§1 · Milestone · M12 · 18 min
§1 Mastery Challenge
Complex Numbers in Quantum
Five problems. No scaffolding. Every §1 skill tested once. A correct answer earns the Phase Master 🌀 badge and unlocks §2.
The rules: Answer all five. Each problem reveals detailed feedback after your attempt — read it, even if you got it right. The last problem (P5) gives you no hints at all. That's the point. If a problem trips you up, the feedback links you back to the relevant lesson.
Progress0 / 5 solved
Given $z = 3 + 4i$, what is $|z|^2$?
Recall: $|z|^2 = z \cdot z^* = a^2 + b^2$ where $z = a + bi$
Convert $z = 1 + i$ to polar form $r e^{i\theta}$. What is $r$?
Type a decimal. Recall: $r = |z| = \sqrt{a^2 + b^2}$. Round to 3 decimal places if needed.
A qubit has amplitude $\alpha = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}$. What is the probability of measuring $|0\rangle$?
Recall: $P(|0\rangle) = |\alpha|^2 = \alpha \cdot \alpha^*$
Which transformation always leaves all measurement probabilities unchanged?
Consider $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ and what happens under each operation.
A qubit state is $|\psi\rangle = \frac{3}{5}|0\rangle + \beta|1\rangle$ where $\beta$ is a positive real number. For the state to be normalized, what must $|\beta|$ equal?
No hints. Normalization means $|\alpha|^2 + |\beta|^2 = 1$. Show your work on paper first.
Phase Master — Badge Earned
You can compute quantum probabilities from complex amplitudes. You understand why global phase is unobservable while relative phase drives interference. That's the foundation of everything in Track 3.
🌀 Phase Master · §1 Complete