Spectral Hamiltonian Neuron

Integrate the construction as the first layer that maps a quantum state to a scalar feature, or as one channel in a hybrid quantum-classical model. Store fixed Hermitian matrices H[1:J], trainable coefficients theta[1:J], input density matrices rho[b], and scalar labels y[b]. For each minibatch, first construct H = sum_j theta[j] * H[j]. Rescale it by H <- H / max(1, spectral_norm(H)) so that its eigenvalues remain in a stable interval. Eigensolve H = U diag(lambda) U^dagger, compute A = U diag(phi(lambda)) U^dagger, and predict yhat[b] = real(trace(rho[b] @ A)). Compute the squared minibatch loss. For every j, transform H[j] into the eigenbasis, construct G[j] using the divided-difference formula, form dA[j] = U @ (G[j] elementwise-multiplied by U^dagger @ H[j] @ U) @ U^dagger, and set grad_theta[j] = 2 times the batch average of (yhat[b]-y[b]) times real(trace(rho[b] @ dA[j])). Update theta with Adam or SGD. A hardware version can estimate trace(rho A) by repeated observable measurements and approximate phi(H) with a truncated polynomial p_K(H) = sum_k c_k H^k implemented through Hamiltonian simulation. The eigendecomposition, divided differences, and gradients are exact in the simulator; shot noise and polynomial truncation are empirical approximations on hardware. Begin with 4 to 8 qubits, random shallow-circuit states, and labels generated by a target observable containing two noncommuting Pauli strings. Compare equal-parameter commuting H_j, noncommuting H_j, and a classical linear-energy neuron. Success is lower test MSE or fewer training examples at the same parameter count, with an ablation confirming that the gain disappears when all H_j commute.