A Spectral Identifiability Threshold for Dissipative Rate Recovery from Truncated Liouvillian Spectra

Human learning is a dissipative dynamical process: mastery accumulates through practice, decays through forgetting, and propagates across interdependent concepts. We model it as a nonlinear dissipative system of ordinary differential equations whose parameters are mechanistically meaningful (a transfer-of-practice matrix, per-concept forgetting rates, and a saturating practice-response gain), and we study when those parameters can actually be recovered from data. For two concepts, we prove a constructive structural identifiability theorem for the associated inverse problem under explicit excitation conditions, together with monotonicity, robustness and L-stability results. We derive an implicit L-stable scheme and a batched solver numerically equivalent to the per-trajectory formulation yet two orders of magnitude faster. The empirical study is two-sided. Under the theorem's excitation conditions, synthetic recovery is exact. On large observational benchmarks it is not. An apparently strong recovery, with forgetting rates correlating with topic error rates at Spearman $ρ= 0.83$, is refuted by four controls: it survives destroying the temporal order of the data, is matched by a classical Bayesian baseline, and is unaffected by removing real timestamps. A single shared forgetting rate attains the same accuracy, and replaying the real schedule with identical true rates reproduces most of the correlation. A positive control shows that the controls detect forgetting where it is observable. We trace the effect to the stationary structure of the model and show that it is consistent with the degeneration anticipated by the theory in the absence of designed excitation.

Inferring the dynamical generator of a many-body quantum system from measurement data is essential for the verification, calibration, and control of quantum processors. When the system is open, this task becomes considerably harder than in the purely unitary case, because coherent and dissipative mechanisms can produce similar measurement statistics and long-time data can be insensitive to coherent couplings. Here we tackle this so-called Lindbladian learning problem of open-system characterisation with maximum-likelihood on Pauli measurements at multiple experimentally friendly transient times, exploiting the richer information content of transient dynamics. To navigate the resulting non-convex likelihood loss-landscape, we augment the physical model neural differential-equation term, which is progressively removed during training to distil an interpretable Lindbladian solution. Our method reliably learns open-system dynamics across neutral-atom (with 2D connectivity) and superconducting Hamiltonians, as well as the Heisenberg XYZ, and PXP models on a spin-1/2 chain. For the dissipative part, we show robustness over phase noise, thermal noise, and their combination. Our algorithm can robustly infer these dissipative systems over noise-to-signal ratios spanning four orders of magnitude, and system sizes up to $N=6$ qubits with fewer than $5 \times 10^5$ shots.

The Lindblad master equation is the standard framework for describing the non-unitary evolution of open quantum systems, where environmental interactions induce dissipation and decoherence. When both the system Hamiltonian and the dissipation rates are partially unknown or explicitly time-dependent, traditional analytical inversion and system-identification techniques become intractable. Recent works have demonstrated that Transformer-based models can infer unknown dissipation rates from observable time series, yet these approaches typically rely on hand-crafted statistical features under idealized and highly restricted conditions. Here we advance the paradigm by introducing a raw time-series Transformer that directly ingests the full trajectories of Pauli expectation values $\langleσ_x(t)\rangle$, $\langleσ_y(t)\rangle$, and $\langleσ_z(t)\rangle$, thereby fully exploiting the self-attention mechanism for temporal modeling. The architecture is further extended to jointly learn unknown Hamiltonian parameters, handle multiple dissipation channels, and operate robustly under realistic measurement noise. Across all tested scenarios the model achieves consistently high reconstruction accuracy while eliminating manual feature engineering. This provides a scalable, robust, and versatile framework for quantum environment sensing in realistic open quantum systems.