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T

T gate tile from the Operations catalog

The T gate applies a smaller phase shift than S - \( \pi/4 \) (45°) instead of \( \pi/2 \). It's also called the √S gate, and is especially important because it's one of the few gates needed to reach any quantum computation when combined with H and CNOT.
Qubits 1
Parameters None
Inverse T† (not self-inverse)
Also called √S, π/8 gate; equal to P(π/4)

What it does

T adds a relative phase of \( \pi/4 \) between \( |1\rangle \) and \( |0\rangle \). \( T^2 = S \) and \( T^4 = Z \) - four T gates in a row bring you back to where two S gates (or one Z) would.

Matrix

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

Action on basis states

  • \( |0\rangle \rightarrow |0\rangle \)
  • \( |1\rangle \rightarrow e^{i\pi/4}\,|1\rangle \)

Phase and Bloch-sphere effect

T rotates the Bloch vector by only \( \pi/4 \) about the Z-axis - an eighth of a full turn, and half of what S does. See the worked H → T → S example on the Phase disks page for how this looks in the app.

Example

H then T on q[0] gives \( \dfrac{1}{\sqrt{2}}(|0\rangle + e^{i\pi/4}|1\rangle) \): identical measurement odds to plain H, with the \( |1\rangle \) component's phase color rotated an eighth of the way around the Phase wheel - half as far as S would move it.

Other notations

Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.

Language Typical form
OpenQASM 2.0 t q[0];
Qiskit circuit.t[q[0]]
Cirq cirq.T(q[0])
Q# T(q[0]);

Notes

T is not one of the Clifford gates (I, X, Y, Z, H, S, CNOT) - adding it is what makes a gate set "universal," able to approximate any quantum computation. This is why T shows up so often in algorithm resource-counting discussions.