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arXiv:2607.12269

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

Dibakar Sigdel

Published Jul 14, 2026• 0 citations
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Abstract

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 $\mathbf{J}$ corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix $\mathbf{R}$ 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 $N\in\{3,6,9\}$. 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.

DOI
10.48550/arxiv.2607.12269

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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\mathbf{J}J corresponds to unitary gate evolution, and the positive-semidefinite dissipation matrix R\mathbf{R}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 NNN-node coupled-phasor networks. In simulation, where every model here was trained, we obtain 1.35%1.35\%1.35% relative energy drift under a symplectic integrator, 100%100\%100% energy monotonicity for the single-oscillator MINL circuit, and 929292--98%98\%98% phase-space energy decay across ring, star and chain networks at N∈{3,6,9}N\in\{3,6,9\}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.
We introduce a measurement-induced quantum neural network (MINN), an adaptive monitored-circuit architecture in which mid-circuit measurement outcomes determine the entangling gates in subsequent layers. In contrast to standard monitored circuits where sites and gates are sampled randomly, the gates are parametrized and variational, producing correlated history-dependent dynamics and injecting nonlinearity through measurement back-action. A generic MINN is not expected to be efficiently classically simulable. To demonstrate feasibility, we study a matchgate MINN that admits exact fermionic simulation and can be trained with gradient estimators. We apply the architecture to continuous optimization, image classification, and ground-state search in the Sherrington-Kirkpatrick spin glass, finding effective training and performance over a broad range of monitoring rates.
Quantum neural networks promise to extend the power of machine learning into the quantum domain, with potential applications ranging from automatic recognition of quantum states to the control of quantum devices. However, their physical implementation and training remain challenging. In particular, the backpropagation algorithm that underpins the efficiency of classical neural networks cannot generally be applied to large quantum systems, as nonlinear quantum dynamics are not efficiently simulable. Instead, variational quantum circuits typically rely on parameter-shift rules or sampling-based gradient estimation. Here we propose a bosonic quantum neural network based on parametrically coupled Gaussian modes. Although the underlying quantum dynamics are linear, nonlinear output features are generated through Fock-basis measurements. Because Gaussian evolution can be efficiently simulated in the Heisenberg representation, the system admits gradient-based optimization by differentiating a classical model of the dynamics, while the forward evolution itself could be implemented on quantum hardware. This hybrid approach enables end-to-end training of physically meaningful parameters without requiring gradient extraction from the experimental device. Such architectures are naturally compatible with circuit quantum electrodynamics platforms featuring tunable parametric couplers, as well as integrated photonic systems with engineered χ(2) or χ(3) nonlinearities. Our results demonstrate that linear bosonic networks combined with nonlinear measurement provide a scalable and trainable route toward experimentally realizable quantum neural networks.
Quantum Neural Networks (QNNs) represent a compelling convergence of quantum mechanics and machine learning, yet their practical deployment remains constrained by noise, limited qubit counts, and the cost of full quantum execution. Hybrid quantum-classical (HQC) training offers a pragmatic path forward: parameterized quantum circuits (PQCs) process information in exponentially large Hilbert spaces while classical optimizers update parameters via gradient methods accessible on today's hardware. We present a modular HQC training framework, formalising the parameter-shift rule for hardware-compatible gradient estimation, characterising the barren-plateau phenomenon through a variance-decay analysis, and introducing noise-aware training strategies for NISQ devices. Empirical results from state-vector and shot-based simulation confirm measurable quantum advantage on quantum-structured tasks, with the phase classification benchmark improving +12.1% over a classical MLP baseline. The accompanying open-source library provides a clean abstraction between quantum simulation and classical optimisation, serving as a foundation for near-term quantum machine learning research.
