Variational quantum algorithm for generalized eigenvalue problems and its application to the finite element method
arXiv:2302.12602 · doi:10.1103/PhysRevA.108.022429
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 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.
arXiv comment19 pages, 10 figures
References in corpus (27)
- A variational eigenvalue solver on a quantum processor
- Quantum algorithm for solving linear systems of equations
- Variational Quantum Algorithms
- Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
- Simulated Quantum Computation of Molecular Energies
- Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
- The Variational Quantum Eigensolver: a review of methods and best practices
- Scalable Quantum Simulation of Molecular Energies
- Variational Quantum Computation of Excited States
- Subspace-search variational quantum eigensolver for excited states
- Structure optimization for parameterized quantum circuits
- Variational Quantum State Diagonalization
- Sequential minimal optimization for quantum-classical hybrid algorithms
- Variational algorithms for linear algebra
- Hardware-efficient variational quantum algorithms for time evolution
- Variational Quantum State Eigensolver
- Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicator
- Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation
- Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
- Variational Quantum Algorithms for Dimensionality Reduction and Classification
- Quantum phase estimation for a class of generalized eigenvalue problems
- Application of a variational hybrid quantum-classical algorithm to heat conduction equation
- Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations
- Simulating Time Evolution with Fully Optimized Single-Qubit Gates on Parameterized Quantum Circuits
- Quantum algorithms for the generalized eigenvalue problem
- Computationally Efficient Quantum Expectation with Extended Bell Measurements
- Excited-state molecular dynamics simulation based on variational quantum algorithms