T phase gate

The $T$ gate is $T=\begin{pmatrix}1&0\\0&e^{i\pi/4}\end{pmatrix}$, applying a $\pi/4$ phase: $T|0\rangle=|0\rangle$, $T|1\rangle=e^{i\pi/4}|1\rangle$. Since $\left(e^{i\pi/4}\right)^2=e^{i\pi/2}=i$, direct matrix multiplication gives $T^2=S$ and $T^8=I$, exactly the power relations this record checks. Unlike $H$, $S$, $X$, $Y$, $Z$, $T$ is not a Clifford gate: conjugating $X$ by $T$ does not map it to another Pauli operator, which is precisely why $\{H, T\}$ is universal for quantum computation while Clifford-only circuits are efficiently classically simulable (Gottesman–Knill). This makes $T$ the standard "expensive" non-Clifford resource in fault-tolerant compilation: magic-state distillation exists specifically to produce high-fidelity $T$ gates from many noisy ones, since — as this record's own caveat notes — the ideal unitary identity above says nothing about a given device's hardware fidelity or noise.