Topological graph states and quantum error correction 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.

21 pages, 7 figures;

References in corpus (8)

• Multi-party entanglement in graph states
• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Quantum Low-Density Parity-Check Codes
• Topological order in a 3D toric code at finite temperature
• Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
• Topological phases and topological entropy of two-dimensional systems with finite correlation length
• Graph state representation of the toric code

Cited by in corpus (7)

• Generation of complete graph states in a spin-$1/2$ Heisenberg chain with a globally optimized magnetic field
• Controlled remote implementation of operations via graph states
• Hardware-Efficient Preparation of Graph States on Near-Term Quantum Computers
• Entanglement in Quantum Systems Based on Directed Graphs
• The additivity of states uniquely determined by marginals
• Robustness of Bell Violation of Graph States to Qubit Loss
• Scaling of Computational Order Parameters in Rydberg Atom Graph States