Topological graph states and quantum error correction codes

Identifying and characterizing topological order is of widespread importance to both condensed matter physics and quantum information theory. In this work, we focus on the graph-state representation of stabilizer topological quantum error correction codes (QECCs). We derive a set of necessary and sufficient conditions for a family of graph states to be in TQO-1, a class of QECC states whose code distance (i.e. the number of protected single-qubit errors) scales macroscopically with the number of physical qubits. Using these criteria, we consider a number of specific graph families and discuss which are topologically ordered and how to construct the codewords. This formalism is then employed to construct several QECCs with macroscopic distance, including a new three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. The results indicate that graph states provide a fruitful approach to the construction and characterization of topological stabilizer QECCs.