On the robustness of topological quantum codes: Ising perturbation

Abstract

We study the phase transition from two different topological phases to the ferromagnetic phase by focusing on points of the phase transition. To this end, we present a detailed mapping from such models to the Ising model in a transverse field. Such a mapping is derived by re-writing the initial Hamiltonian in a new basis so that the final model in such a basis has a well-known approximated phase transition point. Specifically, we consider the toric codes and the color codes on some various lattices with Ising perturbation. Our results provide a useful table to compare the robustness of the topological codes and to explicitly show that the robustness of the topological codes depends on triangulation of their underlying lattices.

arXiv comment15 pages, 12 figures, 1 table, Accepted for publication in Physical Review A

References in corpus (29)

• Fault-tolerant quantum computation by anyons
• Good Quantum Error-Correcting Codes Exist
• Topological quantum memory
• Decoherence, the measurement problem, and interpretations of quantum mechanics
• Topological orders and Edge excitations in FQH states
• An RVB phase in the triangular lattice quantum dimer model
• Topological Quantum Distillation
• Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation
• Topological Order and Conformal Quantum Critical Points
• Superconductors are topologically ordered
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Low-energy effective theory of the toric code model in a parallel field
• Robustness of a perturbed topological phase
• Topological Computation without Braiding
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models
• Strong Resilience of Topological Codes to Depolarization
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Self-duality and bound states of the toric code model in a transverse field
• Adiabatic Preparation of Topological Order
• Phase diagram of the toric code model in a parallel magnetic field
• Breakdown of a perturbed Z_N topological phase
• The critical behaviour of the 2D Ising model in Transverse Field; a Density Matrix Renormalization calculation
• Tricolored Lattice Gauge Theory with Randomness: Fault-Tolerance in Topological Color Codes
• Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices
• Robustness of a topological phase: Topological color code in parallel magnetic field
• Probing topological order with Rényi entropy
• Local response of topological order to an external perturbation
• The Kitaev-Ising model, Transition between topological and ferromagnetic order
• A complete characterization of the spectrum of the Kitaev model on spin ladders

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• Global entanglement in a topological quantum phase transition
• Scalable characterization of localizable entanglement in noisy topological quantum codes
• A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems
• Quantum phase transition in the $Z_3$ Kitaev-Potts model
• Predicting topological quantum phase transition from dynamics via multisite entanglement
• Phase transition in a noisy Kitaev toric code model
• Topological line in frustrated Toric code models
• Classical criticality establishes quantum topological order
• A quantum phase transition from $Z_2\times Z_2$ to $Z_2$ topological order
• Foliated order parameter in a fracton phase transition
• Distinguishing Phases via Non-Markovian Dynamics of Entanglement in Topological Quantum Codes under Parallel Magnetic Field
• Noisy Toric code and random bond Ising model: The error threshold in a dual picture
• General scheme for preparation of different topological states on the cluster states
• Kosterlitz-Thouless phase and $Z_d$ topological quantum phase
• Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries
• Layer-by-layer disentangling two-dimensional topological quantum codes