Kitaev's Z_d-Codes Threshold Estimates

Kitaev’s topological code (KTC) [3] on qubits is the archetypical topological code and has been extensively studied. As explained in Kitaev’s original paper [3], this construction applies to any group. Much less is known about these generalizations, and in this paper we investigate the quantum error correction (QEC) thresholds of the KTCs built with the groups Zd, where d ≥ 2. We label these as Zd-KTC, so the original code on qubits corresponds to Z2-KTC. As explained in [4], Z2-KTC can be decoded by a binary perfect matching algorithm [5], since every particle is its own anti-particle in this model. Because this is not the case for d > 2, other techniques are required and for this purpose we generalize the renormalization group (RG) soft decoder that we introduced in [1, 2]. Our numerical simulations show that the threshold increases monotonically with d and appears to follow the general trend of the qudit hashing bound. This paper is organized as follows. First, we introduce a generalized Pauli group (see [6, 7] for more details), stabilizer codes, and Zd-Kitaev’s toric code. Next, we briefly review the decoding problem of these systems and show how the RG decoder applies in this case. Finally, we present the numerical results and close with a discussion.