Quantum neural networks promise to extend the power of machine learning into the quantum domain, with potential applications ranging from automatic recognition of quantum states to the control of quantum devices. However, their physical implementation and training remain challenging. In particular, the backpropagation algorithm that underpins the efficiency of classical neural networks cannot generally be applied to large quantum systems, as nonlinear quantum dynamics are not efficiently simulable. Instead, variational quantum circuits typically rely on parameter-shift rules or sampling-based gradient estimation. Here we propose a bosonic quantum neural network based on parametrically coupled Gaussian modes. Although the underlying quantum dynamics are linear, nonlinear output features are generated through Fock-basis measurements. Because Gaussian evolution can be efficiently simulated in the Heisenberg representation, the system admits gradient-based optimization by differentiating a classical model of the dynamics, while the forward evolution itself could be implemented on quantum hardware. This hybrid approach enables end-to-end training of physically meaningful parameters without requiring gradient extraction from the experimental device. Such architectures are naturally compatible with circuit quantum electrodynamics platforms featuring tunable parametric couplers, as well as integrated photonic systems with engineered χχχ(2) or χχχ(3) nonlinearities. Our results demonstrate that linear bosonic networks combined with nonlinear measurement provide a scalable and trainable route toward experimentally realizable quantum neural networks.
We present a hybrid quantum-classical recurrent neural network (QRNN) architecture in which the recurrent core is realized as a parametrized quantum circuit (PQC) controlled by a classical feedforward network. The hidden state is the quantum state of an nnn-qubit PQC in an exponentially large Hilbert space C2n\mathbb{C}^{2^n}C2n, which serves as a coherent recurrent quantum memory. The PQC is unitary by construction, making the hidden-state evolution norm-preserving without external constraints. At each timestep, mid-circuit Pauli expectation-value readouts are combined with the input embedding and processed by the feedforward network, which provides explicit classical nonlinearity. The outputs parametrize the PQC, which updates the hidden state via unitary dynamics. The QRNN is compact and physically consistent, and it unifies (i) unitary recurrence as a high-capacity memory, (ii) partial observation via mid-circuit readouts, and (iii) nonlinear classical control for input-conditioned parametrization. We evaluate the model in simulation with up to 14 qubits on sentiment analysis, MNIST, permuted MNIST, copying memory, and language modeling. For sequence-to-sequence learning, we further devise a soft attention mechanism over the mid-circuit readouts and show its effectiveness for machine translation. To our knowledge, this is the first model (RNN or otherwise) grounded in quantum operations to achieve competitive performance against strong classical baselines across a broad class of sequence-learning tasks.
Abstract We introduce and test two measurement-free training primitives for quantum reservoir computers (QRC). Protocol~A encodes the loss gradient in an auxiliary phase that is fed back coherently before a dissipative reset; Protocol~B realizes gradient descent as relaxation of an engineered open system. Both match exact-gradient training in simulation---the ancilla and reset simulated as channels in a de-risk stage, their validated error model applied at the benchmark stage---on a minimal transverse-field Ising reservoir, with the coherent cost oracle emulated classically and its realization costed. A fully coherent update loop (quantum parameter register, unitary controlled update, no readout anywhere) descends the true landscape, its feedback drift attenuated by exactly 2F−12F-12F−1 with the fixed point unchanged. Benchmarked against classical reservoir computing--- including the memory-augmented hybrid---they improve one-step chaotic prediction, though a thirty-seed study finds no autonomous-prediction advantage; we therefore make no quantum-advantage claim. The study instead addresses three questions that gate this route. First, within the channel model studied, the reset fidelity already demonstrated in measurement-free quantum error correction [Butt \textit{et al.}, Nat.\Commun.\\textbf{17}, 995 (2026)] suffices: reset is not the bottleneck. Second, the binding constraint is coherent cost-oracle precision, set by a finite-difference catastrophic cancellation that the exact parameter-shift rule removes, loosening the oracle-precision target 2.52.52.5-fold in a like-for-like comparison. Third, a fault-tolerant realization---arithmetic and synthesis costs taken from real circuits---needs only ∼ ⁣104\sim\!10^{4}∼104 physical qubits and is runtime-, not qubit-bound. On three free IBM Quantum 156156156-qubit Heron backends at three error-mitigation levels the simulation-optimized state survives device noise (RMSE 0.0160.0160.016--0.0570.0570.057, i.e.$7$--26%26\%26% of the signal spread, correlation ≥0.97\ge0.97≥0.97). All code, seeds, and job identifiers are released.