  We develop an alternative boson sampling model operating on single-photon
states followed by linear interferometry and Gaussian measurements. The
hardness proof for simulating such continuous-variable measurements is
established in two main steps, making use of the symmetry of quantum evolution
under time reversal. Namely, we first construct a twofold version of
scattershot boson sampling in which, as opposed to the original proposal, both
legs of a collection of two-mode squeezed vacuum states undergo parallel
linear-optical transformations. This twofold scattershot model yields, as a
corollary, an instance of boson sampling from Gaussian states where photon
counting is hard to simulate. Then, a time-reversed setup is used to exhibit a
boson sampling model in which the simulation of Gaussian measurements -- namely
the outcome of eight-port homodyne detection -- is proven to be computationally
hard. These results illustrate how the symmetry of quantum evolution under time
reversal may serve as a tool for analyzing the computational complexity of
novel physically-motivated computational problems.
