  We study what is arguably the most experimentally appealing Boson Sampling
architecture: Gaussian states sampled with threshold detectors. We show that in
this setting, the probability of observing a given outcome is related to a
matrix function that we name the Torontonian, which plays an analogous role to
the permanent or the Hafnian in other models. We also prove that, provided that
the probability of observing two or more photons in a single output mode is
sufficiently small, our model remains intractable to simulate classically under
standard complexity-theoretic conjectures. Finally, we leverage the
mathematical simplicity of the model to introduce a physically motivated, exact
sampling algorithm for all Boson Sampling models that employ Gaussian states
and threshold detectors.
