  We present a novel scheme for universal quantum computation based on spinless
interacting bosonic quantum walkers on a piecewise-constant graph, described by
the two-dimensional Bose-Hubbard model. Arbitrary X and Z rotations are
constructed, as well as an entangling two-qubit CPHASE gate and a SWAP gate.
Quantum information is encoded in the positions of the walkers on the graph, as
in previous quantum walk-based proposals for universal quantum computation,
though in contrast to prior schemes this proposal requires a number of vertices
only linear in the number of encoded qubits. It allows single-qubit
measurements to be performed in a straightforward manner with localized
operators, and can make use of existing quantum error correcting codes either
directly within the universal gate set provided, or by extending the lattice to
a third dimension. We present an intuitive example of a logical encoding to
implement the seven-qubit Steane code. Finally, an implementation in terms of
ultracold atoms in optical lattices is suggested.
