Pith. sign in

REVIEW 3 major objections 4 minor 44 references

A family of quantum circuits claims to learn classical dynamics with conservation and dissipation enforced by circuit topology, not penalty terms, making measurement itself the source of energy loss.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 06:35 UTC pith:WJ3ALOJ3

load-bearing objection The paper's central passivity guarantee is false — a single MINL step can increase the phase-space energy — so it reads as a proposal with an unproven core. the 3 major comments →

arxiv 2607.12269 v2 pith:WJ3ALOJ3 submitted 2026-07-14 cs.LG

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

Dibakar Sigdel This is my paper
classification cs.LG
keywords port-Hamiltonian systemsquantum neural networksparameterised quantum circuitsmeasurement-induced nonlinearitydissipative dynamicsstructure-preserving learningquantum graph neural networksparameter-shift rule
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Quantum Port-Hamiltonian Neural Networks (Q-pHNNs): parameterised quantum circuits that learn classical mechanical dynamics while preserving the structural laws of port-Hamiltonian systems. Its central claim is that the two algebraic signatures of such systems—the skew-symmetric interconnection matrix J and the positive-semidefinite dissipation matrix R—map directly onto unitary gate evolution and projective measurement with classical feedforward. This mapping, called the Isomorphic Hamiltonian Mapping (IHM), is said to make energy conservation and passivity properties of the circuit topology itself, so the port-Hamiltonian relations cannot be violated for any parameter values. If correct, dissipation is not a modelled extra term but an intrinsic quantum effect: energy leaves through the act of measurement. The paper validates the idea on a nonlinear pendulum, a damped harmonic oscillator, and coupled phasor networks, reporting exact conservation in the conservative mode and monotonically decreasing energy in the dissipative mode.

Core claim

The paper's central claim is that any port-Hamiltonian system ẋ = (J − R)∇H + G u can be realised by a quantum circuit in which J is generated by unitary gates, R is a measurement-induced nonlinear channel, and the Hamiltonian H is read out as a ⟨ZZ⟩ observable. Because unitarity preserves norm and a projective measurement on an ancilla collapses probability irreversibly before renormalisation, the paper argues that the skew-symmetry of J and the positive-semidefiniteness of R are enforced by construction. The dissipative channel is a three-step MINL protocol: a controlled rotation entangles the system with a bath ancilla, the ancilla is measured, and the classical outcome conditionally kick

What carries the argument

The central object is the Isomorphic Hamiltonian Mapping (IHM), an algebraic correspondence that sends the skew-symmetric interconnection matrix J to unitary gate evolution U_J, the positive-semidefinite dissipation matrix R to Measurement-Induced NonLinearity (MINL)—a projective measurement of a bath ancilla followed by a classical conditional kick—and the Hamiltonian H to the expectation value of a two-qubit ZZ observable. The load-bearing mechanism is the MINL channel: a controlled rotation CR_y(θ_R) entangles the system with an ancilla, the ancilla is measured in the computational basis, and the outcome b triggers a rotation R_x(θ_k) on the system. This forms a Kraus map whose complete p

Load-bearing premise

The central claim depends on the unproven assertion that the specific CR_y(θ_R)+R_x(θ_k) measurement channel satisfies the operator inequality Σ_b K_b† Ô K_b ⪯ Ô for the energy observable, so that energy decreases on average for every parameter value; without this, measurement-induced dissipation is an empirical property of the chosen parameters, not a structural guarantee.

