Phase transition in a noisy Kitaev toric code model

Abstract

It is well-known that the partition function of a classical spin model can be mapped to a quantum entangled state where some properties on one side can be used to find new properties on the other side. However, the consequences of the existence of a classical (critical) phase transition on the corresponding quantum state has been mostly ignored. This is particularly interesting since the classical partition function exhibits non-analytic behavior at the critical point and such behavior could have important consequences on the quantum side. In this paper, we consider this problem for an important example of Kitaev toric code model which has been shown to correspond to the two-dimensional (2D) Ising model though a duality transformation. Through such duality transformation, it is shown that the temperature on the classical side is mapped to bit-flip noise on the quantum side. It is then shown that a transition from a coherent superposition of a given quantum state to a non-coherent mixture corresponds exactly to paramagnetic-ferromagnetic phase transition in the Ising model. To identify such a transition further, we define an order parameter to characterize the decoherency of such a mixture and show that it behaves similar to the order parameter (magnetization) of 2D Ising model, a behavior that is interpreted as a robust coherency in the toric code model. Furthermore, we consider other properties of the noisy toric code model exactly at the critical point. We show that there is a relative stability to noise for the toric code state at the critical noise which is revealed by a relative reduction in susceptibility to noise. We close the paper with a discussion on connection between the robust coherency as well as the critical stability with topological order of the toric code model.

10 page, 7 figure, submitted to Physical Review A

References in corpus (41)

• Fault-tolerant quantum computation by anyons
• Topological quantum memory
• Measurement-based quantum computation
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Topological orders and Edge excitations in FQH states
• Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
• The role of quantum information in thermodynamics --- a topical review
• Criticality, the area law, and the computational power of PEPS
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• Quantum memories at finite temperature
• Robustness of a perturbed topological phase
• Fractal symmetries: Ungauging the cubic code
• Error Threshold for Color Codes and Random 3-Body Ising Models
• Simulating Ising Spin Glasses on a Quantum Computer
• Simple universal models capture all classical spin physics
• Wave Function and Strange Correlator of Short Range Entangled states
• Universal Quantum Hamiltonians
• A Quantum Approach to Classical Statistical Mechanics
• Mapping topological to conformal field theories through strange correlators
• On measurement-based quantum computation with the toric code states
• Bringing order through disorder: Localization of errors in topological quantum memories
• Completeness of the classical 2D Ising model and universal quantum computation
• Classical spin models and the quantum stabilizer formalism
• Multi-partite entanglement and quantum phase transition in the one-, two-, and three-dimensional transverse field Ising model
• Statistical Mechanical Models and Topological Color Codes
• Quantum algorithms for spin models and simulable gate sets for quantum computation
• On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
• Unifying all classical spin models in a Lattice Gauge Theory
• Quantum algorithms for classical lattice models
• On the robustness of topological quantum codes: Ising perturbation
• Twins Percolation for Qubit Losses in Topological Color Codes
• Classical Ising model test for quantum circuits
• Dual correspondence between classical spin models and quantum CSS states
• The U(1) Lattice Gauge Theory Universally Connects All Classical Models with Continuous Variables, Including Background Gravity
• An algorithmic proof for the completeness of two-dimensional Ising model
• A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems
• Completeness of classical ${\varphi }^4$ theory on 2D lattices
• A quantum information approach to statistical mechanics
• Quantum simulation of classical thermal states
• Systematic study of the completeness of two-dimensional classical ${\varphi }^4$ theory

Cited by in corpus (7)

• Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field
• Global entanglement in a topological quantum phase transition
• Classical criticality establishes quantum topological order
• Analytical percolation theory for topological color codes under qubit loss
• Noisy Toric code and random bond Ising model: The error threshold in a dual picture
• Conditional global entanglement in a Kosterlitz-Thouless quantum phase transition
• Kosterlitz-Thouless phase and $Z_d$ topological quantum phase