T gate applies a 45° phase to the $|1\rangle$ state. Crucial for universal quantum computation and magic state distillation.

Matrix:

$$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}$$

Action: $T|0\rangle = |0\rangle$, $T|1\rangle = e^{i\pi/4}|1\rangle$ (adds $\pi/4$ phase to $|1\rangle$).

• $T^8 = I$: applying T eight times returns to identity
• $T^4 = Z$: four T gates equal one Z gate
• $T^2 = S$: two T gates equal one S gate
• Inverse: $T^\dagger = e^{-i\pi/4}$ (or equivalently, $T^7$)
• Non-Clifford: T is not in the Clifford group (universality comes from T)

$$T = R_Z(\pi/4) = e^{-i\pi Z/8}$$

$$S = T^2 = R_Z(\pi/2)$$

$$Z = T^4 = R_Z(\pi)$$

• Non-Clifford: T cannot be simulated classically, unlike Clifford gates
• Magic state distillation: fault-tolerant systems require distillation; one logical T costs ~1000 physical qubits
• Bottleneck: primary cost limiting fault-tolerant quantum computation

• Universal computation: T + Clifford gates (Hadamard, CNOT) form universal set
• Variational algorithms: T gates appear in some ansatze but should be minimized
• Exact circuits: Solovay-Kitaev approximation requires T gates

• Superconducting qubits: RZ($\pi/4$) via microwave pulse or virtual rotation; ~20 ns
• Trapped ions: controlled phase via laser
• Photonic: phase shifter
• Cost note: typically ~20 ns on NISQ hardware, but bottleneck for fault-tolerant systems