  The Hubbard model may be the simplest model of particles interacting on a
lattice, but simulation of its dynamics remains beyond the reach of current
numerical methods. In this article, we show that general quantum computations
can be encoded into the physics of wave packets propagating through a planar
graph, with scattering interactions governed by the fermionic Hubbard model.
Therefore, simulating the model on planar graphs is as hard as simulating
quantum computation. We give two different arguments, demonstrating that the
simulation is difficult both for wave packets prepared as excitations of the
fermionic vacuum, and for hole wave packets at filling fraction one-half in the
limit of strong coupling. In the latter case, which is described by the t-J
model, there is only reflection and no transmission in the scattering events,
as would be the case for classical hard spheres. In that sense, the
construction provides a quantum mechanical analog of the Fredkin-Toffoli
billiard ball computer.
