Useful quantum neural networks should not merely explore large Hilbert spaces but should organise their expressive capacity according to the symmetries of the learning problem. We introduce symmetry-organised complexity as an ansatz-level, representation-theoretic trajectory diagnostic for quantum neural networks. The diagnostic combines symmetry-sector organisation, cross-irreducible representation organised complexity, and symmetry metastability into a composite index, which is then multiplied by a compliance factor that penalises apparent complexity arising from symmetry violation. This compliance factor is defined at the level of the implemented trainable generators rather than as a representation-independent channel metric. The representation-theoretic basis of the construction is that, for an exactly equivariant network, the effective trainable operators lie in the commutant of the group action and are controlled by multiplicity dimensions rather than by the full Hilbert-space dimension. We show that joint sector collapse and state freezing force the index to vanish under an explicit multiplicity–purity condition and that networks with identical qubit and parameter counts can have different values of the index. Two analytically tractable four-qubit examples with excitation number and total spin symmetry illustrate how the diagnostic separates sector-collapsed, symmetry-organised, and symmetry-breaking behaviour. A controlled U(1)-compatible teacher–student classification task further shows that, in this validation setting, the ordering of the composite index across equivariant, hybrid, and non-equivariant ansatze agrees with the ordering of generalisation accuracy. The framework is most informative when the relevant symmetry of the learning problem is known.
Ugail et al. (Tue,) studied this question.