Project: A Biased Coin

The previous lesson built a fair coin: apply $H$ to $|0\rangle$ , measure, and each outcome appears with probability $\tfrac{1}{2}$ . But what if you want a coin that lands heads 75% of the time? Adjusting the bias requires a different tool — a rotation gate.

From 50/50 to any split

Every single-qubit state can be parameterised as

|\psi\rangle = \cos\!\tfrac{\theta}{2}\,|0\rangle + \sin\!\tfrac{\theta}{2}\,|1\rangle,

where $\theta \in [0, \pi]$ controls the balance between the two amplitudes (up to an overall phase we can ignore here). The $R_y(\theta)$ gate rotates the Bloch vector by angle $\theta$ around the $y$ -axis, starting from $|0\rangle$ at the north pole:

R_y(\theta)|0\rangle = \cos\!\tfrac{\theta}{2}\,|0\rangle + \sin\!\tfrac{\theta}{2}\,|1\rangle.

By the Born rule, the probability of measuring 1 after this rotation is

P(1) = \left|\sin\!\tfrac{\theta}{2}\right|^2 = \sin^2\!\tfrac{\theta}{2}.

Setting $\theta = \tfrac{\pi}{2}$ recovers the Hadamard result: $\sin^2\!\tfrac{\pi}{4} = \tfrac{1}{2}$ .

Choosing the angle for a 75/25 coin

We want $P(1) = \tfrac{1}{4}$ . Working backwards:

\sin^2\!\tfrac{\theta}{2} = \tfrac{1}{4} \;\Longrightarrow\; \sin\!\tfrac{\theta}{2} = \tfrac{1}{2} \;\Longrightarrow\; \tfrac{\theta}{2} = \tfrac{\pi}{6} \;\Longrightarrow\; \theta = \tfrac{\pi}{3}.

After $R_y\!\left(\tfrac{\pi}{3}\right)$ the state is

R_y\!\left(\tfrac{\pi}{3}\right)|0\rangle = \frac{\sqrt{3}}{2}\,|0\rangle + \frac{1}{2}\,|1\rangle,

giving $P(0) = \left(\tfrac{\sqrt{3}}{2}\right)^2 = \tfrac{3}{4}$ and $P(1) = \left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}$ .

Try it

Build a circuit that produces a biased coin landing on 1 with probability $\tfrac{1}{4}$ (and on 0 with probability $\tfrac{3}{4}$ ). Apply the correctly tuned $R_y$ rotation to qubit 0 and then measure. The grader compares the probability distribution, so the angle must be right.

Run your code to see the quantum state.

After running, open the Probabilities panel. The bar for 0 should be at $0.75$ and the bar for 1 at $0.25$ — a 3:1 coin controlled by a single rotation angle.