Quantum kernel support-vector classifier

A quantum kernel maps a classical input $x$ to a quantum state $|\phi(x)\rangle=U(x)|0\rangle$ via a feature-map circuit $U(x)$, and defines a kernel via the state overlap $K(x,y)=|\langle\phi(x)|\phi(y)\rangle|^2$. The circuit estimates $K(x,y)$ by applying $U(x)$ then $U(y)^\dagger$ and measuring the probability of returning to $|0\rangle^{\otimes n}$: $\Pr[0^n] = |\langle 0|U(y)^\dagger U(x)|0\rangle|^2 = K(x,y)$, so each entry of the kernel matrix costs repeated circuit shots rather than a closed-form evaluation. The resulting Gram matrix $K$ is handed to a standard classical SVM, which still performs the convex optimization over Lagrange multipliers — the quantum circuit only changes the feature map, not the learning algorithm. This record checks that $K$ is symmetric and numerically positive-semidefinite (within tolerance) on a small dataset; as its own caveat states, whether a given feature map's kernel actually separates a dataset better than a tuned classical kernel is a separate, data-dependent question this toy check does not answer.