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S Gate

Definition​

The S gate is a single-qubit gate that rotates the qubit state by 90 degrees around the ZZ 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 Pauli−Z\sqrt{Pauli-Z} gate. The S gate is also known as the Z\sqrt{Z} gate. It is also called as Clifford gate or π2\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+1 to ii and vice versa, where ii is the imaginary unit. (Note that i=eiπ/2i = 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: [100i]\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}

The matrix can also be represented as [100eiπ/2]\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix}

where ii is equal to eiπ/2e^{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⟩|0\rangle, represented as:
∣q0⟩=∣0⟩(1)|q_0\rangle = |0\rangle \tag{1}

The S gate is represented by the following matrix:

S=[100i](2)S = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \tag{2}

To apply the S gate to the qubit ∣q0⟩=∣0⟩|q_0\rangle = |0\rangle, we perform a matrix multiplication of the S gate matrix with the state vector representing ∣0⟩|0\rangle.

S∣q0⟩=[100i][10](3)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∣q0⟩=[1∗1+0∗00∗1+i∗0](4)S|q_0\rangle = \begin{bmatrix} 1*1 + 0*0 \\ 0*1 + i*0 \end{bmatrix} \tag{4}

Simplifying:

S∣q0⟩=[10](5)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⟩|0\rangle.

It does not affect the state of the qubit ∣0⟩|0\rangle as the phase of the qubit is already +1+1. So, the S gate does not introduce any phase change to the qubit ∣0⟩|0\rangle. Instead, it introduces a phase of 9090 degrees to the qubit ∣1⟩|1\rangle.

Therefore, after applying the S gate, the state of the qubit ∣q0⟩|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⟩|1\rangle.

Suppose we have a qubit initially in the state ∣1⟩|1\rangle, represented as:

∣q1⟩=∣1⟩(6)|q_1\rangle = |1\rangle \tag{6}

To apply the S gate to the qubit ∣q1⟩=∣1⟩|q_1\rangle = |1\rangle, we perform a matrix multiplication of the S gate matrix with the state vector representing ∣1⟩|1\rangle.

S∣q1⟩=[100i][01](7)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∣q1⟩=[1∗0+0∗10∗0+i∗1](8)S|q_1\rangle = \begin{bmatrix} 1*0 + 0*1 \\ 0*0 + i*1 \end{bmatrix} \tag{8}

Simplifying:

S∣q1⟩=[0i](9)S|q_1\rangle = \begin{bmatrix} 0 \\ i \end{bmatrix} \tag{9}

Thus, the S gate changes the state of the qubit ∣1⟩|1\rangle to i∣1⟩i|1\rangle.

Therefore, the S gate introduces a phase of 9090 degrees to the qubit ∣1⟩|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. S2=ZS^2 = Z and S∣0⟩=∣0⟩S|0\rangle = |0\rangle and S∣1⟩=i∣1⟩S|1\rangle = i|1\rangle.

Conjugate Transpose​

The conjugate transpose of the S gate is: * S†=[100−i]S^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}

Inverse​

The inverse of the S gate is: * S−1=[100−i]S^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}

Dagger​

The dagger of the S gate is: * S†=[100−i]S^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix}