Checkpoint: Synthesize a Logical Gate

This checkpoint exercises the central skill of the module: turning a target logical rotation into an exact word over the fault-tolerant Clifford+ $T$ gate set. You will synthesize a $z$ -rotation whose angle is a multiple of $\pi/4$ , so the result comes out exactly — no approximation error — letting you focus on the synthesis logic itself.

What you are building

The target is the $z$ -rotation

R_z\!\left(\tfrac{3\pi}{4}\right) = \exp\!\left(-i\,\tfrac{3\pi}{8}\,Z\right) = \mathrm{diag}\!\left(e^{-i 3\pi/8},\; e^{i 3\pi/8}\right).

Recall the two facts you need, both true up to a global phase (which the unitary grader ignores):

R_z\!\left(\tfrac{\pi}{2}\right) \equiv S, \qquad R_z\!\left(\tfrac{\pi}{4}\right) \equiv T .

The synthesis idea

Rotations about the same axis add their angles:

R_z(\alpha)\, R_z(\beta) = R_z(\alpha + \beta).

So decompose the target angle into a sum of $\pi/4$ -multiples and emit the corresponding $S$ / $T$ word:

\tfrac{3\pi}{4} = \tfrac{\pi}{2} + \tfrac{\pi}{4} \;\;\Longrightarrow\;\; R_z\!\left(\tfrac{3\pi}{4}\right) \equiv S\,T .

This is exactly the procedure a real synthesizer follows for angles in the $\tfrac{\pi}{4}\mathbb{Z}$ lattice; for generic angles it would instead call an $\varepsilon$ -approximate routine (Solovay–Kitaev or Ross–Selinger), spending more $T$ gates for more precision. Here a single $T$ suffices, so the synthesized $T$ -count is 1.

Try it

Build the Clifford+ $T$ word that realizes $R_z(3\pi/4)$ . The grader compares your circuit's unitary (up to global phase) against $R_z(3\pi/4)$ — apply $S$ then $T$ .

Run your code to see the quantum state.