Symmetry tests for cyclic groups with quantum linear optics
arXiv:2607.11393
The paper introduces a linear‑optics method to test whether a photonic quantum state is invariant under a cyclic group symmetry, extending circle and SWAP tests and enabling applications such as Hadamard tests in variational and quantum‑machine‑learning algorithms.
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 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.