  We show how to perform universal Hamiltonian and adiabatic computing using a
time-independent Hamiltonian on a 2D grid describing a system of hopping
particles which string together and interact to perform the computation. In
this construction, the movement of one particle is controlled by the presence
or absence of other particles, an effective quantum field effect transistor
that allows the construction of controlled-NOT and controlled-rotation gates.
The construction translates into a model for universal quantum computation with
time-independent 2-qubit ZZ and XX+YY interactions on an (almost) planar grid.
The effective Hamiltonian is arrived at by a single use of first-order
perturbation theory avoiding the use of perturbation gadgets. The dynamics and
spectral properties of the effective Hamiltonian can be fully determined as it
corresponds to a particular realization of a mapping between a quantum circuit
and a Hamiltonian called the space-time circuit-to-Hamiltonian construction.
Because of the simple interactions required, and because no higher-order
perturbation gadgets are employed, our construction is potentially realizable
using superconducting or other solid-state qubits.
