  Aaronson and Arkhipov recently used computational complexity theory to argue
that classical computers very likely cannot efficiently simulate linear,
multimode, quantum-optical interferometers with arbitrary Fock-state inputs
[Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)]. Here we present an
elementary argument that utilizes only techniques from quantum optics. We
explicitly construct the Hilbert space for such an interferometer and show that
its dimension scales exponentially with all the physical resources. We also
show in a simple example just how the Schr\"odinger and Heisenberg pictures of
quantum theory, while mathematically equivalent, are not in general
computationally equivalent. Finally, we conclude our argument by comparing the
symmetry requirements of multiparticle bosonic to fermionic interferometers
and, using simple physical reasoning, connect the nonsimulatability of the
bosonic device to the complexity of computing the permanent of a large matrix.
