We introduce a framework of the equivariant convolutional quantum algorithms which is tailored for a number of machine-learning tasks on physical systems with arbitrary $\mathrm{SU}\left(d\right)$ symmetries. It allows us to enhance a natural model of quantum computation—permutational quantum computing (PQC) [Jordan, Quantum Inf. Comput. 10, 470 (2010)]—and define a more powerful model: $\mathrm{PQC}+$. While PQC was shown to be efficiently classically simulatable, we exhibit a problem which can be efficiently solved on a $\mathrm{PQC}+$ machine, whereas no classical polynomial time algorithm is known, thus providing evidence against $\mathrm{PQC}+$ being classically simulatable. We further discuss practical quantum machine learning algorithms which can be carried out in the paradigm of $\mathrm{PQC}+$.

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• Quantum algorithms
• Quantum information processing
• Quantum information theory