Symmetry tests for cyclic groups with quantum linear optics

How can quantum linear optical systems be utilized to perform symmetry tests on photonic states, and what specific quantum information primitives can be derived from these tests?

The researchers analyze linear optical multiports characterized by scattering matrices $S$ whose eigenvalues are $K$-th roots of unity. By evolving an input photonic state through these multiports and measuring the photon occupation at the output modes, they define a function $f(\vec{n})$ based on the photon counts. The probability distribution of these counts is shown to be the discrete Fourier transform of the moments of the operator $\hat{S}$, allowing for the construction of projectors onto invariant subspaces. This framework is validated using Python simulations via the QOptcraft package.

The study demonstrates that linear optical interferometers can implement a variety of symmetry tests:

• Circle and SWAP Tests: Generalizes previous results to arbitrary photonic states, allowing for state comparison and distinguishability measurements.
• XOR-Permutation Tests: Using Sylvester interferometers, the authors define tests for invariance under bitwise XOR permutations, which are useful for property testing.
• Eigenstate Projection: The method allows for testing whether an input state resides in a specific eigenspace of the evolution operator $\hat{U}$, providing a path for variational methods to find eigenvectors.
• Hadamard Test: For specific cases where $K=2$ or $K=3$, the optical setup reproduces the statistics of the Hadamard test, a fundamental primitive for estimating inner products in quantum machine learning.