The term "π/8 gate" is a historical convention that refers to a specific rotation angle associated with the T gate in quantum computing. While the T gate adds a phase difference of π/4 (or 45 degrees) to the state vector |1>, it is commonly referred to as the π/8 gate because it represents a rotation by π/8 radians.
The reason for this naming convention originates from the relationship between the T gate and other fundamental gates in quantum computing, specifically the Hadamard (H) gate and the S gate. The T gate can be expressed as the combination of the H and S gates as follows:
T = HSH†,
where H† represents the conjugate transpose of the H gate.
If we apply the T gate twice successively, we get:
TT = (HSH†)(HSH†) = H(SS)H† = HRz(π/4)H†,
where Rz(π/4) represents a rotation around the Z-axis by π/4 radians.
From this, we can observe that applying the T gate twice is equivalent to a rotation around the Z-axis by π/4 radians. In terms of angles, π/4 radians is equivalent to 45 degrees. However, if we look at the angle associated with the T gate alone, it corresponds to half of that rotation, which is π/8 radians or 22.5 degrees.
Therefore, although the T gate adds a phase difference of π/4 to the state vector |1>, it is commonly referred to as the π/8 gate due to its relationship with other gates and the corresponding rotation angles. The naming convention helps to maintain consistency and coherence within the field of quantum computing.