T Gate

The “magic” gate that enables universal quantum computation when combined with Clifford gates.

The T gate (also called the π/8 gate) is a single-qubit phase gate needed for universal quantum computation.

Definition

$$T=\left(\begin{array}{cc}1& 0\\ 0& e^{i\pi /4}\end{array}\right)$$

Action: $|0⟩\to |0⟩$, $|1⟩\to e^{i\pi /4}|1⟩$

The name “π/8 gate” comes from writing $T=e^{i\pi /8}{R}_z(\pi /4)$ where the global phase is π/8.

Why T is Special

Non-Clifford

The Clifford gates (H, S, CNOT, and combinations) can be efficiently simulated classically. Adding the T gate makes the gate set universal and computationally powerful.

$$\text{Clifford}+T=\text{Universal}$$

Magic States

T gates can be implemented using “magic states” and Clifford operations: $$|T⟩=T|+⟩=\frac{1}{\sqrt{2}}(|0⟩+e^{i\pi /4}|1⟩)$$

This is important for fault-tolerant quantum computing.

T Gate Count

In fault-tolerant quantum computing, T gates are typically the most expensive operation (requiring complex state distillation). Algorithm efficiency is often measured in T-count: the number of T gates required.

Properties

$$T^2=S\text{(S gate)}$$ $$T^4=Z\text{(Pauli Z)}$$ $$T^8=I\text{(Identity)}$$

T† (T-dagger) is the inverse: $$T^{†}=\left(\begin{array}{cc}1& 0\\ 0& e^{-i\pi /4}\end{array}\right)$$

Bloch Sphere

T is a π/4 (45°) rotation around the Z-axis, an eighth of a full rotation.

In Practice

T gates appear throughout quantum algorithms:

• Toffoli gate decompositions
• Arbitrary rotation synthesis
• Quantum arithmetic circuits
• Phase kickback in phase estimation

See also: Phase Gate, Universal Gate Set, Fault-Tolerant Quantum Computing, Toffoli Gate