Symmetry tests for cyclic groups with quantum linear optics

Abstract

Testing quantum states for symmetries has multiple applications in quantum state comparison or as a primitive in more complex quantum algorithms. We present a method that determines whether an input photonic state is invariant under the action of a cyclic group defined by an operator $\hat{S}$ with eigenvalues which are roots of unity. The results generalize previously known circle and SWAP tests related to suppression laws in Fourier interferometers and offer a new test for permutations defined by a binary shift and a way to search for eigenstates in multiphoton states. Finally, we discuss the scenarios where this measurement can be used to perform Hadamard test, a fundamental primitive in variational and quantum machine-learning algorithms.