Checkpoint: Phase Engineering

Throughout this module you have met the three standard phase tools: the Hadamard gate to create superpositions, the S gate to add a $\frac{\pi}{2}$ relative phase, and the T gate to add a $\frac{\pi}{4}$ relative phase. This checkpoint asks you to combine them to hit a precise target in the complex plane.

The target state

Your goal is the single-qubit state

|\psi\rangle = \frac{1}{\sqrt{2}}\!\left(|0\rangle + e^{i3\pi/4}|1\rangle\right).

The factor $e^{i3\pi/4}$ places the $|1\rangle$ amplitude at $135°$ in the complex plane, squarely in the second quadrant. Working out the Cartesian form,

e^{i3\pi/4} = \cos\!\tfrac{3\pi}{4} + i\sin\!\tfrac{3\pi}{4} = -\tfrac{1}{\sqrt{2}} + \tfrac{i}{\sqrt{2}},

so the two amplitudes are $\frac{1}{\sqrt{2}}$ (real, positive) and $\frac{1}{\sqrt{2}}e^{i3\pi/4}$ (magnitude $\frac{1}{\sqrt{2}}$ , angle $135°$ ). Both have the same magnitude, which means a $Z$ -basis measurement still gives 50/50 — the phase is hidden unless you use interference to reveal it.

Phase as addition

Phase gates accumulate by addition. A phase gate $P(\lambda)$ applied after $P(\mu)$ yields a combined relative phase of $\lambda + \mu$ . Writing the angles of the standard gates,

\underbrace{T}_{\pi/4} \;\text{after}\; \underbrace{S}_{\pi/2} \quad\Rightarrow\quad \phi = \tfrac{\pi}{2} + \tfrac{\pi}{4} = \tfrac{3\pi}{4}.

So the recipe is: put the qubit in an equal superposition with $H$ , then rotate its Bloch vector by $90°$ with $S$ , then by another $45°$ with $T$ .

Why this is useful

In algorithms such as the quantum Fourier transform, specific relative phases of the form $e^{2\pi i k / 2^n}$ are required at each step. Many of those fractions decompose into sums of $\pi/4$ and $\pi/2$ increments — precisely the contributions of $T$ and $S$ . Knowing how to build an exact phase from a small palette of gates is the core skill behind compiling quantum algorithms onto real hardware.

Try it

Engineer the state $\tfrac{1}{\sqrt{2}}(|0\rangle + e^{i3\pi/4}|1\rangle)$ using $H$ , $S$ , and $T$ . The grader checks the full statevector (up to global phase), so both the magnitude and angle of each amplitude must match.

Run your code to see the quantum state.

After running, look at the amplitude panel: the $|0\rangle$ entry should read $\approx 0.707$ on the positive real axis, and the $|1\rangle$ entry should sit at $135°$ in the complex plane — real part $\approx -0.5$ , imaginary part $\approx +0.5$ (each scaled by $\frac{1}{\sqrt{2}}$ ).