In Fock Boson Sampling, one starts with a particle-number state $|n_1, ..., n_m\rangle$ of $m$ modes, sends it to an interferometer effecting a unitary operation $U$ on the creation operators, and measures the output. In Gaussian Boson Sampling, the interferometer is the same, but the input states are Gaussian states (for simplicity, we can consider squeezed vacuum states with no displacement).
Judging by the similarity of the models, one could think that they have the same computational power. My question is whether it is possible, given a FBS setup, simulate the outputs of a GBS setup, or vice versa?
The fact that one outputs a constant number of photons, and the other one outputs a random one, doesn't seem to be a big issue. For example, if I wanted to simulate the FBS setup using the GBS setup, I could carefully choose the unitary and the squeezing parameters, and then postselect the outcome of the measurement on the correct number of photons. Conversely, to simulate GBS using FBS, I would first randomly pick the total number of photons, then send them to a carefully chosen interferometer (I might have a special interferometer for each number of qubits).
