  We present a scheme for implementing high-fidelity quantum logic gates using
the quantum walk of a few interacting bosons on a one-dimensional lattice. The
gate operation is carried out by a single compact lattice described by a
one-dimensional Bose-Hubbard model with only nearest-neighbor hopping and
on-site interactions. We find high-fidelity deterministic logic operations for
a gate set (including the CNOT gate) that is universal for quantum information
processing. We discuss the applicability of this scheme in light of recent
developments in controlling and monitoring cold-atoms in optical lattices, as
well as an implementation with realistic nonlinear quantum photonic devices.
