Functional completeness of planar Rydberg blockade structures

The construction of Hilbert spaces that are characterized by local constraints as the low-energy sectors of microscopic models is an important step towards the realization of a wide range of quantum phases with long-range entanglement and emergent gauge fields. Here we show that planar structures of trapped atoms in the Rydberg blockade regime are functionally complete: Their ground state manifold can realize any Hilbert space that can be characterized by local constraints in the product basis. We introduce a versatile framework, together with a set of provably minimal logic primitives as building blocks, to implement these constraints. As examples, we present lattice realizations of the string-net Hilbert spaces that underlie the surface code and the Fibonacci anyon model. We discuss possible optimizations of planar Rydberg structures to increase their geometrical robustness.