Fast contracted Clebsch--Gordan tensor products for equivariant graph neural networks

We present an $\mathcal{O}(L^3)$ algorithm for evaluating contracted Clebsch–Gordan tensor products in $\mathrm{O}(3)$-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank. Mapping the angular integral to a structured Gauss–Legendre and Fourier tensor-product grid decouples the radial channel contractions from the angular transforms. The antisymmetric parity-odd Clebsch–Gordan channels, unreachable by the symmetric pointwise product on a scalar $S^2$ grid, are recovered through the surface-curl pairing $\widehat{r}\cdot [{\mathrm{\nabla }}_{S^2}A\times {\mathrm{\nabla }}_{S^2}B]$, the spherical Poisson bracket, which supplies the $L=1$ angular momentum on the grid while preserving rotational equivariance. The construction extends to parity-aware equivariant message passing in atomic-cluster-expansion-style architectures and is verified by direct numerical quadrature. The full uncontracted Clebsch–Gordan tensor product remains subject to the $\mathcal{O}(L^4)$ output-size lower bound. A benchmark shows wall-clock scaling empirically as $L^2$ across the practical $l_{\max }$ range. For the on-site contraction this is pre-asymptotic, giving way to $L^3$ at large $l_{\max }$. For message passing it is structural and the runtime is memory-bandwidth bound on $L^2$-sized grid tensors.