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

We introduce Quantum Port-Hamiltonian Neural Networks (Q-pHNNs), a family of parameterised quantum circuits that learn classical dynamics in a structure-preserving manner. The framework relies on the Isomorphic Hamiltonian Mapping (IHM): the skew-symmetric interconnection matrix J corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix R corresponds to Measurement-Induced NonLinearity (MINL) realised via mid-circuit measurement and classical feedforward. This ensures conservation and passivity are enforced by construction rather than penalty terms, and it makes dissipation an intrinsically quantum effect: energy leaves the system through the act of measurement, not through any non-unitary term in the Hamiltonian. We instantiate the IHM in three architectures: (1) a Quantum HNN that learns conservative energy manifolds and extracts Hamilton's equations exactly via the Parameter-Shift Rule; (2) a Q-pHNN that realises dissipation through MINL-mid-circuit measurement of a bath ancilla with conditional feed-forward; and (3) a topology-entangled Quantum Graph Neural Network that lifts both channels to N-node coupled-phasor networks, with one bath ancilla per node. Experiments on the nonlinear pendulum and damped harmonic oscillator, and a network scaling study on GPU, demonstrate: (i) 1.35% relative energy drift with a symplectic integrator and scale correction; (ii) 100% energy monotonicity for the single-oscillator MINL circuit; and (iii) 92--98% phase-space energy decay from measurement-induced dissipation -- monotone at every step -- across ring, star, and chain networks at sizes N\in{3,6,9}, alongside exact machine-precision energy conservation in the conservative mode.