  Recent developments in the field of quantum machine learning have promoted
the idea of incorporating physical symmetries in the structure of quantum
circuits. A crucial milestone in this area is the realization of
$S_{n}$-permutation equivariant quantum neural networks (QNN) that are
equivariant under permutations of input objects. In this work, we focus on
encoding the rotational symmetry of point cloud datasets into the QNN. The key
insight of the approach is that all rotationally invariant functions with
vector inputs are equivalent to a function with inputs of vector inner
products. We provide a novel structure of QNN that is exactly invariant to both
rotations and permutations, with its efficacy demonstrated numerically in the
problems of two-dimensional image classifications and identifying high-energy
particle decays, produced by proton-proton collisions, with the $SO(1,3)$
Lorentz symmetry.
