T† gate

Definition

$$T^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & e^{-i\pi/4} \end{pmatrix}$$

$T^\dagger$ is the conjugate transpose of $T = \mathrm{diag}(1, e^{i\pi/4})$, and since $T$ is unitary, $T^\dagger$ is also its inverse: $TT^\dagger = T^\dagger T = I$.

Action on basis states

$$T^\dagger|0\rangle = |0\rangle, \qquad T^\dagger|1\rangle = e^{-i\pi/4}|1\rangle$$

Applied to $|+\rangle$, $T^\dagger$ rotates the equatorial phase by $-\pi/4$, the mirror image of what $T$ does.

Key identities

• $(T^\dagger)^2 = S^\dagger$ and $(T^\dagger)^8 = I$: the phase ladder $\{I, T^\dagger, S^\dagger, \dots\}$ steps down in $-\pi/4$ increments exactly as $\{I,T,S,\dots\}$ steps up.
• $T^\dagger = P(-\pi/4) = R_Z(-\pi/4)$ up to global phase.
• $T^\dagger = T^7$ since $T^8=I$, so seven applications of $T$ implement one $T^\dagger$ when a compiler's native gate set exposes only $T$.
• $T^\dagger$ is not Clifford: conjugating $X$ by $T^\dagger$ does not return a Pauli operator, which is exactly why $\{H, S, \mathrm{CX}, T^\dagger\}$ (or $T$) is needed for universal fault-tolerant quantum computation.