  Quantum neural network architectures that have little-to-no inductive biases
are known to face trainability and generalization issues. Inspired by a similar
problem, recent breakthroughs in machine learning address this challenge by
creating models encoding the symmetries of the learning task. This is
materialized through the usage of equivariant neural networks whose action
commutes with that of the symmetry. In this work, we import these ideas to the
quantum realm by presenting a comprehensive theoretical framework to design
equivariant quantum neural networks (EQNN) for essentially any relevant
symmetry group. We develop multiple methods to construct equivariant layers for
EQNNs and analyze their advantages and drawbacks. Our methods can find unitary
or general equivariant quantum channels efficiently even when the symmetry
group is exponentially large or continuous. As a special implementation, we
show how standard quantum convolutional neural networks (QCNN) can be
generalized to group-equivariant QCNNs where both the convolution and pooling
layers are equivariant to the symmetry group. We then numerically demonstrate
the effectiveness of a SU(2)-equivariant QCNN over symmetry-agnostic QCNN on a
classification task of phases of matter in the bond-alternating Heisenberg
model. Our framework can be readily applied to virtually all areas of quantum
machine learning. Lastly, we discuss about how symmetry-informed models such as
EQNNs provide hopes to alleviate central challenges such as barren plateaus,
poor local minima, and sample complexity.
