Topological graph states and quantum error correction codes

TL;DR

This paper establishes criteria for identifying topologically ordered graph states, enabling the construction of quantum error correction codes with macroscopic distance and logical qubits, advancing topological quantum computing.

Contribution

It derives necessary and sufficient conditions for topological order in graph states and constructs new codes with large distance and multiple logical qubits.

Findings

Criteria for TQO-1 topological order in graph states

Construction of 3D topological codes with local stabilizers

Identification of graph families suitable for topological quantum codes

Abstract

Deciding if a given family of quantum states is topologically ordered is an important but nontrivial problem in condensed matter physics and quantum information theory. We derive necessary and sufficient conditions for a family of graph states to be in TQO-1, which is a class of quantum error correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, we consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a 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 quantum error correction codes.