A localization constraint is sufficient to recover nontrivial semiclassical organization from spectral data and at the same time periodic orbits confirm their fundamental role in the structure of quantum chaotic eigenfunctions.
Abstract
Physics-informed neural networks and neural quantum states have consolidated a new paradigm to analyze and discover physical phenomena through constrained neural parametrizations. In this context, we investigate whether the semiclassical structure of the eigenfunctions of a quantum chaotic system can be unveiled through unsupervised learning. To this end, we train a"quantum dictionary", formulated as an overcomplete autoencoder, that sparsely represents the eigenstates of the system, using as an illustration the quantum baker map. The only explicit physical information imposed on the dictionary atoms is their localization in phase space, without providing any kind of information about the periodic orbits of the corresponding classical system. The model achieves high fidelity in reconstructing eigenstates not used during training. By comparing the learned atoms with an independently constructed"semiclassical dictionary", we find that they spontaneously localize on the periodic orbits and develop scar-like structures. This result is interesting in two ways: a localization constraint is sufficient to recover nontrivial semiclassical organization from spectral data and at the same time periodic orbits confirm their fundamental role in the structure of quantum chaotic eigenfunctions. More generally, our proposed architecture opens a new route to learning representations whose atoms optimize other chosen physical properties.
Abstract The numerical simulation of quantum systems with non-trivial topology typically requires specialized coordinate charts or symplectic integrators. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative, but standard implementations struggle with periodic boundary conditions. We present a robust PINN architecture for the Quantum Pendulum (a particle on S 1) that enforces strict periodicity via a Trigonometric Coordinate Embedding layer. This “Hard Constraint” approach eliminates the need for unstable boundary penalty terms. We successfully reconstruct the unitary evolution of the Mathieu ground state under a non-linear cosine potential, achieving a norm conservation > 0.99 and initial condition error of �� ( 10 − 6 ). The method captures subtle quantum phenomena, including macroscopic tunneling tails and phase space delocalization. Validation is performed against the exact analytical solution via Mathieu functions. This work bridges the gap between modern Scientific Machine Learning and canonical quantum mechanics, serving as a tutorial on embedding topological constraints into neural solvers.
José Carlos Ferreira Bastos, Glendo de Freitas Guimarães, Maximilian Esper· Revista Brasileira de Ensino...· 0 citations
Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.
In this work, we introduce a complementary optimization method for progressive and adaptive state search (COMPASS) based on biorthogonal adaptive recurrent neural quantum states. Our approach combines an adaptive autoregressive architecture with a biorthogonal variational Monte Carlo scheme as well as a complementary optimization scheme that alternates between energy and variance minimization. This enables the stable convergence to ground-state eigenpairs, while avoiding Markov chain sampling through exact autoregressive generation. We demonstrate that for parity-time(PT)-symmetric Hamiltonians, unconstrained complex ansatze can spontaneously break PT symmetry during optimization, even in the unbroken phase, leading to spurious imaginary energies. Real-valued ansatze, on the other hand, naturally constrain the optimization to the correct physical manifold. Conversely, for generic non-Hermitian (NH) Hamiltonians without symmetry protection and complex spectra, complex ansatze are essential for capturing complex ground-state properties. Our results establish that physically-informed ansatz selection is crucial for reliable NH simulations. By combining adaptive architectures, biorthogonal optimization, and symmetry-aware modeling, this framework enables a direct study of 1D and 2D NH many-body systems without Hermitian embeddings or adiabatic continuation. Applying this framework to systems with frustrated magnetism, we show that gap frustration provides a quantitative shield against NH spectral instability, with the frustration gap setting a critical threshold for PT-symmetry breaking. Also, complexifying the frustration coupling itself generates a new topologically nontrivial network of diabolic level crossings, controlled by the phase of the complex coupling, that has no Hermitian analog. We term this novel spectral topology in NH frustrated systems the diabolic ring.
