$T$
The $T$ gate applies a phase of $\mathrm{e}^{i\pi/4}$ to $|1\rangle$.
$$ T = \begin{bmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i\frac{\pi}{4}} \end{bmatrix} $$
Properties
- Non-Clifford; together with Clifford gates yields universal quantum computation.
- $T^2 = S$, $T^4 = Z$, and $T^8 = I$.
- $P(\pi/4) = T$.
- Often referred to as the $\pi/8$ gate because when viewed as a rotation around the $Z$-axis of the Bloch sphere, it can be written as $T = \mathrm{e}^{i\pi/8}R_z(\pi/4)$.
Usage
- Resource gate in fault-tolerant protocols (magic state distillation).
- Phase rotations and compiling arbitrary single-qubit unitaries.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis.
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| SDK | Name |
|---|---|
| Qiskit | qiskit.circuit.library.TGate |
| PennyLane | pennylane.T |
| Cirq | cirq.T |
| Q# | Std.Intrinsic.T |
| PyQuil | pyquil.gates.T |
| Braket | braket.circuits.gates.T |
| BQSKit | bqskit.ir.gates.TGate |
| Qibo | qibo.gates.T |
| pytket | pytket.circuit.OpType.T |
| Stim | — |
| OpenQASM | stdgates.inc: t |