Quantum algorithms for the generalized eigenvalue problem

Abstract

The generalized eigenvalue (GE) problems are of particular importance in various areas of science engineering and machine learning. We present a variational quantum algorithm for finding the desired generalized eigenvalue of the GE problem, $\mathcal{A}|\psi ⟩=\lambda \mathcal{B}|\psi ⟩$, by choosing suitable loss functions. Our approach imposes the superposition of the trial state and the obtained eigenvectors with respect to the weighting matrix $\mathcal{B}$ on the Rayleigh-quotient. Furthermore, both the values and derivatives of the loss functions can be calculated on near-term quantum devices with shallow quantum circuit. Finally, we propose a full quantum generalized eigensolver (FQGE) to calculate the minimal generalized eigenvalue with quantum gradient descent algorithm. As a demonstration of the principle, we numerically implement our algorithms to conduct a 2-qubit simulation and successfully find the generalized eigenvalues of the matrix pencil $(\mathcal{A},\mathcal{B})$. The numerically experimental result indicates that FQGE is robust under Gaussian noise.

References in corpus (8)

• Quantum algorithm for solving linear systems of equations
• Quantum random access memory
• Simulating Hamiltonian dynamics with a truncated Taylor series
• Quantum Computation of Electronic Transitions using a Variational Quantum Eigensolver
• Simulating chemistry efficiently on fault-tolerant quantum computers
• Quantum Anomaly Detection with Density Estimation and Multivariate Gaussian Distribution
• Avoiding local minima in Variational Quantum Algorithms with Neural Networks
• Incoherent quantum algorithm dynamics of an open system with near-term devices

Cited by in corpus (6)

• Sharing Quantum Nonlocality in Star Network Scenarios
• Quantum gradient descent algorithms for nonequilibrium steady states and linear algebraic systems
• Variational quantum algorithm for generalized eigenvalue problems and its application to the finite element method
• Quantum Information Masking in Non-Hermitian Systems and Robustness
• Improved iterative quantum algorithm for ground-state preparation
• Tighter Constraints of Multipartite Systems in terms of General Quantum Correlations