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:

$$|q_0\rangle = |0\rangle \tag{1}$$

The S gate is represented by the following matrix:

$$S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \tag{2}$$

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 $|0\rangle$.

$$S|q_0\rangle = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \tag{3}$$

Performing the matrix multiplication:

$$S|q_0\rangle = \begin{bmatrix} 1*1 + 0*0 \\ 0*1 + i*0 \end{bmatrix} \tag{4}$$

Simplifying:

$$S|q_0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \tag{5}$$

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:

$$|q_1\rangle = |1\rangle \tag{6}$$

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 $|1\rangle$.

$$S|q_1\rangle = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \tag{7}$$

Performing the matrix multiplication:

$$S|q_1\rangle = \begin{bmatrix} 1*0 + 0*1 \\ 0*0 + i*1 \end{bmatrix} \tag{8}$$

Simplifying:

$$S|q_1\rangle = \begin{bmatrix} 0 \\ i \end{bmatrix} \tag{9}$$

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:

• $S^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}$