What would settle it

Sweep the MINL circuit's parameters θ_R and θ_k over a grid and simulate the channel from a set of initial states, computing the expected value of the energy observable after one step; if any parameter choice and initial state yield an average energy increase, the structural passivity claim is false. Alternatively, directly evaluate the Kraus operator sum for the circuit and check whether Σ_b K_b† Ô K_b ⪯ Ô holds.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the IHM claim holds, conservative dynamics can be learned with energy conservation guaranteed at every parameter value—no loss penalty, no post-hoc projection.
  • Dissipative dynamics can be learned with monotone energy decrease enforced by the measurement channel itself, replacing the matrix projections and Cholesky decompositions that classical port-Hamiltonian networks need.
  • The MINL channel is a genuine discrete-time completely-positive map, so the same circuit trained in simulation could run on mid-circuit-measurement hardware without a separate noise model for dissipation.
  • The topology-entangled QGNN extends the guarantees to coupled networks, suggesting that arbitrary known coupling graphs can be hard-coded into circuit entanglement so conservation and dissipation scale with the graph.
  • Exact analytic gradients from the parameter-shift rule on data-encoding gates mean Hamilton's equations can be extracted in four circuit evaluations, potentially enabling fast, gradient-based training of structure-preserving models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the passivity guarantee truly holds for all parameters, a natural next step is to parameterise the damping rate as a function of the measured state, turning the channel into state-dependent dissipation without changing the circuit topology.
  • The mapping of R to a projective measurement suggests a concrete two-way bridge between port-Hamiltonian learning and open quantum system identification: the learned ancilla angle could be read as a Lindblad jump rate, which could be verified against independently measured decoherence rates on real hardware.
  • The paper's experiments measure energy monotonicity on specific trajectories; a decisive extension would be to test the passivity inequality on adversarial initial states and parameter values, since the claimed structural guarantee is intended to hold universally, not just on the tested rollouts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces Quantum Port-Hamiltonian Neural Networks (Q-pHNNs), parameterized quantum circuits intended to learn classical conservative and dissipative dynamics with hard structural guarantees. The central claim is the Isomorphic Hamiltonian Mapping (IHM): skew-symmetric pH interconnection J is realized by unitary gates, and positive-semidefinite dissipation R is realized by measurement-induced nonlinearity (MINL) via mid-circuit measurement and classical feedforward. Three architectures are instantiated: a conservative Q-HNN, a single-oscillator Q-pHNN with an ancilla-based MINL channel, and a topology-entangled QGNN for N-node networks. Experiments on the nonlinear pendulum and damped harmonic oscillator, plus network sweeps over ring/star/chain topologies, report low energy drift, 100% monotone energy decay, and exact machine-precision conservation. The paper asserts that passivity and conservation are enforced by circuit construction and that the constitutive relations are impossible to violate regardless of parameter values.

Significance. If valid, the IHM would be an appealing contribution: a circuit-level mechanism for dissipation that is a genuine CPTP map, exact analytic gradients from the parameter-shift rule on data-encoding gates, and graph topology hard-encoded into the entangler layout. The experimental methodology is transparent (exact statevector simulation, fixed seeds, stated hyperparameters), and the QGNN energy readout is elegant. However, the central structural-guarantee claim is not established and, as shown below, is false in its current form. The paper reduces to an empirical demonstration that certain trained parameter choices yield monotone energy decay on selected initial conditions, which is not the advertised construction-level guarantee.

