# Review Quantum Circuit Synthesis for [PERSON]'s Algorithm Oracle
## 2 Background  
### Quantum Computing  
Quantum computing is a field that relies on the principles of quantum mechanics to solve problems more efficiently than classical computers [4]. Unlike classical computers, where information is represented by the binary states 1 and 0, quantum mechanics allows the extension of classical systems by using the superposition of these two states. This new representation receives the name of qubit, the fundamental unit of information in quantum computing [14].  
The qubit representation can be described using the Dirac notation; two vectors represent the binary states \(\ket{0}\) and \(\ket{1}\), as seen in Equation (1).  
\[\ket{0}=\begin{bmatrix}1\\ 0\end{bmatrix};\ \ket{1}=\begin{bmatrix}0\\ 1\end{bmatrix} \tag{1}\]  
The linear combination of these two vectors can produce a new vector \(\psi\), defined as a new quantum state in an arbitrarily unknown state using the quantum superposition principle [15]. The complex coefficients accompanying this expression are \(\alpha\) and \(\beta\), as shown in Equation (2).  
\[\ket{\psi}=\alpha\ket{0}+\beta\ket{1} \tag{2}\]  
The superposition principle states that the linear combination of two or more state vectors is another state vector in the same Hilbert space and describes another system state [16]. This is why every quantum circuit corresponds to a particular unitary operator \(U\) in the Hilbert space that transforms an initial quantum state into another quantum state, meeting the criteria shown in Equation (3) for maintaining the superposition principle [15].  
\[U^{\dagger}U=UU^{\dagger}=1 \tag{3}\]  
In addition to the superposition principle, the Born rule states that the modulus square of the amplitude of a state is the probability of that state's resulting after measurement [16]. This implies that the coefficient present in Equation (2) can be operated to obtain the likelihood of the states \(\ket{0}\) and \(\ket{1}\), and because all the possible states are present in Equation (2), they must meet a normality condition where the sum of the probabilities of the states is equal to 1, as shown in Equation (4).  
\[|\alpha|^{2}+|\beta|^{2}=1 \tag{4}\]  
Then, based on the definition of the quantum circuit as \(U\) matrix, we can decompose this matrix into smaller matrices representing part of the circuit, called quantum gates. These gates can interact with a single qubit, as in the case of the \(X\) gate that is shown in Equation (5).  
\[X=\begin{bmatrix}0&1\\ 1&0\end{bmatrix} \tag{5}\]  
This gate is a unary gate that is responsible for flipping the states of a single qubit, as it is shown in Equation (6).  
\[X\ket{0}=\ket{1};X\ket{1}=\ket{0} \tag{6}\]  
There also exist gates that can interact with multiple qubits at the same time. One of them is the \(CX\) (\(CNOT\)) Gate, shown in Equation (7), where an \(X\) gate is applied depending on the value of a control qubit.  
\[CX=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\end{bmatrix} \tag{7}\]A quantum circuit can compromise multiple gates for its construction, as shown in Figure 1. Each horizontal line represents a qubit, and the symbols on the lines represent the basic operations applied to that qubit. The circuit is organized chronologically from left to right.