Lavoisier Wah, Flore K. Kunst, Mohamed Hibat-Allah· 3 citations· ⚡1
Understanding and coherently controlling the properties of interacting quantum many-body systems is a central challenge in non-equilibrium quantum physics. While in the past decades, a wide range of many-body Hamiltonians have been introduced to study quantum chaos, atypical eigenstates, and quantum resources, systematically engineering and continuously tuning these properties within a single microscopic model remains largely unexplored. Here, we employ a pulse engineering scheme to construct an effective Hamiltonian that continuously interpolates between a chaotic Heisenberg (XXX) chain with a local impurity and the ZX Hamiltonian. Along this interpolation, we identify several families of analytically tractable atypical eigenstates embedded in the excited-state spectrum with distinct entanglement and nonstabilizerness properties. In the XXX limit, these states exhibit exact plateaus in both entanglement and stabilizer R\'enyi entropy and correspond to coherent superpositions of long-range valence-bond solid (VBS) states. As the pulse strength increases, the effective Hamiltonians exhibit a hierarchy of new set of approximate entanglement plateaus in the low-energy spectrum. Interestingly, in the fully pulse-engineered ZX limit, we uncover a distinct pair of long-range entangled stabilizer eigenstates, corresponding to rainbow scar states. We further show that pulse engineering preserves the distinct chaotic and non-chaotic regimes of the original model, while that is largely absent in the dynamical generation of entanglement and nonstabilizerness. The pulse-engineered models generate nearly identical quantum resources in both regimes, revealing a partial decoupling between quantum chaos and quantum-resource generation. Our results establish pulse engineering as a versatile framework for generating many-body Hamiltonians with structured eigenstates and tunable quantum resources.
Prasant Mallik, A. Sil, Sudipto Singha Roy· 0 citations
We propose a physics-informed neural network (PINN) scheme for designing shortcuts to adiabaticity in a three-level non-Hermitian quantum system. The PINN is used to solve an inverse control problem in which the state amplitudes and the auxiliary driving field are learned simultaneously from the Schrödinger residual, the initial and target population constraints and a probability conservation constraint on the control pulse. The learned compensation field counteracts the loss of the intermediate state and enables high-fidelity population inversion in an open-system setting. Importantly, the imposed probability conservation is treated as an auxiliary constraint along the learned trajectory rather than as an intrinsic property of the non-Hermitian Hamiltonian. Under this constrained evolution, the Hamiltonian expectation value evaluated on the obtained state remains real within numerical accuracy. Numerical simulations and independent propagation with the fitted control field verify the population inversion and exhibit strong generalization capability over a range of coupling strengths and dissipation rates, as verified by retraining the network independently for each parameter set. The results demonstrate that PINNs provide a flexible inverse design tool for non-Hermitian shortcut to adiabaticity protocols when the governing dynamics and physical constraints are explicitly incorporated into the loss function.
Ming Liu, Feng Huang, Siqi Zhang et al.· Entropy· 0 citations
Grokking, the delayed transition from memorization to generalization, is a fundamental phenomenon in gradient-based learning, yet its dynamics within variational quantum machine learning (QML) remain largely unexamined. In this work, we report the empirical observation of both the grokking transition and epoch-wise double descent in a two-qubit quantum neural network (QNN) under a complete parameterization of the SU(4) manifold. We demonstrate that overparameterization via increased circuit depth improves the probability of successful generalization. Notably, these architectures frequently exhibit an epoch-wise double descent in test error, degrading at a critical epoch before recovering into a generalizing state. Crucially, we identify a generalization decay in late-stage training, where the test error increases significantly despite a stagnant training loss. Bridging this behavior with algorithmic stability theory, our analysis reveals that this decay correlates with an unconstrained increase of the weight-norm, drifting away from sparse, phase-aligned harmonic solutions toward overfitted solutions in the Hilbert space. We analyze the underlying temporal dynamics of this transition, demonstrating how the onset of generalization is linked to optimization hyperparameters such as learning rate and weight decay. Finally, to mitigate late-stage decay, we introduce a weak explicit weight-norm regularization into the loss function. We demonstrate that this structural anchor stabilizes the post-grokking phase and permanently preserves generalization gains, providing a robust framework for training overparameterized quantum circuits.
