Two qubits across Hamiltonians.
Exact evolution, MixFunn and parameter-conditioned temporal continuation.
Time evolution
State fidelity
F = 1 denotes identical normalized states up to a global phase.
State snapshot
Fidelity in the selected interval
Summary statistics are evaluated on 801 uniformly spaced samples of the selected viewing interval. The exact reference remains unchanged when its curve is hidden. Trajectories are interpolated between the 598 evaluation points. Values depend on both the selected couplings and viewing interval.
| Method | Mean fidelity | Minimum fidelity | Maximum error 1 − F |
|---|---|---|---|
| MixFunn | |||
| MixFunn + FNO |
Methods and numerical evaluation
Parameter-conditioned representation
The Hamiltonian contains variable diagonal interactions Jxx, Jyy and Jzz and fixed fields hx = hy = hz = 0.5. Exchange symmetry restricts the evolution from |00⟩ to the triplet subspace {|00⟩, (|01⟩ + |10⟩)/√2, |11⟩}. Two MixFunn layers [1], with widths 64 and 6, map (Jxx, Jyy, Jzz, t) to the real and imaginary parts of the triplet amplitudes. The first projection includes pairwise input products. The network contains 3,492 parameters and uses sine, cosine and identity functions.
Training domain and supervision
Training uses 512 Sobol-sampled Hamiltonians in [−1, 1]³ and 200 temporal points on [0, 4]. The objective combines Schrödinger-residual, initial-condition, normalization and supervised exact-observable losses. Optimization uses 550 outer LBFGS epochs. Predictions are normalized before probabilities, correlators and fidelities are evaluated. The parameter box describes the sampling domain; it does not guarantee uniform approximation accuracy throughout that domain.
Fourier-operator continuation
The temporal operator follows the time-domain FNO approach [2], conditioned on the same three interaction parameters. Its eleven input channels contain six real amplitude components, two sinusoidal positional coordinates and the three couplings. The architecture has three Fourier layers, 32 latent channels and 16 retained modes, totaling 103,110 real parameters. FNO training uses MixFunn input trajectories and exact target trajectories.
Autoregressive inference starts from the first 64 MixFunn samples and advances 32 samples per step, with Δt = 4/199. Predicted amplitudes are normalized, and overlapping output windows are blended before final normalization. The first output window begins near t = 0.643, within the training interval. The next FNO input uses the preceding normalized prediction before overlap blending. Temporal extrapolation, marked by shading, begins at t = 4.
Evaluation and numerical accuracy
The reference is the unitary solution ψ(t) = exp(−iHt)|00⟩, evaluated in the triplet subspace and independently validated against QuTiP. Both neural models are evaluated for each selected coupling triplet; no retraining is performed. The evaluation grid contains 598 points up to t = 12. Displayed curves and instantaneous values use linear interpolation, and the summary statistics use 801 uniformly spaced points within the selected viewing interval. Fidelity is evaluated against the exact state even when the reference curve is hidden.
Fidelity is displayed without performance-based filtering. Errors can grow with time and vary across Hamiltonians; successful parameter conditioning does not imply accurate long-time evolution. Changing the viewing interval preserves the initial condition at t = 0.
References
- Farias, T. de S., de Lima, G. G., Maziero, J. & Villas-Boas, C. J. MixFunn: A Neural Network for Differential Equations with Improved Generalization and Interpretability. arXiv 2503.22528 (2025). doi:10.48550/arXiv.2503.22528
- Shah, F., Patti, T. L., Berner, J., Tolooshams, B., Kossaifi, J. & Anandkumar, A. Fourier neural operators for learning dynamics in quantum spin systems. Communications Physics 9, 226 (2026). doi:10.1038/s42005-026-02644-1