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Phase Gate

Definition​

The phase gate is a single-qubit gate used to change the phase of the qubit. Also known as the RϕR_\phi gate, it alters the phase of the qubit along the ZZ axis of the Bloch sphere, performing a phase-flip operation. It is one of the most commonly used gates in quantum computing after the Pauli gates and the Hadamard gate.

Effect on qubit​

The phase gate changes the phase of the qubit from +1+1 to −1-1 and vice versa.

PS: You might be thinking that this operation is performed by the Pauli-Z gate as well. Indeed, both gates perform the same operation, but the Phase gate is a generalization of the Pauli-Z gate. The distinction lies in the phase change introduced: Pauli-Z gate introduces a phase change of π\pi (180180 degrees) to the ∣1⟩|1\rangle state, while the Phase gate introduces a phase change of π/2\pi/2 (9090 degrees) to the ∣1⟩|1\rangle state.

Types​

The phase gate has only one type.

Matrix representation​

The matrix representation of the phase gate is:

  • Phase gate: [100eiπ/2]\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix}

Circuit representation​

The phase gate is represented as ───Rz─── in the circuit.

Example​

Let's demonstrate the phase gate with an example.

  • Phase Gate (Rz) 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 phase gate is represented by the following matrix:

Rz=[100eiπ/2](2)Rz = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix} \tag{2}

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

Rz∣q0⟩=[100eiπ/2][10](3)Rz|q_0\rangle = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \tag{3}

Performing the matrix multiplication:

Rz∣q0⟩=[1∗1+0∗00∗1+eiπ/2∗0](4)Rz|q_0\rangle = \begin{bmatrix} 1*1 + 0*0 \\ 0*1 + e^{i\pi/2}*0 \end{bmatrix} \tag{4}

Simplifying:

Rz∣q0⟩=[10](5)Rz|q_0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \tag{5}

Therefore, after applying the phase gate, the state of the qubit ∣q0⟩|q_0\rangle remains unchanged, indicating that the phase gate only changes the phase of the qubit.

Let's take the phase of the qubit to be ϕ=π/2\phi = \pi/2. Then, the matrix representation of the phase gate is:

Rz=[100eiπ/2]=[100i](6)Rz = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \tag{6}

After applying the phase gate, the state of the qubit ∣q0⟩|q_0\rangle transforms to:

∣q1⟩=Rz∣q0⟩=[100i][10]=[1∗1+0∗00∗1+i∗0]=[10](7)|q_1\rangle = Rz|q_0\rangle = \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1*1 + 0*0 \\ 0*1 + i*0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \tag{7}

The ket representation of the final state is:

∣q1⟩=[10]=∣0⟩(8)|q_1\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = |0\rangle \tag{8}

The circuit representation of this operation is as follows:

Initial state: ∣0⟩|0\rangle

───Rz───

Final state: ∣0⟩|0\rangle

Properties​

Conjugate Transpose​

The conjugate transpose of the phase gate is:

  • Phase gate: [100eiπ/2]\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix}

Inverse​

The inverse of the phase gate is:

  • Phase gate: [100eiπ/2]\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/2} \end{bmatrix}

Dagger​

The dagger of the phase gate is:

  • Phase gate: Rz†=RzRz^\dagger = Rz