  A symmetry of a state $\vert \psi \rangle$ is a unitary operator of which
$\vert \psi \rangle$ is an eigenvector. When $\vert \psi \rangle$ is an unknown
state supplied by a black-box oracle, the state's symmetries provide key
physical insight into the quantum system; symmetries also boost many crucial
quantum learning techniques. In this paper, we develop a variational hybrid
quantum-classical learning scheme to systematically probe for symmetries of
$\vert \psi \rangle$ with no a priori assumptions about the state. This
procedure can be used to learn various symmetries at the same time. In order to
avoid re-learning already known symmetries, we introduce an interactive
protocol with a classical deep neural net. The classical net thereby
regularizes against repetitive findings and allows our algorithm to terminate
empirically with all possible symmetries found. Our scheme can be implemented
efficiently on average with non-local SWAP gates; we also give a less efficient
algorithm with only local operations, which may be more appropriate for current
noisy quantum devices. We simulate our algorithm on representative families of
states, including cluster states and ground states of Rydberg and Ising
Hamiltonians. We also find that the numerical query complexity scales well with
qubit size.
