# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
The results showed superior accuracy, scalability, and faster execution times compared to PDE solvers based on Variational Quantum Algorithms (VQAs) on both noise-free and noisy simulators, as well as hardware emulators, and real quantum hardware from IBM simulators. We have also provided a generalized algorithm that computes the coefficient matrix and the vector representing the source of the Poisson equation up to four dimensions. With minor modifications, this algorithm can be extended to handle any number of dimensions and can solve various types of PDEs. We also tested various noise mitigation techniques, especially Pauli Twirling and concluded that these techniques are still not mature enough to significantly improve circuit execution results on quantum emulators or real quantum hardware. However, noise mitigation techniques are still an active area of research and are continuously improving. With ongoing research in the field of data encoding techniques, continuously improving compilers, rapid progress in noise mitigation techniques, and advancements in quantum hardware, we anticipate that our proposed algorithm will be further optimized on quantum simulators, emulators, and real quantum hardware. Conceptualization: M.C., K.E.-A., A.N. and E.E.-A. ; Methodology: M.C., K.E.-A., A.N., V.J., D.K. and E.E.-A. ; Software: M.C., K.E.-A., A.N. and E.E.-A. ; Validation: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B. and E.E.-A. ; Formal analysis: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B., S.K. and E.E.-A. ; Investigation: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B., S.K. and E.E.-A. ; Resources: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B., S.K. and E.E.-A. ; Data curation: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B. and E.E.-A. ; Writing--original draft preparation: M.C., K.E.-A., A.N., V.J., D.K., I.I. and E.E.-A. ; Writing--review and editing: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B. and E.E.-A. ; Visualization: M.C., K.E.-A., A.N., V.J., D.K., I.I., M.S., S.O., B.P., K.E., S.T., D.B. and E.E.-A. ; Supervision: E.E.-A. ; Supervision: E.E.-A. ; Project administration: E.E.-A. ; Funding acquisition: E.E.-A. ; All authors have read and agreed to the published version of the manuscript. This research received no external funding. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. This research used resources from the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00 OR22725. The authors declare no conflicts of interest. ## Appendix A Quantum Computing Basics

### Quantum Bits and States

A quantum bit or a _qubit_ is the fundamental unit of quantum information. Unlike a classical bit, which can only be 0 or 1, a qubit can exist in superposition of basis states, \(|0\rangle\)and \(|1\rangle\), which can be represented by a state vector \(|\psi\rangle\), see (A1) [50]. Here, \(c_{0}\) and \(c_{1}\) are the complex amplitudes of the basis states. \[|\psi\rangle=c_{0}|0\rangle+c_{1}|1\rangle= \begin{bmatrix}c_{0}\\ c_{1}\end{bmatrix} ,\;\text{where}\;|c_{0}|^{2}+|c_{1}|^{2}=1,\,c_{i}\in\mathbb{C}\] (A1)

For any \(n\)-qubit quantum system, the number of possible states in the statevector is \(N=2^{n}\). As shown in (A2), each element \(c_{j}\) corresponds to the amplitude of the \(j^{\text{th}}\) element in \(|\psi\rangle\) or, the basis state \(|j\rangle\). \[|\psi\rangle=\sum_{j=0}^{N-1}c_{j}\cdot|j\rangle,\;\text{where}\;\sum_{j=0}^{N- 1}|c_{j}|^{2}=1,\,\text{and}\;0\leq j<N\] (A2)

### Quantum Gates

Quantum gates are unitary operations that transform the state of a qubit. In this section, we present an overview of single- and multi-qubit gates relevant to our proposed algorithm.