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.

翻译：状态 $\vert \psi \rangle$ 的对称性定义为使 $\vert \psi \rangle$ 成为其特征向量的幺正算符。当 $\vert \psi \rangle$ 由黑盒预言机提供的未知状态时，其对称性可提供关于量子系统的关键物理洞见；此外，对称性还能增强许多重要的量子学习技术。本文提出一种变分混合量子-经典学习方案，可在无任何先验假设的条件下系统探测 $\vert \psi \rangle$ 的对称性。该过程可同时学习多种对称性。为避免重复学习已知对称性，我们引入一种与经典深度神经网络交互的协议。经典网络通过正则化防止重复发现，使算法在经验上能找出所有可能的对称性后终止。我们的方案平均上可通过非局域SWAP门高效实现；同时给出一种仅依赖局部操作的效率较低算法，更适合当前含噪声量子设备。我们在代表性态族（包括团簇态、里德伯哈密顿量基态和伊辛哈密顿量基态）上模拟了算法，并发现数值查询复杂度随量子比特规模扩展性良好。