Absence of Fibonacci anyons in Rydberg chains

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.

Figure 1

Left: A chain of Fibonacci anyons in the fusion tree basis. Each vertical leg carries charge $\tau$, while the horizontal bond $i$ with label $Z_i$ represents the net fusion outcome of fusing the anyons $1,...,i$ with the boundary label $Z_0$. The Hilbert space is described by the labels $Z_i$ for $1\le i\le N-1$ which obey the constraint in Eq. (4) in each boundary condition sector $(Z_0,Z_N)$. Right: A chain of Rydberg atoms in the blockaded regime which obey the same constraint [Eq. (1)] in each boundary condition sector $(n_0,n_N)$.

Figure 2

Left: Space-time depiction of braiding process that realizes $P_N^1$. A pair of Fibonacci anyons is created from the vacuum extremely far from the anyonic cluster of interest. One member of the pair (${\tau }_0$) is taken around the cluster, remaining at all times extremely far away from any anyons in the cluster. The pair is then brought back together and annihilated. Right: By deforming the world lines, the process is represented as a closed loop encircling the anyon cluster.

Figure 3

Pictorial depiction of various operators of the Fibonacci chain. The operator $P_i^1$ projects the fusion outcome of the first $i$ anyons in the Fibonacci chain to the vacuum channel, while $P_{N-i}^1(i+1,...,N)$ projects the outcome of the anyons $i+1,...,N$. The value of $Z_i$ can be measured through a braid experiment around the boundary leg $Z_0$ and the anyons $1,...,i$, or through a braid around the boundary leg $Z_N$ and the anyons $i+1,...,N$. By deforming the world lines of either braid experiment, we obtain the loop encircling the bond $i$.

Figure 4

Illustration of loops representing different braiding operations on the Fibonacci chain. As explained in detail in Ref. [53], loops that can slide past each other freely (such as those associated with (a) the projectors $P_N^1$ and $P_m^1(i_1,...,i_m)$) commute, while loops that cannot slide freely past each other (such as those associated with (b) $P_N^1$ and ${\widehat{Z}}_i$) do not.

Figure 5

An example of a qubit composed of one segment of Fibonacci chain, with two well-separated Fibonacci anyons. Information is encoded in the net topological charge of the two-anyon system, which can take on values of 1 or $\tau$.

Figure 6

Identities required to evaluate diagrams in the Fibonacci anyon model. Here $\overline{R}\equiv R^{*}$. Unless otherwise specified, solid lines represent $\tau$ anyons. All diagrams are invariant under arbitrary planar deformations of the graphs.

Figure 7

Sliding a loop over a fusion cluster.