  Quantum walks provide a framework for understanding and designing quantum
algorithms that is both intuitive and universal. To leverage the computational
power of these walks, it is important to be able to programmably modify the
graph a walker traverses while maintaining coherence. Here, we do this by
combining the fast, programmable control provided by optical tweezer arrays
with the scalable, homogeneous environment of an optical lattice. Using this
new combination of tools we study continuous-time quantum walks of single atoms
on a 2D square lattice, and perform proof-of-principle demonstrations of
spatial search using these walks. When scaled to more particles, the
capabilities demonstrated here can be extended to study a variety of problems
in quantum information science and quantum simulation, including the
deterministic assembly of ground and excited states in Hubbard models with
tunable interactions, and performing versions of spatial search in a larger
graph with increased connectivity, where search by quantum walk can be more
effective.
