Variational Quantum Linear Solver for Multiphysics Problems (Final)
Mostafa Atallah
Cairo University
∂x2∂2T=0
\frac{\partial^2 T}{\partial x^2} = 0
The Laplace Equation
Ti+1−2Ti+Ti−1=0
T_{i+1} - 2T_{i}+ T_{i-1} = 0
apply FDM
−201000000−201000010−201000010−211000011−201000010−201000010−200000010−2T0T1T2T3T4T5T6T7=00000000
\begin{pmatrix}
-2 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & -2 & 0 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & -2 & 0 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & -2 & 1 & 1 & 0 & 0\\
0 & 0 & 1 & 1 & -2 & 0 & 1 & 0\\
0 & 0 & 0 & 1 & 0 & -2 & 0 & 1\\
0 & 0 & 0 & 0 & 1 & 0 & -2 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & 0 & -2
\end{pmatrix}
\begin{pmatrix}
T_{0} \\
T_{1} \\
T_{2} \\
T_{3} \\
T_{4} \\
T_{5} \\
T_{6} \\
T_{7}
\end{pmatrix}
=\begin{pmatrix}
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0
\end{pmatrix}
Ax=b
A x = b
A=2(I−0.125X0X1X2+0.125X0Y1Y2+0.125Y0X1Y2−0.125Y0Y1X2−0.25X1X2−0.25Y1Y2−0.5X2)
A = 2 \left( I -0.125\; X_0 X_1 X_2 + 0.125 \; X_0 Y_1 Y_2 + 0.125 \; Y_0 X_1 Y_2 - 0.125 \; Y_0 Y_1 X_2 - 0.25 \; X_1 X_2 - 0.25 \; Y_1 Y_2 - 0.5 \; X_2 \right)
VQLS
Ry(αi)=e−αiY/2
R_y(\alpha_i) = \mathbf{e}^{-\imath \; \alpha_i Y / 2}
V(α)
V(\alpha)
H=A†(I−∣b><b∣)A
C=<x∣H∣x>
Cost Convergence
Cost vs. Shots
Goals Checklist
- Study and analyze the performance of the Variational Quantum Linear Solver (VQLS) in general.
- Solve some Multiphysics problems such as the Heat equation, etc. using VQLS.
- Execute the results using Qiskit on both a simulator and a device.
- Work with higher dimensions and employ Qiskit runtime and the local cost approach.
- Study quantum chip qubits connectivity and use the error mitigation techniques which can significantly improve expectation values and therefore the final fidelity.
Variational Quantum Linear Solver for Multiphysics Problems (Final) Mostafa Atallah Cairo University