Capstone: train a data re-uploading classifier on the circle

Complete the TypeScript function

trainCapstone(Xtrain: number[][], ytrain: number[]): { template: '3' | '4' | '5' | '6'; weights: Record<string, number> }

The data. Xtrain holds $100$ points [x0, x1] of the unit square $[0,1)^2$; ytrain[m] is $+1$ if $(x_0-\frac12)^2+(x_1-\frac12)^2<0.15$ (inside the circle of radius $\sqrt{0.15}\approx0.387$ around the centre) and $-1$ otherwise. The training set has $50$ points of each class. The held-out set has $100$ further points from the same generator ($49$ inside).

The model. For a depth $L$, dataReuploadingClassifier(L) from 'quantumcore/qml' is exactly the circuit the grader uses: two qubits, data parameters x0, x1 and the $4L$ weights w_l_q_k (layer $\ell$, qubit $q$, $k=0$ for $R_Y$, $k=1$ for $R_Z$). Qubit $0$ is the most significant bit and $R_P(\theta)=e^{-i\theta P/2}$. Layer $\ell=0,\dots,L-1$ applies, in time order,

$R_Y^{(0)}(\pi x_0),\ R_Y^{(1)}(\pi x_1),\ R_Y^{(0)}(w_{\ell,0,0}),\ R_Z^{(0)}(w_{\ell,0,1}),\ R_Y^{(1)}(w_{\ell,1,0}),\ R_Z^{(1)}(w_{\ell,1,1}),\ \mathrm{CNOT}(0,1).$

The output is $f_w(x)=\langle Z_0Z_1\rangle$ (Pauli word ZZ, i.e. model.observable) and the prediction is $\mathrm{sign}\,f_w(x)$; $f=0$ counts as wrong. Evaluate $f$ with expectationValue(model.template, model.observable, bindings) from 'quantumcore', where bindings holds x0, x1 and every weight.

What to write. A training loop that lowers a loss over the training set, for example the mean squared error $L(w)=\frac1M\sum_m\big(f_w(x_m)-y_m\big)^2$ with $\partial L/\partial w_k=\frac2M\sum_m\big(f_w(x_m)-y_m\big)\,\partial f_w(x_m)/\partial w_k$. Compute $\partial f/\partial w_k$ with your own parameter-shift rule: each weight occurs in exactly one $R_Y$ or $R_Z$ gate, so $\partial f/\partial w_k=\frac12\big[f(w_k+\frac\pi2)-f(w_k-\frac\pi2)\big]$ holds exactly (lesson). One epoch then costs $M(2P+1)$ circuit evaluations with $P=4L$, i.e. $4100$ for $L=5$. Any optimiser is allowed: gradient descent, Adam, restarts. Pick the template by changing TEMPLATE in the starter.

What to return. { template, weights } where template is one of '3', '4', '5', '6' and weights has exactly the keys dataReuploadingClassifier(L).weightParams of that template ($12$, $16$, $20$ or $24$ finite numbers), no more and no fewer.

Grading. The function runs once, with the training set as its arguments, within a time limit of 120 seconds. The reference solution ($L=5$, Adam with $\eta=0.1$ for $80$ epochs, $80\cdot4100=328\,000$ circuit evaluations) needs about $16$ seconds in Node.js. The check shape verifies the template name and the weight keys. The check accuracy binds your weights into the chosen circuit on the server and computes the accuracy on the $100$ held-out points, which are never sent to your browser. The required accuracy is at least $0.85$; the reference reaches $1.00$ on the training set and $0.96$ on the held-out set.

About the shortcut. 'quantumcore/qml' also exports vqcGradient, vqcLossAndGradient and trainVQC, and 'quantumcore' exports parameterShiftGradient; the lesson uses them. Calling them would pass this lab without doing the exercise. The grader only sees the returned weights, so this rule is not enforced; the exercise only works if you write the shift rule and the loop yourself.