## Boson Sampling with a single beamsplitter

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In Boson Sampling as first proposed by Aaronson and Arkhipov in arxiv.org/abs/1011.3245, the interferometer is made up of phase shifters and beamsplitters. As these gates are universal, drawing the interferometer at random (just like in Google's random circuit sampling) gives rise to an approximately Haar-random $$m \times m$$ unitary $$U$$ acting on the single photon subspace of the $$m$$-mode Fock states.

I wonder whether one could also perform BosonSampling with a single beamsplitter. If I understand correctly, a later paper by Aaronson and Bouland arxiv.org/abs/1310.6718 shows that a single non-trivial beamsplitter, say a 50:50 beamsplitter, is universal on the single-photon subspace omitting any extra phaseshifters.

My question is thus to confirm whether I could equally well just draw a sufficiently long (how long?) sequence of 50:50 beamsplitters and make up the interferometer in BosonSampling just out of these?

This claim to me seems very suprising, as I would intuitively think that one needs phaseshifters to get all unitaries, thus I would like to confirm this.