major comments (3)
  1. [§IV.C, Theorem IV.2(iii), §VIII.A] The passivity guarantee is conditional and is never verified for the implemented circuit. Theorem IV.2(iii) requires the Kraus inequality Σ_b K_b† Ô K_b ⪯ Ô for a positive observable Ô, but the paper never proves this condition for the CR_y(θ_R) + R_x(θ_k) MINL channel. Moreover, the energy actually used in the experiments, E_ps = ½Σ(⟨X_i⟩²+⟨Y_i⟩²), is a quadratic function of expectation values, not a linear observable, so the theorem cannot be applied to it. The claim in §VIII.A that the port-Hamiltonian constitutive relations are 'impossible to violate, regardless of the parameter values' is directly contradicted by a simple example: for the single-qubit MINL channel with θ_R=θ_k=π/2 and initial state |1⟩, the Bloch map gives r_y'=0.5, r_z'=−0.5, so E_ps increases from 0 to 0.125 in one step. Thus the asserted structural passivity is false as stated; the reported 100% monotonicity is a
  2. [§VI.D.d, §VII.B, §VII.E] The monotone-dissipation results are computed for a quantity that is not shown to be a circuit observable satisfying the theorem's hypotheses. E_ps is a classical function of measured expectation values; a CPTP map need not decrease such a function. No theorem is given that the multi-ancilla MINL channel decreases E_ps for all parameters and all initial states. The network scaling results (Table IV) report 100% monotone decay, but this is an empirical observation over the specific settings tested, not a structural guarantee. In addition, the single-oscillator energy H(t) used in the f_mono metric is never defined (Section VI.D.c and Section VII.B); the Q-pHNN readout is q̂(t)=⟨σ_x⟩(t), and no Hamiltonian or energy observable is specified. Therefore '100% energy monotonicity' lacks a precise formal meaning.
  3. [Definition IV.1, Theorem IV.2] The IHM is presented as a 'structural isomorphism' but the mapping is not a mathematical isomorphism between pH systems and quantum circuits. A pH system has state-dependent matrices J(x), R(x) on ℝ^{2n}; the circuit has fixed unitary gates, a fixed measurement map, and a fixed observable. Sending a skew-symmetric matrix J to a unitary e^{−iHt} is not an isomorphism of algebraic structures: the exponential map from so(n) to the unitary group is not bijective, and unitary evolution on density matrices is linear, while the pH flow ẋ=(J−R)∇H is generally nonlinear. Theorem IV.2(i)-(ii) establish only norm preservation of a unitary and norm decrease before renormalization in a projective measurement; they do not imply that the circuit reproduces any pH equation of the form (1). The correspondence is at best a design heuristic, and the central claim that conservation and passivity hold 'by ci
minor comments (4)
  1. [§V.B] The Kraus operator notation 'K_b = P_b ⊗ R_x(θ_k)^b' omits the controlled rotation U_CR(θ_R) and the ancilla initial state; as written it does not reproduce the circuit in Fig. 2. Please write the full map acting on system⊗ancilla, including the outcome-dependent post-measurement state.
  2. [§III.C, Eq. (2)] The generator condition is stated as G²=G, but the subsequent use with G=σ_y/2 has G²=I/4. The parameter-shift formula is correct, but the stated condition should be reconciled (e.g., eigenvalues ±1/2).
  3. [§VII.A.d and Table III] The text says the Q-HNN residual drift is due solely to the circuit approximation, not the integrator, while Table III attributes the 7.6% network drift to integrator error. Please clarify the decomposition of integrator error vs. approximation error in both cases.
  4. [§VI.D.d, §VII.E] E_ps is introduced as 'phase-space energy', but its relation to the port-Hamiltonian energy H or to the ZZ observable H_θ is never stated. Please define the normalization and physical dimensions, and specify how E_ps relates to the claimed pH energy.

Circularity Check

2 steps flagged

Passivity guarantee is a conditional restated as an unconditional conclusion; the 'impossible to violate' claim is not derived and is contradicted by a reachable MINL parameter choice.

specific steps
  1. self definitional [Theorem IV.2(iii); Definition IV.1; Section VIII.A]
    "For any observable O≥0 representing energy, Tr(O E(ρ)) ≤ Tr(Oρ) holds when the Kraus operators satisfy Σ_b K_b† O K_b ⪯ O—the quantum analogue of the passivity inequality Ḣ≤y^⊤u. ... The Q-pHNN therefore produces a learning architecture in which the port-Hamiltonian constitutive relations are impossible to violate, regardless of the parameter values optimised by BFGS or COBYLA."

    The theorem's desired conclusion (monotone energy decrease) is made to follow from the exact condition Σ K†OK ⪯ O, which is itself the passivity inequality for the chosen observable. The paper never proves this condition for the CR_y(θR)+Rx(θk) MINL circuit or for the actual energy proxies used (⟨ZZ⟩ or E_ps); Section VIII.A then drops the condition and asserts the constitutive relations are impossible to violate for all parameters. In addition, E_ps = ½Σ(⟨Xi⟩²+⟨Yi⟩²) is a quadratic function of expectation values, not a linear observable, so the theorem cannot even be applied to the network energy. The universal passivity claim is therefore the proof's hypothesis restated as the paper's conclusion, not a derived circuit property.

