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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 Pauli−Z\sqrt{Pauli-Z} gate. The T gate is used to change the phase of the qubit in terms of the ZZ 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+1 to eiπ/4e^{i\pi/4} and vice versa, where ii is the imaginary unit.

Types​

The T gate has only one type.

Matrix representation​

The matrix representation of the T gate is:

  • T gate: [100eiπ/4]\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{bmatrix}

Circuit representation​

The T gate is represented as ───T─── in the circuit.

Example​

Let's take an example to demonstrate the T gate.

  • T 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 T gate is represented by the following matrix:

T=[100eiπ/4](2)T = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{bmatrix} \tag{2}

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

T∣q0⟩=[100eiπ/4][10](3)T|q_0\rangle = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \tag{3}

Performing the matrix multiplication:

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

Simplifying:

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

Thus, the T 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 T gate does not introduce any phase change to the qubit ∣0⟩|0\rangle. Instead, it introduces a phase of 4545 degrees to the qubit ∣1⟩|1\rangle.

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

Properties​

The T gate is self-adjoint, i.e., T†=TT^{\dagger} = T. The T gate is also its own inverse, i.e., T2=IT^2 = I, where II is the identity matrix.

Conjugate Transpose​

The conjugate transpose of the T gate is:

  • T†=[100e−iπ/4]T^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{bmatrix}

Inverse​

The inverse of the T gate is:

  • T−1=[100e−iπ/4]T^{-1} = \begin{bmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{bmatrix}

Dagger​

The dagger of the T gate is:

  • T†=[100e−iπ/4]T^{\dagger} = \begin{bmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{bmatrix}