A Class of Cyclic Quantum Codes

TL;DR

The paper introduces bipartite cyclic cluster (BCC) codes, a class of cyclic quantum codes constructed from bipartite cluster states that become CSS codes after a Hadamard on one sublattice. It connects this framework to rotated toric codes, shows how to realize additional small-distance codes via computer search, and analyzes stabilizer structure and automorphisms that enable logical Hadamard and Hadamard-SWAP operations. The authors provide fault-tolerant preparation strategies, including ancilla-assisted checks, and demonstrate that these circuits can maintain low logical error rates up to distance 5 with favorable scaling. The work offers a circuit-centric approach to code design with potential hardware advantages for neutral-atom systems, balancing preparation simplicity with nontrivial distance and logical-operation capabilities.

Abstract

We introduce a class of cyclic quantum codes, basing the construction not on the simplicity of the stabilizers, but rather on the simplicity of preparation of a code state (at least in the absence of noise). We show how certain known codes, such as a certain family of rotated two-dimensional toric codes, fall into this class, and we also give certain other examples at small sizes found by computer search. We finally discuss fault tolerant preparation of these codes.