  2. other [Section VII.B; Section VII.E.c; Table IV]
    "The key result is that the energy monotone fraction is 100%: every one of the 30 MINL trajectory runs produces energy-decreasing behaviour... This confirms that Born-rule measurement effectively dissipates energy from the system qubit, realising the R-channel of the IHM."

    The 100% monotonicity and the 92–98% network decays are empirical values obtained from one trained parameter vector and fixed initial conditions, with the network study using hand-set θR,i values and no optimizer. They are used as evidence for the unconditional structural guarantee in Section VIII.A, but they cannot establish a guarantee 'regardless of parameter values.' The reported metrics are selected-case observations read back as confirmation of the missing theorem, rather than consequences of the theorem. A reachable violation (θR=θk=π/2, initial |1⟩, where E_ps increases from 0 to 0.125) confirms the claim is not structural.

full rationale

The conservative side of the paper is independent and not circular: Theorem IV.3's parameter-shift gradient is a standard exact identity applied to data-encoding gates, and the Q-HNN energy-conservation claim follows from Hamiltonian structure plus symplectic integration; the 1.35% drift is an honest approximation residual. The circularity is concentrated in the dissipative channel. Theorem IV.2(iii) states only a sufficient condition for passivity, yet the paper repeatedly presents the Q-pHNN as guaranteeing ˙H≤0 'by construction' and 'impossible to violate, regardless of parameter values.' The needed Kraus condition is never verified for the implemented circuit or for the quadratic E_ps energy, and the universal claim is in fact false for allowed parameters. The reported 100% monotone results are parametric observations, not structural guarantees. No self-citation load-bearing issue is present; the problem is the conditional proof being converted into an unconditional conclusion. Because the paper's central advertised guarantee reduces to this undischarged premise, the circularity score is elevated to 7.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 1 invented entities

The central 'isomorphism' is an analogy between algebraic properties (skew-symmetry ↔ unitarity, PSD ↔ probability) that does not automatically transfer the pH passivity inequality to the quantum circuit. The missing verification of the Kraus condition is the main uncharged premise; the other listed parameters and assumptions support the empirical numbers rather than the advertised guarantees.

free parameters (4)
  • energy scale s (and offset b) = s*=1.335, b*=0
    Learned jointly with the Q-HNN circuit weights to rescale the bounded ⟨ZZ⟩ observable to match pendulum gradient magnitudes; the reported 1.35% energy drift uses this correction.
  • Q-pHNN parameters θ_J, θ_R, θ_k = [-0.048, 0.624, 0.474]
    Trained by COBYLA on a 6-step damped-oscillator trajectory; the resulting 100% energy monotonicity is a property of these fitted parameters.
  • network MINL angles θ_R,i, θ_k,i (or damping rates γ_i) = not reported
    The per-node ancilla angles and kick angles in the QGNN scaling study are chosen by the formula θ_R,i = 2 arcsin√(1−e^{−γ_i}), but the actual γ_i (and θ_k,i) values are not listed; the 92–98% decay depends on them.
  • graph energy readout weights a_i, w_ij and scale s = not reported
    The QGNN's read-out weights and scale are trained to fit the Kuramoto vector field; the trained values are not provided in the text.
axioms (4)
  • standard math Standard parameter-shift rule for rotation gates with generator having two distinct eigenvalues differing by 1
    Borrowed from refs [14,15]; used in Theorem IV.3 to compute ∂H/∂q and ∂H/∂p from four circuit evaluations.
  • domain assumption The ZZ expectation value is a sufficiently expressive energy proxy for the target Hamiltonians
    The paper asserts the 2-qubit ZZ observable captures cross-terms like sin(q)sin(p) for the pendulum; this is a modeling choice, and the 1.35% drift shows it is approximate.
  • ad hoc to paper The measurement + feed-forward Kraus map satisfies the passivity condition Σ_b K_b† Ô K_b ⪯ Ô
    This is the load-bearing premise of Theorem IV.2(iii), stated conditionally but never verified for the specific CR_y + Rx(θ_k) circuit; it is exactly what would connect measurement to guaranteed energy monotonicity.
  • ad hoc to paper The damping-rate mapping θ_R = 2 arcsin√(1−e^{−γ})
    Imported from the amplitude-damping channel formula without derivation for this circuit; used to set per-node damping rates in the network study.
invented entities (1)
  • Isomorphic Hamiltonian Mapping (IHM) no independent evidence
    purpose: A conceptual dictionary mapping J to unitary gates, R to measurement-induced nonlinearity, H to a ZZ observable, and ports to control gates; used to justify structural guarantees 'by construction'.
    The IHM is a postulated framework introduced in this paper. It is not independently evidenced: the claimed structural guarantees are not derived, and the mapping is analogical rather than a proven isomorphism of the relevant dynamical structures.

