Functional completeness of planar Rydberg blockade structures

Abstract

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.

arXiv comment33 pages, 14 figures, v2: fixed typos, added additional references and comments

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