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+.

本研究提出了基于不同 SU (d) 对称性物理系统的 equivariant convolutional algorithms 框架，它能够增强量子计算模型，如 permutational quantum computing (PQC)，定义了一个更强大的模型，PQC+。我们进一步讨论了可在 PQC + 范式下执行的实用量子机器学习算法。