Realistic photon-number resolution in Gaussian Boson Sampling

Gaussian Boson Sampling (GBS) is the model of non-universal quantum computation that has already demonstrated quantum supremacy in experiments. This model entails sampling photocounting events from a multimode Gaussian state at the outputs of a linear interferometer. In this scheme, collision events -- those with more than one photon for each mode -- are infrequent. However, they are still used for validation purposes. Therefore, the limitation of realistic detectors to perfectly resolve adjacent photon numbers becomes pivotal. We have derived the photocounting probability distribution in the GBS schemes, which is applicable for use with general detectors and photocounting techniques. This probability distribution is expressed in terms of functionals of the field-quadrature covariance matrix -- e.g., Hafnian and Torontonian in the well-known special cases of photon-number resolving and on-off detectors, respectively. Based on our results, we have considered a GBS validation technique involving detectors with realistic photon-number resolution.