Capstone: Original Research Demo

The track ends as research does: with a question of your own, reduced to something you can actually run and check. This capstone walks one complete miniature project — posing a question, designing a circuit, building it, and self-verifying the result — as a template for the original demo you will design next.

The question

Can I prepare a maximally entangled four-qubit GHZ state with a linear-depth circuit, and confirm I succeeded without doing full state tomography?

The Greenberger–Horne–Zeilinger state generalizes the Bell pair to $n$ qubits:

|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}\bigl(|0\rangle^{\otimes n} + |1\rangle^{\otimes n}\bigr).

It is the canonical "all or nothing" entangled state — measuring any one qubit instantly fixes all the others — and a standard benchmark for how many qubits a device can entangle coherently (Monz et al. reached 14 trapped ions this way).

The design

A naive thought is to entangle qubit $0$ with each of the others separately, but a cheaper, shallower construction chains the entanglement down a ladder:

|\mathrm{GHZ}_4\rangle = \texttt{CX}(2,3)\,\texttt{CX}(1,2)\,\texttt{CX}(0,1)\,H_0\,|0000\rangle.

The Hadamard seeds a superposition on qubit $0$ ; each CNOT then copies the $|1\rangle$ branch one qubit further along, so the two global branches $|0000\rangle$ and $|1111\rangle$ grow together. The circuit has depth $\mathcal{O}(n)$ and uses only $n - 1$ two-qubit gates.

The verification

Full tomography of $n$ qubits costs exponentially many measurement settings — far too much for a quick check. Instead we use two cheap diagnostics, both implied by the GHZ structure:

Populations. Measuring in the computational basis must give only $0^n$ and $1^n$ , each with probability $\tfrac{1}{2}$ . Any weight in another bin signals a bit-flip error in the ladder.
Coherence. The populations alone cannot distinguish the entangled GHZ from a classical mixture of $|0000\rangle$ and $|1111\rangle$ ; the relative phase must be checked too (e.g. via parity oscillations in the $X$ basis). A genuine GHZ has both equal populations and a definite phase.

Here the grader checks the populations directly and confirms the amplitudes of $|0000\rangle$ and $|1111\rangle$ are real and equal to $1/\sqrt{2}$ — pinning down the coherent superposition, not a mixture.

Try it

Build the four-qubit GHZ ladder. The grader runs your self-verification: the populations of $|0000\rangle$ and $|1111\rangle$ each equal $\tfrac{1}{2}$ , all other bins vanish, and the two surviving amplitudes are real and equal to $1/\sqrt{2}$ .

Run your code to see the quantum state.