  We introduce a framework of the equivariant convolutional algorithms which is
tailored for a number of machine-learning tasks on physical systems with
arbitrary SU($d$) symmetries. It allows us to enhance a natural model of
quantum computation--permutational quantum computing (PQC) [Quantum Inf.
Comput., 10, 470-497 (2010)] --and defines a more powerful model: PQC+. While
PQC was shown to be effectively classically simulatable, we exhibit a problem
which can be efficiently solved on PQC+ machine, whereas the best known
classical algorithms runs in $O(n!n^2)$ time, thus providing strong evidence
against PQC+ being classically simulatable. We further discuss practical
quantum machine learning algorithms which can be carried out in the paradigm of
PQC+.
