A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods

Abstract

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Keywords: Poisson equation; finite-element methods; heat equation; quantum algorithms; quantum computing; quantum variational algorithm.

Figure 1

A four-qubit example of the fixed ansatz used for this study.

Figure 2

A quantum circuit representing ${\overrightarrow{f}}^T=[0.1,2,2,2,2,2,2,0.1]$ found using Qiskit’s Isometry command.

Figure 3

Two-qubit VQLS cost function results for the reduced Poisson problem with homogeneous Dirichlet boundary conditions. The results were averaged over 20 trial runs. Variances are shown by respective bars.

Figure 4

Wall clock time in seconds versus the number of layers for the two-qubit VQLS reduced Poisson problem with homogeneous Dirichlet boundary conditions.

Figure 5

Three-qubit VQLS cost function results for the reduced Poisson problem with homogeneous Dirichlet boundary conditions. The results were averaged over 10 trial runs. Variances are shown by respective bars.

Figure 6

Three-qubit (eight-node) VQLS results (filled circles with dashed lines) for reduced Poisson problem with homogeneous Dirichlet boundary conditions. The classical discrete solution is shown with a solid black line.

Figure 7

The root mean squared solution error versus the number of layers for the three-qubit VQLS reduced Poisson problem with homogeneous Dirichlet boundary conditions. Here, the errors were averaged over all 10 runs for each layer.

Figure 8

Wall clock time in seconds versus the number of layers for the three-qubit VQLS reduced Poisson problem with homogeneous Dirichlet boundary conditions. The three-layer run does not converge.

Figure 9

The COBYLA cost convergence for a range of shots in the two-qubit VQLS reduced Poisson problem with homogeneous Dirichlet boundary conditions.

Figure 10

Two-qubit, two-layer solution (filled circles) along with the analytic solution (solid line) of Case 2: the cubic Poisson problem.

Figure 11

VQLS mean cost versus iteration or optimization count over a range of layers for the cubic Poisson problem. Variances are shown as curve error bars.

Figure 12

Analytic solution (solid line) versus two-qubit VQLS-based finite element results (dashed line with open circles) for the time-dependent heat equation at each time step.

Figure 13

The COBYLA iteration count over time for the two-qubit solution of the heat equation.

Figure 14

Analytic solution (solid line) versus two-qubit VQLS-based finite element results (dashed line with open circles) for the time-dependent wave equation at each time step.

Figure 15

The COBYLA iteration count over time for the two-qubit solution of the wave equation.