Homological Stabilizer Codes

Abstract

In this paper we define homological stabilizer codes which encompass codes such as Kitaev's toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and show that under a small set of constraints the only homological stabilizer codes without local logical operators are equivalent to Kitaev's toric code or to the topological color codes.

Some statements clarified and typos fixed

References in corpus (16)

• Fault-tolerant quantum computation by anyons
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Fault-tolerant quantum computation with high threshold in two dimensions
• Topological Quantum Distillation
• Topological fault-tolerance in cluster state quantum computation
• A fault-tolerant one-way quantum computer
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Quantum codes on a lattice with boundary
• Stabilizer Formalism for Operator Quantum Error Correction
• Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory
• Stabilizing Quantum Information
• Tradeoffs for reliable quantum information storage in 2D systems
• Error Threshold for Color Codes and Random 3-Body Ising Models
• Clifford Gates by Code Deformation
• Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes