Parafermion stabilizer codes

We define and study parafermion stabilizer codes, which can be viewed as generalizations of Kitaev's one-dimensional (1D) model of unpaired Majorana fermions. Parafermion stabilizer codes can protect against low-weight errors acting on a small subset of parafermion modes in analogy to qudit stabilizer codes. Examples of several smallest parafermion stabilizer codes are given. A locality-preserving embedding of qudit operators into parafermion operators is established that allows one to map known qudit stabilizer codes to parafermion codes. We also present a local 2D parafermion construction that combines topological protection of Kitaev's toric code with additional protection relying on parity conservation.

Figure 1

A portion of the lattice place on a torus, where each dot represents four parafermion modes (the index $j\ge 0$ uniquely denotes the lattice point). On the right, parafermion star and plaquette operators $A_s$ and $B_p$ are given in detail [$p^l$ is a prime power; further details are given in Eq. (28)].

Figure 2

Loops corresponding to the logical operators of the toric code.