EXPERIMENT 01 · SPARSE REPRESENTATION

Parameterized single-qubit dynamics.

Sparse MixFunn predictions evaluated against exact unitary evolution.

Reference verified against QuTiP

Loading models and sparsity masks…

Mean fidelity · selected interval——

Minimum fidelity · selected interval—Reference: exact evolution

Active parameters after pruning—Including biases · 1,284 dense parameters

Time evolution

Observable

ExactMixFunn

Curves and colors

Exact solution

MixFunn

Shaded region: t > 2, beyond the reference intervalInitial state fixed at t = 0

State fidelity

F = |⟨ψ_exact|ψ_model⟩|²

Fidelity is measured against the exact state regardless of curve visibility.

State snapshot

t = 1Within [0, 2]

Fidelity—

Norm² · unnormalized network—

Probabilities and fidelity are evaluated using the normalized state.

Model architecture and sparsity

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The parameterized MixFunn representation [1] maps four inputs to two complex state amplitudes through 32 hidden neurons. Binary masks constrain function mixing, input projection and output projection. The dense architecture contains 1,284 parameters; 305 remain active in the data-assisted variant and 366 in the variant without supervised data.

Mask counts for the selected variant. All 100 bias parameters remain unmasked.
Parameter group	Active	Total	Removed

The data-assisted variant removes 90% of input-projection coefficients and 39.84% of output-projection coefficients. The corresponding fractions without supervised data are 85% and 29.69%. Both variants retain one of three function-mixing coefficients per hidden neuron. Model parameters and sparsity patterns remain fixed throughout evaluation.

Methods, reference solution and evaluation domain

Hamiltonian and exact propagator

A single qubit evolves under H = hₓσₓ + hᵧσᵧ + h_zσ_z, with ℏ = 1 and initial state |+⟩. For a time-independent field, the reference state is ψ(t) = exp(−iHt)|+⟩. The closed-form propagator is evaluated directly and has been independently checked against QuTiP sesolve for seven Hamiltonians, including H = 0, over [0, 50].

Parameterized neural representation

The MixFunn architecture [1] uses mixed activation functions and second-order input features. The input vector (hₓ, hᵧ, h_z, t) is augmented by its six distinct pairwise products, excluding individual squares. Sine, cosine and identity functions form the nonlinear basis. Variant-specific masks enforce sparsity in function mixing and in the input and output projections. The four outputs encode the real and imaginary components of the two state amplitudes.

Domain and statistical interpretation

Field components vary continuously within [−1, 1]. The nominal sampling grid consists of nine values per axis: −1, −0.75, −0.5, −0.25, 0.01, 0.25, 0.5, 0.75 and 1, giving 729 Hamiltonians. Intermediate fields assess parameter interpolation. The interval [0, 2] is used as a temporal reference; the shaded region marks evaluation beyond this interval. This boundary does not certify the complete temporal coverage of all fine-tuning stages.

Probabilities, Pauli expectations and fidelity are computed from normalized states. The unnormalized squared norm is reported separately. Mean and minimum fidelities use 801 samples in the selected interval for the selected Hamiltonian; they do not constitute an aggregate assessment over the full Hamiltonian family. Changing the displayed 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