Continuous families of bipartite 2-unitary matrices: a Paradigm for quantum convolution gates

As quantum computing develops, the problem of implementing entangling anddisentangling quantum gates in a controllable manner reemerges in multiplecontexts. One of the newest applications of such disentangling channels arequantum convolutional neural networks, where the core idea lies in thesystematic decrease of qudit numbers without loss of information encoded inentangled states. In this work, we focus on quantum analogues of convolution and pooling -basic building block for convolutional networks - and construct andcharacterize parametrizable ``quantum convolution'' channels as coherificationsof permutation tensors. Operations constructed in this manner genericallyprovide high (dis)entangling power. In particular, we identify conditionsnecessary for the convolution channels constructed using our method to possessmaximal entangling power. Based on this, we establish new, continuous classesof bipartite 2-unitary matrices of dimension $d^2$ for $d = 7$ and $d = 9$,with $2$ and $4$ free nonlocal parameters, corresponding to perfect tensors ofrank $4$ or $4$-partite absolutely maximally entangled states. The newlyestablished families may serve as the prototype for trainableconvolution/pooling layers in quantum convolutional neural networks.