# Solving Multidimensional Partial Differential Equations Using Efficient Quantum Circuits
## 4 Materials and Methods
### Multidimensional Poisson Equation
#### 4.2.4 4D Poisson Equation
#### Hadamard Gate

The Hadamard gate is a single-qubit quantum gate that creates an equal superposition of \(|0\rangle\) and \(|1\rangle\) states when applied to a qubit [50], see Figure A1. Controlled-NOT (CNOT) Gate

The CNOT gate is a two-qubit quantum gate that consists of a control and a target qubit. It facilitates entanglement between qubits [50]. The unitary matrix and the symbol of the CNOT gate are shown in Figure A1. #### SWAP Gate

The SWAP gate is a two-qubit quantum gate that exchanges the states of two qubits [50]. It can be implemented by applying three CNOT gates. The unitary matrix representation and the symbol of the SWAP gate are shown in Figure A1. #### Y-Rotation Gate

The \(R_{y}(\theta)\) rotation gate is a single-qubit gate that rotates the quantum state \(|\psi\rangle\) by an angle \(\theta\) around the \(y\)-axis of the Bloch sphere [50]. The unitary matrix representation and the symbol of \(R_{y}(\theta)\) are shown in Figure 11.

#### Z-Rotation Gate

The \(R_{z}(\phi)\) gate rotates a qubit \(|\psi\rangle\) by an angle \(\phi\) around the \(z\)-axis of the Bloch sphere [50]. The unitary matrix representation and the symbol of \(R_{z}(\phi)\) are shown in Figure 11.

#### Quantum Measurement

The quantum measurement or measurement gate is required to obtain classical results from a quantum circuit. This gate collapses the qubit state \(|\psi\rangle\) to its computational basis states, i.e., \(|0\rangle\) or \(|1\rangle\)[50], and is non-unitary and irreversible. As shown in Figure 11, the classical data \(\psi^{\text{classical}}_{\text{decoded-data}}\) after measurement are represented by double line rails. ## Appendix B Classical Preprocessing

### Finite Difference Method (FDM)

The Finite Difference Method (FDM) is a numerical technique for solving Partial Differential Equations (PDEs) by approximating them with difference equations [15]. This technique converts a continuous differential equation into a finite number of grid points, see (A3). Here, \(A\) is the coefficient matrix, \(\vec{u}\) is the vector of unknown function, and \(\vec{b}\) represents the discretized source term. \[A\vec{u}=\vec{b}\] (A3)

### Polar Decomposition

Polar decomposition is a matrix decomposition technique that decomposes a complex square matrix \(A\), see (A3) into positive semi-definite symmetric matrix \(P\) and a unitary matrix \(U\)[51; 52]. Using polar decomposition we, therefore, can express \(A\) as shown in (A4). We have used polar decomposition as a preprocessing step in our proposed algorithm (discussed in detail in Section 4.1)

\[A=PU\] (A4)

### Singular Value Decomposition (SVD)

Singular Value Decomposition (SVD) [38] decomposes any matrix into three components: an orthogonal matrix \(W\), a diagonal matrix \(\Sigma\), and the conjugate transpose of the orthogonal matrix \(V^{\dagger}\). In quantum computing, SVD assists with encoding classical data into the quantum domain [53]. Our algorithm employs SVD as a pre-processing step to decompose matrix \(A\) (discussed in Section 4.1) into two orthogonal matrices, \(W\) and \(V^{\dagger}\), and a diagonal matrix, \(\Sigma\), as shown in (A5). \[A=W\Sigma V^{\dagger}\] (A5)

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