Recent experiments have focused attention on the properties of chains of atoms in which the atoms are either in their ground states or in highly excited Rydberg states which block similar excitations in their immediate neighbors. As the low-energy Hilbert space of such chains is isomorphic to that of a chain of Fibonacci anyons, they have been proposed as a platform for topological quantum computation and for simulating anyon dynamics. We show that generic local operators in the Rydberg chain correspond to nonlocal anyonic operators that do not preserve a topological symmetry of the physical anyons. Consequently, we argue that Rydberg chains do not yield Fibonacci anyons and quantum computation with Rydberg atoms is not topologically protected.
Bibliographical noteFunding Information:
We are grateful to M. Zaletel for several discussions on the connections between the Rydberg-Fibonacci map and the Jordan-Wigner map. We are also grateful to C. R. Laumann for valuable discussions on anyons. This work was supported by NSF Grant No. DMR-1752759 (A.C.), NSF Grant No. DMR-1352271 (F.J.B.), and the U.S. Department of Energy Grant No. DE-SC0016244 (S.L.S.). A.C. acknowledges support from the Sloan Foundation through the Sloan Research Fellowship.
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