eigenvalue problems (GEPs) play a significant role in many practical applications, such as mechanical engineering. Based on the generalized Schur decomposition, we propose a variational quantum algorithm for solving the GEPs in non-Hermitian systems. The algorithm transforms the generalized eigenvalue problem into a process of searching for unitary transformation matrices. We demonstrate a method for evaluating both the loss function and its gradients on near-term quantum devices. We validate numerically the algorithm's performance through simulations, and demonstrate its application to GEPs in ocean acoustics. The algorithm's robustness is further confirmed through noise simulations.
Many eigenvalue problems arising in practice are often of the generalized form A x = λ B x . One particularly important case is symmetric, namely A , B are Hermitian and B is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than the standard one based on quantum phase estimation. Moreover, it works for a wider case than symmetric: B is invertible, B − 1 A is diagonalizable and all the eigenvalues are real.
Polynomial eigenvalue problems (PEPs) arise in a variety of science and engineering applications, and many breakthroughs in the development of classical algorithms to solve PEPs have been made in the past decades. Here we attempt to solve PEPs in a quantum computer. Firstly, for generalized eigenvalue problems (GEPs) $A\x = \lambda B\x$ with A,B symmetric, and B positive definite, we give a quantum algorithm based on block-encoding and quantum phase estimation. In a more general case when B is invertible, B−1A is diagonalizable and all the eigenvalues are real, we propose a quantum algorithm based on the Fourier spectral method to solve ordinary differential equations (ODEs). The inputs of our algorithms can be any desired states, and the outputs are superpositions of the eigenpairs. The complexities are polylog in the matrix size and linear in the precision. The dependence on precision is optimal. Secondly, we show that when B is singular, any quantum algorithm uses at least Ω(n) queries to compute the eigenvalues, where n is the matrix size. Thirdly, based on the linearization method and the connection between PEPs and higher-order ODEs, we provide two quantum algorithms to solve PEPs by extending the quantum algorithm for GEPs. We also give detailed complexity analysis of the algorithm for two special types of quadratic eigenvalue problems that are important in practice. Finally, under an extra assumption, we propose a quantum algorithm to solve PEPs when the eigenvalues are complex.