pith-pipeline@v1.3.0-alltime-deepseek · 19848 in / 16449 out tokens · 169321 ms · 2026-08-02T06:35:58.511165+00:00 · methodology

0 comments
read the original abstract

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

Figures

Figures reproduced from arXiv: 2607.12269 by Dibakar Sigdel.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 10 linked inside Pith

  1. [1]

    1.35% energy drift over 300 steps(nonlinear pendulum, Q-HNN): the St¨ ormer–Verlet symplectic integrator paired with a learnable energy scale s∗ = 1.335 recovers the correct oscillation frequency and achieves a 9 .1× improvement over the first-order Euler baseline (12.3% drift), with convergence in only 8 BFGS iterations

  2. [2]

    On the damped oscillator the MINL channel yields 100% energy monotonicity across all measurement trajectories

    Dissipation from Born-rule measurement(Q-pHNN, MINL): measurement-induced nonlinearity enforces ˙H≤ 0 at every time step without any non-unitary term in the Hamiltonian and without a passivity penalty in the loss—a structural guarantee that classical pHNNs can only approximate through matrix projection [ 6]. On the damped oscillator the MINL channel yield...

  3. [3]

    Measurement-induced dissipation scales across networks(QGNN, MINL): the multi-ancilla MINL channel decays the network phase-space energy by 92–98% over eight Trotter steps—monotonically at every step—for ring, star, and chain topologies at every size N∈ {3, 6, 9}, produced entirely by Born-rule measurement and feed-forward, one bath ancilla per node with ...

  4. [4]

    0 (exact)

    Scale-free structural conservation for N -body networks(QGNN): placing two- qubit entanglers on graph edges only yields exact machine-precision energy conservation (| ˙H|= 0) at every node countNand topology, with no architectural changes required. 4 C. Broad Impact and Applications The Q-pHNN framework is not limited to toy physical systems. Its core con...

  5. [5]

    This is inherent to the quantum computing paradigm and enables exact BFGS convergence in 8–13 iterations

    Exact analytic gradients from data-encoding gates.The Parameter-Shift Rule provides exact partial derivatives of the circuit observable with respect to phase-space coordinates from two circuit evaluations, without automatic differentiation or finite differences. This is inherent to the quantum computing paradigm and enables exact BFGS convergence in 8–13 ...

  6. [6]

    No post-processing or noise modelling is needed

    Hardware-native Lindblad dynamics.The MINL channel is not a software approximation of dissipation: it is a genuine discrete-time CPTP map, physically 30 realised on any device that supports mid-circuit measurement and classical feedforward (IBM Eagle/Heron, IonQ Aria, Quantinuum H2). No post-processing or noise modelling is needed

  7. [7]

    Long-term quantum advantage for large systems.As quantum hardware matures, the same Q-pHNN circuit—currently run in classical statevector simulation— can be executed on real hardware, potentially exploiting genuine quantum parallelism for high-dimensional phase spaces where classical structure-preserving methods are computationally expensive. E. Limitatio...

