Simulating lossy and partially distinguishable quantum optical circuits: theory, algorithms, and applications to experiment validation and state preparation

To understand quantum optics experiments, we must perform calculations that consider the principal sources of noise, such as losses, spectral impurity, and partial distinguishability. In both discrete and continuous variable systems, these can be modeled as mixed Gaussian states over multiple modes. The modes are not all resolved by photon-number measurements and so require calculations on coarse-grained photon-number distributions. Existing methods can lead to a combinatorial explosion in the time complexity, making this task unfeasible for even moderately sized experiments of interest. We prove that the computation of this type of distribution can be done in exponential time, providing a combinatorial speed-up. We develop numerical techniques that allow us to determine coarse-grained photon number distributions of Gaussian states, as well as density matrix elements of heralded non-Gaussian states prepared in the presence of spectral impurity and partial distinguishability. These results offer significant speed-up and accuracy improvements to validation tests of both Fock and Gaussian boson samplers that rely on binned probability distributions. Moreover, they pave the way to a more efficient simulation of realistic photonic circuits, unlocking the ability to perform exact calculations at scales that were previously out of reach. In addition, our results, including loop Hafnian master theorems, may be of interest to the fields of combinatorics and graph theory.