The T Gate

Every single-qubit gate can be understood as a rotation or reflection on the Bloch sphere. The T gate is a rotation about the $z$-axis by $\frac{\pi}{4}$ radians — exactly 45°. Its matrix is

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

It leaves $|0\rangle$ untouched and multiplies the amplitude of $|1\rangle$ by the complex number $e^{i\pi/4} = \tfrac{1}{\sqrt{2}} + \tfrac{i}{\sqrt{2}}$.

Relative phase vs. global phase

A global phase on the whole state vector is physically unobservable: $e^{i\theta}|\psi\rangle$ and $|\psi\rangle$ give identical measurement probabilities for any observable. A relative phase between two components of a superposition is different — it genuinely changes the quantum state and can be revealed by a later gate.

For the equal superposition $|+\rangle = \tfrac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, applying $T$ gives

$T|+\rangle = \frac{1}{\sqrt{2}}\bigl(|0\rangle + e^{i\pi/4}|1\rangle\bigr).$

The $|0\rangle$ amplitude is still $\tfrac{1}{\sqrt{2}}$; the $|1\rangle$ amplitude has rotated by 45° in the complex plane. Measuring right now still gives 50/50 — the Born-rule probabilities only depend on magnitudes. The phase difference becomes physically meaningful when you follow $T$ with another gate that mixes the two components (such as a second Hadamard).

The $T^\dagger$ gate

The inverse of $T$ is called $T^\dagger$ (or $\text{Tdg}$) and applies a $-45°$ phase to $|1\rangle$:

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

You will need $T^\dagger$ whenever you want to undo a $T$ step, for example when uncomputing an ancilla register.

Try it

Starting from $|0\rangle$, the starter code applies $H$ to create $|+\rangle$. Your task is to follow it with a $T$ gate so the final state is $\tfrac{1}{\sqrt{2}}\bigl(|0\rangle + e^{i\pi/4}|1\rangle\bigr)$. After pressing Run, look at the amplitude of $|1\rangle$ in the state-vector panel: it should point at $45°$ in the complex plane rather than along the positive real axis.

Notice that the probabilities remain 50/50 — phase does not show up in a straight measurement. The visible difference is in the complex amplitude of $|1\rangle$: its angle has rotated from $0°$ to $45°$.

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