Hamiltonian Learning

Hamiltonian neural networks provide a principled framework for learning conservative dynamics. A scalar Hamiltonian is parameterized by a neural network and the vector field is derived using Hamilton's equations. However, preserving Hamiltonian structure does not guarantee that the energy function itself is represented accurately. When modeled by a conventional multilayer perceptron, the Hamiltonian inherits the spectral bias of coordinate-based neural representations, favoring smooth over fine-scale structure. These fine-scale errors can be amplified by differentiation when computing the dynamics. To address this limitation, we introduce Local Multiscale Hamiltonian Neural Network (LM-HNN), which restructures the Hamiltonian according to its physical interaction structure. LM-HNN decomposes the energy into kinetic and potential components and represents them as sums of shared local functions over multiple interaction neighborhoods. This replaces a single high-dimensional approximation problem with lower-dimensional, reusable interaction laws, making fine-scale energy structure easier to resolve while preserving Hamiltonian dynamics. Across low-dimensional, interacting, and fast--slow systems, LM-HNN consistently achieves more accurate Hamiltonian derivatives and long-horizon rollouts than competing structure-preserving models, while substantially extending the effective spectral range of the learned energy. The same structured representation also generalizes more reliably to energy levels not observed during training. These results show that accuracy depends on the Hamiltonian representation as well as its structure, and that incorporating locality, decomposition, and shared interaction laws provides a simple and effective route to more accurate and generalizable Hamiltonian learning.

Exactly symplectic neural networks are built almost exclusively from shear maps and gain expressivity only through depth. We ask what else a single explicit symplectic layer can be. Observing that every layer in use is the exact flow of a Hamiltonian that is a function of commuting integrals, we obtain three families of layers: a learned-type layer, in which every conjugate pair is rotated, squeezed or sheared by a state-dependent amount and the type is found by training; moving-plane shears, residual symplectic maps outside the known ridge-function class; and fiber-preserving layers, which combine an unconstrained point transformation with a fiber translation at the cost of one linear solve. We characterize residual symplectic layers $x \mapsto x + J\nabla H(x)$: in one degree of freedom they are exactly the shears along a line, while from two degrees of freedom on a strictly larger class exists. Conjugating a learned-type layer by a symplectic network yields an exact one-parameter group with an exactly conserved learned Hamiltonian. In experiments on sixteen systems with one to forty degrees of freedom, against SympNets, HénonNets and learned-plane shear networks, layers built on nonlinear integrals are far more accurate when the Hamiltonian contains functions of those integrals. On nonlinear Schrödinger lattices with six to forty sites, twist layers alternating with linear symplectic maps are over a hundred times more accurate than every shear-based network, and with an on-site potential of learned form three to four orders of magnitude more accurate. On chains of Tao oscillators with the step size as an input, stretch layers are 45 to 70 times more accurate than learned-plane shears and recover the Hamiltonian to 1%. On generic non-separable systems, fiber layers improve on learned-plane shears by at most a factor of two, and widening the networks does not help any architecture.

Morohoshi, Nakayama, Manabe, and Mitarai proposed a physically motivated quantum machine learning problem in which the goal is to predict quantities of the form $\operatorname{Tr}[f(H)ρ]$ from classical descriptions of a Hamiltonian $H$ and a quantum state $ρ$, where $f$ is an unknown function. We call this problem Hamiltonian function learning in this paper. They constructed an efficient quantum learning algorithm under suitable conditions, while leaving a rigorous proof of average-case classical hardness open. In this paper, we rigorously prove the average-case classical hardness for two distribution-specific Hamiltonian function learning problems for $f_{\cos,π}(λ)=\cos(πλ)$ and $f_{\exp,β}(λ)=e^{-βλ}$ discussed in the paper of Morohoshi et al. under the assumption of the average-case hardness of factoring random RSA moduli. More specifically, we show that an efficient classical randomized learner under squared loss whose output hypotheses are evaluable in classical polynomial time for either problem would yield a classical randomized polynomial-time algorithm for factoring random RSA moduli.

Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.

