Gaussian boson sampling with click-counting detectors

Gaussian boson sampling constitutes a prime candidate for an experimental demonstration of quantum advantage within reach with current technological capabilities. The original proposal employs photon-number-resolving detectors, however, these are not widely available. Nevertheless, inexpensive threshold detectors can be combined into a single click-counting detector to achieve approximate photon-number resolution. We investigate the problem of sampling from a general multimode Gaussian state using click-counting detectors and show that the probability of obtaining a given outcome is related to a matrix function which is dubbed as the Kensingtonian. We show how the Kensingtonian relates to the Torontonian and the Hafnian, thus bridging the gap between known Gaussian boson sampling variants. We then prove that, under standard complexity-theoretical conjectures, the model cannot be simulated efficiently.