Detailed study of Gaussian boson sampling

Since the development of boson sampling, there has been a quest to construct more efficient and experimentally feasible protocols to test the computational complexity of sampling from photonic states. In this paper, we interpret and extend the results presented previously [Phys. Rev. Lett. 119, 170501 (2017)]. We derive an expression that relates the probability to measure a specific photon output pattern from a Gaussian state to the Hafnian matrix function and use it to design a Gaussian boson sampling protocol. Then, we discuss the advantages that this protocol has relative to other photonic protocols and the experimental requirements for Gaussian boson sampling. Finally, we relate it to the previously most general protocol, scattershot boson sampling [Phys. Rev. Lett. 113, 100502 (2014)].