Put your understanding to work

For every investigation, record your prediction, calculation, circuit, seed, shots, observed counts and conclusion. Explain the difference between an ideal model and a claim about hardware. These investigations support independent, self-paced learning.

P1 · Reversible computation

Write the truth table for NOT and XOR. Explain why NOT can be reversed and why overwriting two inputs with one XOR output loses information. Build a reversible controlled-X version in the gate playground.

P2 · State validation

For amplitudes $(1/2, i\sqrt{3}/2)$, calculate the norm and measurement probabilities. Explain why amplitudes $(1,1)$ do not describe a normalised state. Use the probability lesson.

P3 · State construction

Prepare a single qubit with $P(1)=0.75$. Give a polar angle and two distinct relative phases. Check that the Z probabilities agree while Bloch directions differ.

P4 · Circuit repair

A circuit intended to create a Bell pair applies H to q1 and then CX with q0 as the control. Predict its output. Repair it, then compare the original and corrected probabilities in the gate playground.

P5 · Entanglement evidence

Compare the Bell state and independent $\lvert+\rangle$ states in Z and X bases. Identify which observations distinguish them. Explain why matching Z outcomes alone would not certify an arbitrary unknown state as entangled.

P6 · Oracle building

Write a truth table for $f(x)=x_1$ on two input bits. Describe how it becomes a phase oracle. Separate the cost of calling an oracle from constructing it.

P7 · Deutsch

Enumerate the four one-bit Boolean functions. Classify them as constant or balanced. Use the first complete algorithm to explain the role of its $\lvert-\rangle$ ancilla.

P8 · Deutsch–Jozsa

Run all three oracle presets. State the promise, prediction and output-register observation. Give an example function outside the promise and explain why the algorithm’s conclusion no longer follows.

P9 · Grover

With one marked item among four, verify one ideal iteration. Repeat with eight items and record success probabilities for 0–6 iterations. Compare the best iteration count with the cost of building and querying the oracle.

P10 · Noise, shots and bounded claims

Run the shot uncertainty lab at 100, 1,024 and 10,000 shots, keeping the prepared state fixed. Repeat with another seed. Use the intervals to discuss uncertainty. In BB84, compare physical channel noise and intercept–resend errors. State what each model excludes.

Extend the investigation

Explore alternative cases, compare competing explanations, and design an independent, reproducible question. Keep enough detail for someone else to repeat your investigation and understand the limits of your conclusion.