Variational quantum eigensolver for ground-state energy

VQE estimates a ground-state energy using the variational principle: for any normalized trial state $|\psi(\theta)\rangle=U(\theta)|0\rangle$ and Hamiltonian $H$, the Rayleigh quotient obeys $\langle\psi(\theta)|H|\psi(\theta)\rangle \ge E_0$, the true ground energy, with equality only when $|\psi(\theta)\rangle$ is an exact ground eigenstate. The circuit prepares $|\psi(\theta)\rangle$ with a parameterized ansatz $U(\theta)$, measures each Pauli term of $H=\sum_k c_k P_k$ to build $\langle H\rangle(\theta)$, and hands that scalar to a classical optimizer that updates $\theta$ to push the energy down — a hybrid quantum/classical loop repeated to convergence. This record checks that expectation-value loop for a small Pauli Hamiltonian against exact diagonalization, matching its own caveat: the variational bound is guaranteed by the math above, but reaching a chemically useful energy still depends on ansatz expressivity, optimizer convergence, and shot noise this toy fixture does not certify.