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.

arXiv comment21 pages, 7 figures;

References in corpus (36)

• Topological insulators and superconductors
• Fault-tolerant quantum computation by anyons
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• A generic new platform for topological quantum computation using semiconductor heterostructures
• Topological quantum memory
• Non-Abelian statistics and topological quantum information processing in 1D wire networks
• Multi-party entanglement in graph states
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Stabilizer Codes and Quantum Error Correction
• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum error-correcting codes associated with graphs
• Sparse Graph Codes for Quantum Error-Correction
• Interacting topological insulators: a review
• Topological quantum order: stability under local perturbations
• Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure
• Graphical description of the action of local Clifford transformations on graph states
• Quantum Low-Density Parity-Check Codes
• Topological order in an exactly solvable 3D spin model
• Tradeoffs for reliable quantum information storage in 2D systems
• Topological order in a 3D toric code at finite temperature
• Fault-Tolerance of "Bad" Quantum Low-Density Parity Check Codes
• Spurious topological entanglement entropy from subsystem symmetries
• Topological order in 1D Cluster state protected by symmetry
• Characterization of topological states via dual multipartite entanglement
• Irreducible multi-particle correlations in states without maximal rank
• Braiding of Atomic Majorana Fermions in Wire Networks and Implementation of the Deutsch-Josza Algorithm
• Foliated fracton order in the checkerboard model
• Sorting topological stabilizer models in three dimensions
• Topological phases and topological entropy of two-dimensional systems with finite correlation length
• The Quantum PCP Conjecture
• Graph state representation of the toric code
• Irreducible many-body correlations in topologically ordered systems
• A Construction of Quantum LDPC Codes from Cayley Graphs

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