  In 1994, Reck et al. showed how to realize any unitary transformation on a
single photon using a product of beamsplitters and phaseshifters. Here we show
that any single beamsplitter that nontrivially mixes two modes, also densely
generates the set of unitary transformations (or orthogonal transformations, in
the real case) on the single-photon subspace with m>=3 modes. (We prove the
same result for any two-mode real optical gate, and for any two-mode optical
gate combined with a generic phaseshifter.) Experimentally, this means that one
does not need tunable beamsplitters or phaseshifters for universality: any
nontrivial beamsplitter is universal for linear optics. Theoretically, it means
that one cannot produce "intermediate" models of linear optical computation
(analogous to the Clifford group for qubits) by restricting the allowed
beamsplitters and phaseshifters: there is a dichotomy; one either gets a
trivial set or else a universal set. No similar classification theorem for
gates acting on qubits is currently known. We leave open the problem of
classifying optical gates that act on three or more modes.
