Checkpoint: Beat the Standard Limit

Time to put the whole module together. You will build a complete entangled-probe sensing protocol and watch it resolve a phase twice as fast as the single-qubit probe from the standard-quantum-limit lesson — a direct, measurable demonstration of beating the SQL.

The protocol

The 2-qubit GHZ Ramsey protocol has four stages:

Prepare the GHZ probe $|\mathrm{GHZ}_2\rangle = (|00\rangle + |11\rangle)/\sqrt2$ with $H$ then $\mathrm{CX}(0,1)$ .
Encode the phase by applying $P(\theta)$ to both qubits. The two branches separate by the amplified phase $2\theta$ : $|\mathrm{GHZ}_2\rangle \to \frac{|00\rangle + e^{2i\theta}|11\rangle}{\sqrt2}.$
Decode by reversing the entangler — $\mathrm{CX}(0,1)$ then $H(0)$ — which disentangles the pair and maps the amplified relative phase onto qubit 0 alone, returning qubit 1 to $|0\rangle$ .
Measure.

The amplified fringe

Working the decode through, the population of qubit 0 oscillates as

p_1^{(q_0)}(\theta) = \frac{1 - \cos(2\theta)}{2}.

Compare this with the single-qubit Ramsey fringe $p_1 = (1-\cos\theta)/2$ from Lesson 2. The GHZ probe oscillates at twice the rate — its slope $\partial_\theta p_1 = \sin(2\theta)$ is doubled, so the same phase error produces twice the change in counts. That doubled slope is precisely the $N = 2$ factor in the $1/N$ Heisenberg scaling.

The headline comparison

In the SQL lesson a single-qubit probe at $\theta = \pi/3$ gave outcome 1 with probability $0.25$ . Here, with the entangled probe, you reach the same $0.25$ signal at only $\theta = \pi/6$ — half the phase — because $2\theta = \pi/3$ does the work. The entangled probe extracts the same information from a smaller phase, which is exactly what "more precise" means.

Try it

Implement the full four-stage protocol at $\theta = \pi/6$ and measure both qubits. The grader checks the exact distribution: $0.75$ on outcome 00 and $0.25$ on outcome 10, with the other two outcomes at $0$ . Getting this right means you have built — and beaten the SQL with — an entangled quantum sensor.

Run your code to see the quantum state.