added S gate

docs/quantum-gates-circuits/s-gate.md

@@ -0,0 +1,133 @@
+---
+sidebar_position: 5
+sidebar_label: 6. S Gate
+---
+
+# S Gate
+
+## Definition
+The S gate is a single-qubit gate that rotates the qubit state by 90 degrees around the $Z$ of the Bloch sphere. The relationship between the S gate and the Pauli-Z gate is similar to the relationship between the T gate and the $\sqrt{Pauli-Z}$ gate. The S gate is also known as the $\sqrt{Z}$ gate. It is also called as Clifford gate or $\frac{\pi}{2}$-gate. 
+
+
+## Effect on qubit
+The S gate changes the phase of the qubit by 90 degrees. The S gate changes the phase of the qubit from $+1$ to $i$ and vice versa, where $i$ is the imaginary unit. (Note that $i = e^{i\pi/2}$) [An exact behaviour of T gate, just the difference in the phase change]
+
+## Types
+The S gate has only one type.
+
+## Matrix representation
+The matrix representation of the S gate is:
+    * `S gate`: $\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}$ 
+
+The matrix can also be represented as $\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix}$
+
+where $i$ is equal to $e^{i\pi/2}$.
+
+## Circuit representation
+The S gate is represented as ` ───S─── ` in the circuit.
+
+## Example
+
+Let's take an example to demonstrate the S gate.
+
+
+*   S Gate
+Suppose we have a qubit initially in the state $|0\rangle$, represented as:
+
+$$
+|q_0\rangle = |0\rangle
+\tag{22}
+$$
+
+The S gate is represented by the following matrix:
+
+$$
+S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}
+\tag{23}
+$$
+
+To apply the S gate to the qubit $|q_0\rangle = |0\rangle$, we perform a matrix multiplication of the S gate matrix with the state vector representing $|0\rangle$.
+
+$$
+S|q_0\rangle = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix}
+\tag{24}
+$$
+
+Performing the matrix multiplication:
+
+$$
+S|q_0\rangle = \begin{bmatrix} 1*1 + 0*0 \\ 0*1 + i*0 \end{bmatrix}
+\tag{25}
+$$
+
+Simplifying:
+
+$$
+S|q_0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}
+\tag{26}
+$$
+
+Thus, the S gate does not change the state of the qubit $|0\rangle$.
+
+It does not affect the state of the qubit $|0\rangle$ as the phase of the qubit is already $+1$. So, the S gate does not introduce any phase change to the qubit $|0\rangle$. Instead, it introduces a phase of $90$ degrees to the qubit $|1\rangle$.
+
+Therefore, after applying the S gate, the state of the qubit $|q_0\rangle$ remains unchanged, indicating that the S gate only changes the phase of the qubit.
+
+Now, let's see the effect of the S gate on the qubit $|1\rangle$.
+
+Suppose we have a qubit initially in the state $|1\rangle$, represented as:
+
+$$
+|q_1\rangle = |1\rangle
+\tag{27}
+$$
+
+To apply the S gate to the qubit $|q_1\rangle = |1\rangle$, we perform a matrix multiplication of the S gate matrix with the state vector representing $|1\rangle$.
+
+$$
+S|q_1\rangle = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix}
+\tag{28}
+$$
+
+Performing the matrix multiplication:
+
+$$
+S|q_1\rangle = \begin{bmatrix} 1*0 + 0*1 \\ 0*0 + i*1 \end{bmatrix}
+\tag{29}
+$$
+
+Simplifying:
+
+$$
+S|q_1\rangle = \begin{bmatrix} 0 \\ i \end{bmatrix}
+
+\tag{30}
+$$
+
+Thus, the S gate changes the state of the qubit $|1\rangle$ to $i|1\rangle$.
+
+Therefore, the S gate introduces a phase of $90$ degrees to the qubit $|1\rangle$.
+
+Hence, the S gate changes the phase of the qubit by 90 degrees.
+
+The S gate is used in various quantum algorithms and quantum circuits to introduce phase changes to the qubits.
+
+## Properties
+This gate has exact same properties as the T gate ie. $S^2 = Z$ and $S|0\rangle = |0\rangle$ and $S|1\rangle = i|1\rangle$.
+
+## Conjugate Transpose
+The conjugate transpose of the S gate is:
+    * $S^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}$
+
+## Inverse
+The inverse of the S gate is:
+    * $S^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}$
+
+## Dagger
+The dagger of the S gate is:
+    * $S^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}$
+
+
+
+
+

docs/quantum-gates-circuits/t-gate.md

@@ -8,7 +8,7 @@ sidebar_label: 5. T Gate
 ## Definition
 Similar to the Phase gate, the T gate is a single-qubit gate that introduces a phase of 45 degrees. The T gate is also known as the $\sqrt{Pauli-Z}$ gate. The T gate is used to change the phase of the qubit in terms of the $Z$ axis of the Bloch sphere. 
 The T gate is the most commonly used gate in quantum computing after the Pauli gates, Hadamard gate, and the Phase gate.
-
+    
 ## Effect on qubit
 The T gate changes the phase of the qubit by 45 degrees. The T gate changes the phase of the qubit from $+1$ to $e^{i\pi/4}$ and vice versa, where $i$ is the imaginary unit.