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arXiv:2010.15027 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

Changpeng Shao, Jin-Peng Liu

Published Jun 1, 2022• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1098/rspa.2021.0797

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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.
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix MMM is sss-sparse, the complexity is O~(sρ2κ2/ε2)\widetilde{O}(sρ^2 κ^2/ε^2)O(sρ2κ2/ε2) for diagonalizable matrices that only have real eigenvalues, and O~(sρ∥M∥max⁡/ε2)\widetilde{O}(sρ\|M\|_{\max} /ε^2)O(sρ∥M∥max​/ε2) for normal matrices. Here ρρρ is an upper bound of the eigenvalues, κκκ is the conditioning of the eigenvalue problem, and εεε is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.
Computing eigenvalues of matrices is ubiquitous in numerical linear algebra problems. Currently, fast quantum algorithms for estimating eigenvalues of Hermitian and unitary matrices are known. However, the general case is far from fully understood in the quantum case. Based on a quantum algorithm for solving linear ordinary differential equations, we show how to estimate the eigenvalues of diagonalizable matrices that only have real eigenvalues. The output is a superposition of the eigenpairs, and the overall complexity is polylog in the dimension for sparse matrices. Under an assumption, we extend the algorithm to diagonalizable matrices with complex eigenvalues.
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,BA,BA,B symmetric, and BBB positive definite, we give a quantum algorithm based on block-encoding and quantum phase estimation. In a more general case when BBB is invertible, B−1AB^{-1}AB−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 BBB is singular, any quantum algorithm uses at least Ω(n)\Omega(\sqrt{n})Ω(n​) queries to compute the eigenvalues, where nnn 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.
Quantum phase estimation provides a path to quantum computation of solutions to Hermitian eigenvalue problems $Hv=\ensuremath{\lambda}v$, such as those occurring in quantum chemistry. It is natural to ask whether the same technique can be applied to generalized eigenvalue problems $Av=\ensuremath{\lambda}Bv$, which arise in many areas of science and engineering. We answer this question affirmatively. A restricted class of generalized eigenvalue problems could be solved as efficiently as standard eigenvalue problems. A paradigmatic example is provided by Sturm-Liouville problems. Another example comes from linear ideal magnetohydrodynamics, where phase estimation could be used to determine the stability of magnetically confined plasmas in fusion reactors.
Eigenproblem arises in a large number of disciplines of sciences and engineering. Combining quantum algorithms to solve linear differential equations, quantum singular value estimation and quantum phase estimation, we propose quantum algorithms to compute the eigenvalues of diagonalizable matrices of two types: matrices that only have real eigenvalues and normal matrices. The quantum algorithms return a quantum state such that the first register stores eigenvalues and the second register stores the corresponding eigenvectors. For normal matrices, the complexity to obtain this quantum state is dominated by the complexity to perform quantum singular value estimation. For the other case, the complexity is determined by the solving of a linear system of differential equations. Under certain conditions (e.g. sparse), the complexities are polylog on the size nnn of the given matrix. If we perform measurements, then we can obtain all the eigenvalues classically. If MMM is sss sparse, then the complexity is O~(sn∥M∥max⁡κ2/ϵ2)\widetilde{O}(sn\|M\|_{\max}\kappa^2/\epsilon^2)O(sn∥M∥max​κ2/ϵ2), where κ\kappaκ is the condition number of the matrix of eigenvectors.
Quantum algorithms for estimating the eigenvalues of matrices, including the phase estimation algorithm, serve as core subroutines in a wide range of quantum algorithms, including those in quantum chemistry and quantum machine learning. The standard quantum eigenvalue (phase) estimation algorithm accepts a Hermitian (unitary) matrix and a state in an unknown superposition of its eigenstates as input, and coherently records the estimates for real eigenvalues (eigenphases) in an ancillary register. Extension of quantum eigenvalue and phase estimation algorithms to the case of non-normal input matrices is obstructed by several factors such as non-orthogonality of eigenvectors, existence of generalized eigenvectors and the fact that eigenvalues may lie anywhere on the complex plane. In this work, we propose a novel approach for estimating the eigenvalues of non-normal matrices based on preparation of a state that we call the "resolvent state". We construct the first efficient algorithm for estimating the phases of the unimodular eigenvalues of a given non-unitary matrix. We then construct an efficient algorithm for estimating the real eigenvalues of a given non-Hermitian matrix, achieving complexities that match the best known results while operating under significantly relaxed assumptions on the non-real part of the spectrum. The resolvent-based approach that we introduce also extends to estimating eigenvalues that lie on a parametrized complex curve, subject to explicitly stated conditions, thereby paving the way for a new paradigm of parametric eigenvalue estimation.