Two-qubit quantum dynamics.

Symmetry reduction and neural representation

The exchange-symmetric Hamiltonian preserves the triplet subspace spanned by |T₊⟩ = |00⟩, |T₀⟩ = (|01⟩ + |10⟩)/√2 and |T₋⟩ = |11⟩. A MixFunn network [1] maps time to the real and imaginary components of these three amplitudes. The 1 → 16 → 6 architecture contains 246 parameters and uses sine, cosine and identity functions. Predicted states are normalized before evaluating probabilities, correlators and state fidelity.

Exact reference and temporal continuation

The reference trajectory is ψ(t) = exp(−iHt)|00⟩, evaluated by spectral decomposition with ℏ = 1. MixFunn optimization uses 4,000 time points on [0, 10] and 50 outer LBFGS epochs. The data-assisted variant includes a QuTiP observable loss. The variant without supervised data uses the Schrödinger residual, initial-condition and norm losses; QuTiP provides an independent evaluation reference. The temporal continuation adopts the time-domain neural-operator approach studied by Shah et al. [2], adapted here to a hybrid MixFunn–FNO pipeline. MixFunn trajectories provide both the input and target windows for FNO training.

The FNO has three Fourier layers, 32 latent channels and 16 retained modes. Autoregressive inference uses 512-point windows, a 256-point shift and Δt = 10/3999, with blending across overlapping windows. Training combines trajectory error with state-normalization, fidelity, Schrödinger-residual, overlap and temporal-smoothness terms. The available continuation extends to t = 50 for the data-assisted variant and t = 20 for the variant without supervised data, with fixed parameters in both cases.

Evaluation protocol

The data-assisted FNO trajectory contains 19,996 time points over [0, 50]. The variant without supervised data is evaluated at 4,000 points over [0, 20]. Displayed FNO observables and fidelities are linearly interpolated between stored samples. Each plot samples the selected interval at 801 points, while the state-inspection control evaluates MixFunn directly at the selected time. Full-interval summary statistics refer to the selected variant’s complete available interval; the training-domain card reports its evaluation within [0, 10]. The two full-interval scores cover different horizons and should not be interpreted as a comparison on identical time grids. Changing the displayed interval preserves the initial condition at t = 0. Fidelity is always measured against the exact reference, independently of curve visibility.