  Particle transport and localization phenomena in condensed-matter systems can
be modeled using a tight-binding lattice Hamiltonian. The ideal experimental
emulation of such a model utilizes simultaneous, high-fidelity control and
readout of each lattice site in a highly coherent quantum system. Here, we
experimentally study quantum transport in one-dimensional and two-dimensional
tight-binding lattices, emulated by a fully controllable $3 \times 3$ array of
superconducting qubits. We probe the propagation of entanglement throughout the
lattice and extract the degree of localization in the Anderson and
Wannier-Stark regimes in the presence of site-tunable disorder strengths and
gradients. Our results are in quantitative agreement with numerical simulations
and match theoretical predictions based on the tight-binding model. The
demonstrated level of experimental control and accuracy in extracting the
system observables of interest will enable the exploration of larger,
interacting lattices where numerical simulations become intractable.
