"We propose to build a two-dimensional (2D) discrete quantum simulator based on ensembles of ultracold neutral atoms. In this system all degrees of freedom will be controlled at the quantum limit: the number and positions of the atoms, as well as their internal (qubit) and vibrational states. The dynamics is implemented by discrete steps of spin-dependent transport combined with controlled cold collisions of the atoms.Although numerous theoretical studies have considered this architecture as the most promising route to quantum simulation, it has not yet been realized experimentally in all essential aspects.This simulator allows us to study dynamical properties of single-particle and many-body systems in engineered 2D environments. In single particle discrete systems, also known as quantum walks, we plan to investigate transport properties connected to graphene-like Dirac points, and localization phenomena associated with disorder. In the many-particle setting we will realize 2D cluster states as needed for measurement-based quantum computation, as well as simple quantum cellular automata."