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arXiv:2302.12602 · Physical Review A

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

Yuki Sato, Hiroshi C Watanabe, Rudy Raymond, Ruho Kondo, Kaito Wada, Katsuhiro Endo, Michihiko Sugawara, Naoki Yamamoto

Published Aug 28, 2023• 10 citations
arXivHTMLPDF

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 or maximization problem of the fractional form of the expectations of two Hermitians. We then show that the fractional objective function can be analytically minimized or maximized with respect to a single-qubit gate by solving a GEP of a $4\ifmmode\times\else\texttimes\fi{}4$ matrix. Second, we show that a system of linear equations 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.

DOI
10.1103/physreva.108.022429

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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 or maximization problem of the fractional form of the expectations of two Hermitians. We then show 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\ifmmode×\else\texttimes\fi44\ifmmode\times\else\texttimes\fi{}44\ifmmode×\else\texttimes\fi4 matrix. Second, we show that a system of linear equations 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.
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.
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.
The variational quantum eigensolver (VQE) is a hybrid algorithm to find the minimum eigenvalue/vector of the given Hamiltonian by optimizing a parameterized quantum circuit (PQC) using a classical computer. Sequential optimization methods, which are often used in quantum circuit tensor networks, are popular for optimizing the parameterized gates of PQCs. In this paper, we focus on the case where the components to be optimized are single-qubit gates, in which the analytic optimization of a single-qubit gate is sequentially performed. The analytical solution is given by diagonalization of a matrix whose elements are computed from the expectation values of observables specified by a set of predetermined parameters, which we refer to as the parameter configurations. In this paper, we first show that the optimization accuracy significantly depends on the choice of parameter configurations owing to the statistical errors in the expectation values. We then identify a metric that quantifies the optimization accuracy of a parameter configuration for all possible statistical errors, named configuration overhead/cost or C-cost. We theoretically provide the lower bound of C-cost and show that, for the minimum size of parameter configurations, the lower bound is achieved if and only if the parameter configuration satisfies the so-called equiangular line condition. Finally, we provide numerical experiments demonstrating that the optimal parameter configuration exhibits the best result in several VQE problems. We hope that this general statistical methodology will enhance the efficacy of sequential optimization of PQCs for solving practical problems with near-term quantum devices.
In this study, we propose the Quantum Gradient Flow Algorithm (QGFA), a novel quantum algorithm for solving symmetric positive definite (SPD) linear systems based on the variational formulation and time-evolution dynamics. Conventional quantum linear solvers, such as the quantum matrix inverse algorithm (QMIA), focus on approximating the matrix inverse through quantum signal processing (QSP). However, QMIA suffers from a crucial drawback: its computational efficiency deteriorates as the condition number increases. In contrast, classical SPD linear solvers, such as the steepest descent and conjugate gradient methods, are known for their fast convergence, which stems from the variational optimization principle of SPD systems. Inspired by this, we develop QGFA, which obtains the solution vector through the gradient-flow process of the corresponding quadratic energy functional. To validate the proposed method, we apply QGFA to the displacement-based finite element method (FEM) for two-dimensional linear elastic problems under plane stress conditions. The algorithm demonstrates accurate convergence toward classical FEM solutions even with a moderate number of QSP phase factors. Compared with QMIA, QGFA achieves lower relative errors and faster convergence when initialized with suitable initial states, demonstrating its potential as an efficient preconditioned quantum linear solver. The proposed framework provides a physically interpretable connection between classical iterative solvers and quantum computational paradigms. These findings suggest that QGFA can serve as a foundation for future developments in Quantum Computer-Aided Engineering (Quantum CAE), including nonlinear and multiphysics simulations.
We propose an algorithm for variational quantum algorithms (VQAs) to optimize the structure of parameterized quantum circuits (PQCs) efficiently. The algorithm optimizes the PQC structure on-the-fly in VQA by sequentially replacing a single-qubit gate with the optimal one to minimize the objective function. To directly find the optimal gate, our method uses the factorization of matrices whose elements are evaluated on a set of the slightly-modified circuits. The matrix factorization enables us to not only unify the existing sequential methods for further extension but also provide rigorous proofs of their limitation and potential in comparison with conventional gradient-based optimizers. Firstly, when the circuits are sufficiently deep, the sequential methods encounter a barren plateau that the spectrum of the matrix concentrates on a single value exponentially fast with respect to the number of qubits. Secondly, if the objective functions are local observables, they can avoid barren plateaus as long as the depth of the n-qubit PQCs is O(log⁡n)O(\log{n})O(logn). Although the family of these optimizers does not directly employ gradients of the objective function, our results establish their connection with conventional optimizations providing a consistent picture of the barren plateau. We also perform numerical experiments showing the advantages over conventional VQAs and confirm the successful optimization getting over the barren plateau in the ground state problem of the mixed field Ising model up to 12 qubits.
We propose a variational method for constructing the eigenvalues and generalized eigenvalues for an arbitrary N×NN\times NN×N complex matrix. The quantum part of our algorithm is based on encoding the matrix elements into the pure state of a quantum system and expressing the loss function with optimization parameters in terms of certain probability amplitudes in the superposition state. The principal step of this algorithm is the measurement of the ancilla state that removes all extra terms from the above superposition and allows to probabilistically construct the required loss function along with its derivatives with respect to the optimization parameters. These output data are used to find the new values of optimization parameters for the next iteration of the loss function in the gradient optimization method. The depth and size of the circuit for this algorithm are, respectively, O(N2log⁡N)O(N^2 \log N)O(N2logN) and O(log⁡N)O(\log N)O(logN).