Checkpoint: Design a Better Ansatz

This checkpoint pulls the module together. You will design an ansatz that hits a specific chemistry ground state exactly, while respecting the physical symmetry that makes the problem tractable in the first place. The lessons you are combining: expressibility (can the ansatz reach the target?), barren plateaus (is it trainable?), and symmetry preservation (does it stay physical?).

The target

The ground state of a minimal two-electron model — the spin singlet — is

|\psi^\star\rangle = \frac{1}{\sqrt{2}}\bigl(|01\rangle - |10\rangle\bigr).

Two features define it. First, it is an equal-weight superposition of the two single-excitation states. Second, it carries a relative minus sign, the antisymmetry that marks a singlet rather than a triplet. Both must come out right.

Why a hardware-efficient ansatz is the wrong tool here

A generic two-qubit hardware-efficient ansatz can certainly reach this state — it is highly expressible. But that very freedom is a liability:

It can also produce $|00\rangle$ and $|11\rangle$ components with the wrong particle number, so the optimizer wastes effort in unphysical regions and can converge to a lower-energy state with the wrong electron count.
More layers mean more parameters and a flatter, plateau-prone landscape — exactly what the barren-plateau lesson warned against.

The better design

Restrict the search to the one-excitation subspace from the outset with a particle-number- preserving Givens rotation. Starting from the single-excitation reference $|01\rangle$ , the Givens gate $G(\theta)$ rotates within $\mathrm{span}\{|01\rangle, |10\rangle\}$ and never leaks amplitude into $|00\rangle$ or $|11\rangle$ :

G(\theta)\,|01\rangle = \cos\tfrac{\theta}{2}\,|01\rangle - \sin\tfrac{\theta}{2}\,|10\rangle.

The minus sign is built in. All that remains is the angle. Equal weight requires $\cos(\theta/2) = \sin(\theta/2)$ , i.e. $\theta/2 = \pi/4$ , so

\theta = \frac{\pi}{2} \quad\Longrightarrow\quad G(\tfrac{\pi}{2})\,|01\rangle = \frac{1}{\sqrt{2}}\bigl(|01\rangle - |10\rangle\bigr) = |\psi^\star\rangle.

One parameter, one entangling structure, and the answer is exact — a vivid illustration that a small, symmetry-respecting ansatz can beat a large generic one.

Try it

Build the particle-number-preserving ansatz, choose the correct angle, and prepare the singlet. The grader verifies four things at once: zero amplitude on $|00\rangle$ and $|11\rangle$ (particle number conserved), equal probability $1/2$ on $|01\rangle$ and $|10\rangle$ (correct weights), and the $+/-$ sign structure of the amplitudes (correct singlet phase).

Run your code to see the quantum state.

If any check fails, inspect the amplitudes: a nonzero $|00\rangle$ or $|11\rangle$ means your circuit broke the symmetry; a wrong sign means you used a state-preparation that does not match the Givens convention.