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

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## Abstract

**:**

## 1. Introduction

**any problem**faster on a quantum computer, regardless of its real-world applicability [2].

## 2. The Variational Quantum Linear Solver

#### 2.1. The Variational Ansatz

#### 2.2. Matrix Pauli Decomposition

#### 2.3. Right-Hand Side Preparation

## 3. Computational Details

## 4. Training Algorithm

## 5. Applications

#### 5.1. Application 1: The Poisson Equation

- qc = QuantumCircuit (4)
- U = [0.1,2,2,2,2,2,2,0.1]
- U /= np.linalg.norm (U)
- qc.isometry (U, [0, 1, 2], [])
- qc = transpile (qc, basis_gates = [’u3’, ’cx’], optimization_level=3)

#### 5.1.1. Poisson Case 1: Parabolic Solution with Homogeneous Boundary Conditions

#### 5.1.2. Poisson Case 2: Cubic Solution with Non-Homogeneous B.C.

#### 5.2. Application 2: The Heat Equation

#### 5.3. Application 3: The Wave Equation

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Preparation of the Stiffness Matrix

#### Appendix A.1. Implementing the Recursion Using GHZ States

#### Appendix A.2. Preparation of the m=3 Stiffness Matrix

#### Appendix A.3. Preparation of a General m Qubit Stiffness Matrix

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**Figure 2.**A quantum circuit representing ${\overrightarrow{f}}^{T}=[\phantom{\rule{0.166667em}{0ex}}0.1,2,2,2,2,2,2,0.1\phantom{\rule{0.166667em}{0ex}}]$ 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 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.

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**MDPI and ACS Style**

Trahan, C.J.; Loveland, M.; Davis, N.; Ellison, E. A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods. *Entropy* **2023**, *25*, 580.
https://doi.org/10.3390/e25040580

**AMA Style**

Trahan CJ, Loveland M, Davis N, Ellison E. A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods. *Entropy*. 2023; 25(4):580.
https://doi.org/10.3390/e25040580

**Chicago/Turabian Style**

Trahan, Corey Jason, Mark Loveland, Noah Davis, and Elizabeth Ellison. 2023. "A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods" *Entropy* 25, no. 4: 580.
https://doi.org/10.3390/e25040580