Two-detector reconstruction of multiphoton states in linear optical networks

We propose a method for partial state reconstruction of multiphoton states in multimode ($N$-photon $M$-mode) linear optical networks (LONs) employing only two bucket photon-number-resolving (PNR) detectors. The reconstructed Heisenberg-Weyl-reduced density matrix captures quantum coherence and symmetry with respect to Heisenberg-Weyl operators. Employing deterministic quantum computing with one qubit (DQC1) circuits, we reduce the detector requirement from $M$ to $2$, while the requirement on measurement configurations is retained $2M^{3}-2M$. To ensure physicality, maximum likelihood estimation (MLE) is incorporated into the DQC1 reconstruction process, with numerical simulations demonstrating the efficiency of our approach and its robustness against interferometer noises. This method offers a resource-efficient solution for state characterization in large-scale LONs to advance photonic quantum technologies.