The Hong-Ou-Mandel test estimates the overlap between spectral functions characterizing the internal degrees of freedom of two single photons. It can be viewed as a photon-native protocol that implements the well-known quantum SWAP test. Here, we propose a native linear-optical protocol that efficiently estimates multivariate traces of quantum states called Bargmann invariants, which are ubiquitous in quantum mechanics. Our protocol may be understood as a photon-native version of the cycle test in the circuit model, which encompasses many-photon multimode quantum states. We show the protocol is sample-efficient and discuss applications, such as generalized suppression laws, efficient quantum kernel estimation for quantum machine learning, eigenspectrum estimation, and the characterization of multiphoton indistinguishability.
@article{arxiv.2601.14204,
title = {Native linear-optical protocol for efficient multivariate trace estimation},
author = {Leonardo Novo and Marco Robbio and Ernesto F. Galvão and Nicolas J. Cerf},
journal= {arXiv preprint arXiv:2601.14204},
year = {2026}
}