Coupled-cluster theory defines the accuracy standard for molecular electronic-structure properties but scales too steeply for routine application, whereas density-functional theory is affordable yet systematically biased. We resolve this trade-off with a single equivariant network, HARP (Hamiltonian Read-out for Properties), that predicts an effective one-electron Hamiltonian from one inexpensive B3LYP/def2-SVP calculation and derives a broad suite of properties from it (energy, optical gap, dipole, quadrupole, polarizability, Mulliken atomic charges, and Mayer bond orders) at coupled-cluster accuracy across nine main-group elements, including the under-served phosphorus, sulfur, and chlorine chemistries. The model is trained on a new in-house dataset of multi-property labels computed at the CCSD(T) level for all nine elements. On a held-out test set, it reduces the error of every property by a factor of 3.8 to 270 relative to semi-local, hybrid, and double-hybrid DFT (referenced to composite CCSD(T)/cc-pVTZ), while adding only ~0.1 s wall time per molecule, delivering coupled-cluster-quality predictions at the cost of a single DFT calculation. Critically, deriving every property from a predicted Hamiltonian rather than pooling per-atom features builds the correct size-scaling into the model architecture: on pi-conjugated oligothiophenes it matches finite-field CCSD polarizability to ~1% and the EOM-CCSD optical gap to ~3% at the largest sizes where those references remain affordable (44 and 37 atoms, where a single CCSD field point already costs ~500x the model's entire inference) and extrapolates the corrected trends to 58-atom chains, a regime where pooling-based architectures fail by construction. Accurate extrapolation is therefore set by the model's inductive bias rather than by the training data.

Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works proposed protocols have achieved the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian $H$ with $||H||\leqΛ$ in total evolution time of $Θ(\fracΛ{ε^2}\log(\fracΛε))$. The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of $Ω(\fracΛ{ε^2}\log(\fracΛε))$. Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on $Λ$ instead of $\varepsilon$, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.

Adaptive Hamiltonian learning is central to calibrating and characterizing quantum devices. In an adaptive controller, choosing the next experiment is itself a computation. Bayesian design rules are recomputed after every posterior update, and that step can take seconds. Across hundreds of shots, those seconds become a significant wall-clock cost for adaptivity. We introduce SymQNet, an amortized reinforcement-learning approach for low-latency adaptive Hamiltonian learning. SymQNet learns a posterior-conditioned acquisition policy offline, then uses a fast policy forward pass online while retaining Bayesian posterior feedback. On transverse-field Ising benchmarks, SymQNet substantially reduces acquisition latency relative to bounded Fisher-information search and bounded two-step Bayesian active learning by disagreement (BALD). At five qubits, it reduces acquisition-only decision latency by $47.1\times$ and $72.6\times$ relative to these online baselines; at twelve qubits, full simulated steps take $1.02$ s for SymQNet versus $13.27$ s for bounded two-step BALD. Overall, we show that learned acquisition can make adaptive Hamiltonian learning practical for repeated low-latency workloads.

Autonomous neutron spectroscopy must solve three distinct tasks: detection (where is the signal?), inference (which Hamiltonian governs it?), and refinement (what are the parameters?). No single controller solves all three equally well. We present TAS-AI, a hybrid agnostic-to-physics-informed framework for autonomous triple-axis spin-wave spectroscopy that separates these tasks explicitly. In blind reconstruction benchmarks, model-agnostic methods such as random sampling, coarse grids, and Gaussian-process mappers reach a global error threshold more reliably and with fewer measurements than physics-informed planning, supporting the claim that discovery and inference are distinct tasks requiring distinct controllers. Once signal structure is localized, the physics-informed stage performs in-loop Hamiltonian discrimination and parameter refinement: in a controlled square-lattice test between nearest-neighbor-only and J1-J2 Hamiltonians, TAS-AI reaches a decisive AIC-derived evidence ratio (>100) in fewer than 10 measurements, while motion-aware scheduling cuts wall-clock time by 32% at a fixed measurement budget. We also identify a failure mode of posterior-weighted design, algorithmic myopia, in which the planner over-refines the current leading model while under-sampling low-intensity falsification probes. A constrained falsification channel sharply reduces time spent committed to the wrong model and accelerates correct model selection without modifying the Bayesian inference engine. In controlled two-model ablations, both a deterministic top-two max-disagreement rule and an LLM-based audit committee achieve this gain under identical constraints. We demonstrate the full workflow in silico using a high-fidelity digital twin and provide an open-source Python implementation.