Quantum Port-Hamiltonian Neural Networks: Learning Conservative and Dissipative Dynamics via Measurement-Induced Nonlinearity

We introduce Quantum Port-Hamiltonian Neural Networks (Q-pHNNs), parameterised quantum circuits that learn classical dynamics in a structure-preserving manner. The framework rests on the Isomorphic Hamiltonian Mapping (IHM): the skew-symmetric interconnection matrix J corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix R to Measurement-Induced NonLinearity (MINL), realised by mid-circuit measurement with classical feedforward. Conservation and passivity are then enforced by construction rather than by penalty terms, and dissipation becomes an intrinsically quantum effect: energy leaves through the act of measurement. We instantiate the IHM in three architectures: a Quantum HNN that extracts Hamilton's equations via the Parameter-Shift Rule; a Q-pHNN that dissipates through MINL; and a topology-entangled Quantum Graph Neural Network lifting both channels to N-node coupled-phasor networks. In simulation, where every model here was trained, we obtain 1.35% relative energy drift under a symplectic integrator, 100% energy monotonicity for the single-oscillator MINL circuit, and 92--98% phase-space energy decay across ring, star and chain networks at N∈3,6,9. On an IBM Heron processor the trained energy surface and its parameter-shift gradients reproduce their simulated values, with an error budget dominated by readout rather than gate infidelity; the dissipative channel executes natively, but its decay is not separable from measurement back-action at these depths.