Quantum Logic Gates

T Gate (π/8, ⁴√Z)

The T gate is the non-Clifford π/8 gate — the ingredient that lifts a quantum computer from classically simulable to universal.

Matrix

Unitary matrix

100e^{iπ/4}

Circuit symbol

Kronos motion — thermal gate

Action on basis states

input	output
\|0⟩	\|0⟩
\|1⟩	e^{iπ/4}\|1⟩

How it works

T applies a π/4 phase (T²=S, T⁴=Z). It is the canonical non-Clifford gate: Clifford+T is universal, and by the Gottesman–Knill theorem Clifford-only circuits are classically efficient, so T-count is the standard measure of a fault-tolerant algorithm's cost.

In code (Qiskit)

python

qc = QuantumCircuit(1)
qc.t(0)
qc.tdg(0)  # T-dagger

single-qubitnon-Clifforduniversalphase