  8. [8]

    Greydanus, M

    S. Greydanus, M. Dzamba, and J. Yosinski, Hamiltonian neural networks, Advances in Neural Information Processing Systems32(2019), arXiv:1906.01563

    Pith/arXiv arXiv 2019

  9. [9]

    Cranmer, S

    M. Cranmer, S. Greydanus, S. Hoyer, P. Battaglia, D. Speilman, and S. Ho, Lagrangian neural networks, ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations (2020), arXiv:2003.04630. 35

    Pith/arXiv arXiv 2020

  10. [10]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Duvenaud, Neural ordinary differential equations, Advances in Neural Information Processing Systems31(2018), arXiv:1806.07366

    Pith/arXiv arXiv 2018

  11. [11]

    T´ oth, D

    P. T´ oth, D. J. Rezende, A. Jaegle, S. Racani` ere, A. Botev, and I. Higgins, Hamiltonian generative networks, International Conference on Learning Representations (2020), arXiv:1909.13789

    Pith/arXiv arXiv 2020

  12. [12]

    P. Jin, Z. Zhang, A. Zhu, Y. Tang, and G. E. Karniadakis, SympNets: Intrinsic structure- preserving symplectic networks for identifying Hamiltonian systems, Neural Networks132, 166 (2020)

    2020

  13. [13]

    S. A. Desai, M. Mattheakis, D. Sondak, P. Protopapas, and S. J. Roberts, Port-Hamiltonian neural networks for learning explicit time-dependent dynamical systems, Physical Review E 104, 034312 (2021)

    2021

  14. [14]

    Duong and N

    T. Duong and N. Atanasov, Hamiltonian-based neural ODE networks on the SE(3) manifold for learning robot dynamics, arXiv preprint arXiv:2109.12325 (2021)

    Pith/arXiv arXiv 2021

  15. [15]

    Y. D. Zhong, B. Dey, and A. Chakraborty, Dissipative SymODEN: Encoding Hamiltonian dynamics with dissipation and control into deep learning, arXiv preprint arXiv:2002.08860 (2020)

    Pith/arXiv arXiv 2002

  16. [16]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics378, 686 (2019)

    2019

  17. [17]

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proceedings of the National Academy of Sciences113, 3932 (2016)

    2016

  18. [18]

    Rackauckas, Y

    C. Rackauckas, Y. Ma, J. Martensen, C. Warner, K. Zubov, R. Supekar, D. Skinner, A. Ra- madhan, and A. Edelman, Universal differential equations for scientific machine learning, arXiv preprint arXiv:2001.04385 (2020)

    Pith/arXiv arXiv 2001

  19. [19]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nature Reviews Physics3, 625 (2021)

    2021

  20. [20]

    Biamonte, P

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature549, 195 (2017)

    2017

  21. [21]

    Mitarai, M

    K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Physical Review A98, 032309 (2018). 36

    2018

  22. [22]

    Schuld, V

    M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Physical Review A99, 032331 (2019)

    2019

  23. [23]

    Schuld, R

    M. Schuld, R. Sweke, and J. J. Meyer, Effect of data encoding on the expressive power of variational quantum machine-learning models, Physical Review A103, 032430 (2021)

    2021

  24. [24]

    Hairer, C

    E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration: Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31, 10.1007/3-540-30666-8 (2006)

    doi:10.1007/3-540-30666-8 2006

  25. [25]

    Leimkuhler and S

    B. Leimkuhler and S. Reich,Simulating Hamiltonian Dynamics(Cambridge University Press, 2004)

    2004

  26. [26]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru- Guzik, Noisy intermediate-scale quantum algorithms, Reviews of Modern Physics94, 015004 (2022)

    2022

  27. [27]

    Karlebach and R

    G. Karlebach and R. Shamir, Modelling and analysis of gene regulatory networks, Nature Reviews Molecular Cell Biology9, 770 (2008)

