No. T applied 8 times equals the identity (T⁸ = I). T² = S (the phase gate). T's inverse is T-dagger.

T Gate (π/8 phase): Interactive Explainer

Q: Why is the T gate important?

Together with the Hadamard and CNOT gates, T forms a universal gate set: any quantum operation can be approximated to arbitrary precision using just H, T, and CNOT.

Q: Why is the T gate sometimes called the π/8 gate?

Old convention: it was originally written with eigenvalues e^(±iπ/8) (after factoring out a global phase), giving the name. The actual rotation about the z-axis is π/4 (45°).

The T gate rotates the qubit 45° about the z-axis of the Bloch sphere. It leaves |0⟩ alone and multiplies |1⟩ by e^iπ/4, a complex phase that's invisible on basis states but very real for superpositions, which is why the demo above starts at |+⟩.

T is the third member of the H–T–CNOT universal gate set. Any unitary operation, on any number of qubits, can be built from those three to whatever precision you want.

Pauli-Z: a 180° z-rotation (Z = T⁴).
Hadamard gate: the second member of the universal set.
CNOT gate: the two-qubit member of the universal set.
What is a quantum gate? The umbrella concept.

Frequently asked questions

Why is the T gate important?

Together with H and CNOT, the T gate forms a universal gate set: any quantum operation can be approximated to arbitrary precision using just these three.

Is T self-inverse?

No. T⁸ = I, and T² = S (the phase gate). The inverse of T is T^†, which rotates by −π/4 about z.

Why is the T gate sometimes called the π/8 gate?

Historical convention. After factoring out a global phase, the matrix can be written with eigenvalues e^±iπ/8. The actual rotation about the z-axis is π/4 (45°).

The T gate