Toward superpolynomial quantum speedup of equivariant quantum algorithms with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>SU</mml:mi> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> </mml:math> symmetry

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 $\mathrm{SU}(d)$ 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 $\mathrm{PQC}+$.