    2008

  28. [28]

    J. Gao, B. Barzel, and A.-L. Barab´ asi, Universal resilience patterns in complex networks, Nature530, 307 (2016)

    2016

  29. [29]

    K. V. Shenoy, M. Sahani, and M. M. Churchland, Cortical control of arm movements: A dynamical systems perspective, Annual Review of Neuroscience36, 337 (2013)

    2013

  30. [30]

    Pandarinath, D

    C. Pandarinath, D. J. O’Shea, J. Collins, R. Jozefowicz, S. D. Stavisky, J. C. Kao, E. M. Trautmann, M. T. Kaufman, S. I. Ryu, L. R. Hochberg, J. M. Henderson, K. V. Shenoy, L. F. Abbott, and D. Sussillo, Inferring single-trial neural population dynamics using sequential auto-encoders, Nature Methods15, 805 (2018)

    2018

  31. [31]

    Rebentrost, B

    P. Rebentrost, B. Gupt, and T. R. Bromley, Quantum computational finance: Monte Carlo pricing of financial derivatives, Physical Review A98, 022321 (2018)

    2018

  32. [32]

    Or´ us, S

    R. Or´ us, S. Mugel, and E. Lizaso, Quantum computing for finance: Overview and prospects, Reviews in Physics4, 100028 (2019)

    2019

  33. [33]

    N. A. Trayanova, Whole-heart modeling: Applications to cardiac electrophysiology and elec- tromechanics, Circulation Research108, 113 (2011)

    2011

  34. [34]

    Y. D. Zhong, B. Dey, and A. Chakraborty, Symplectic ODE-net: Learning Hamiltonian dynamics with control, International Conference on Learning Representations (2020), 37 arXiv:1909.12077

    Pith/arXiv arXiv 2020

  35. [35]

    Mattheakis, P

    M. Mattheakis, P. Protopapas, D. Sondak, M. Di Giovanni, and E. Kaxiras, Physical symmetries embedded in neural networks, arXiv preprint arXiv:1904.08991 (2022)

    Pith/arXiv arXiv 1904

  36. [36]

    J. N. Kutz and S. L. Brunton, Parsimony as the ultimate regularizer for physics-informed machine learning, Nonlinear Dynamics107, 1801 (2022)

    2022

  37. [37]

    J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature Communications9, 4812 (2018)

    2018

  38. [38]

    Scarselli, M

    F. Scarselli, M. Gori, A. C. Tsoi, M. Hagenbuchner, and G. Monfardini, The graph neural network model, IEEE Transactions on Neural Networks20, 61 (2009)

    2009

  39. [39]

    L. C. G. Govia, G. J. Ribeill, G. E. Rowlands, H. K. Krovi, and T. A. Ohki, Quantum reservoir computing with a single nonlinear oscillator, Physical Review Research3, 013077 (2021)

    2021

  40. [40]

    Popovych, K

    Z. Popovych, K. Jacobs, G. Korpas, J. Marecek, and D. I. Bondar, Quantum open system identification via global optimization: Optimally accurate markovian models of open systems from time-series data, arXiv preprint arXiv:2203.17164 (2022)

    Pith/arXiv arXiv 2022

  41. [41]

    Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathe- matical Physics48, 119 (1976)

    G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathe- matical Physics48, 119 (1976)

    1976

  42. [42]

    van der Schaft and D

    A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Foundations and Trends in Systems and Control1, 173 (2014)

    2014

  43. [43]

    M. J. D. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, Advances in Optimization and Numerical Analysis , 51 (1994)

    1994

  44. [44]

    Appendix A: Hyperparameters and Experimental Details All experiments are fully specified by the architectures and settings of Section VI

    Qiskit contributors, Qiskit: An open-source framework for quantum computing, https:// qiskit.org(2023). Appendix A: Hyperparameters and Experimental Details All experiments are fully specified by the architectures and settings of Section VI. Table V collects all hyperparameters; no hyperparameter tuning was performed—all values were fixed before running t...

    2023