Decoherence and wavefunction deformation of $D_4$ non-Abelian topological order

Abstract

The effect of decoherence on topological order (TO) has been most deeply understood for the toric code, the paragon of Abelian TOs. We show that certain non-Abelian TOs can be analyzed and understood to a similar degree, despite being significantly richer. We consider both wavefunction deformations and quantum channels acting on $D_4$ TO, which has recently been realized on a quantum processor. By identifying the corresponding local statistical mechanical spin or rotor model with $D_4$ symmetry, we find a remarkable stability against proliferating non-Abelian anyons. This is shown by leveraging a reformulation in terms of the tractable O$(2)$ loop model in the pure state case, and $n$ coupled O$(2)$ loop models for Rényi-$n$ quantities in the decoherence case -- corresponding to worldlines of the proliferating anyon with quantum dimension $2$. In particular, we find that the purity ($n=2$) remains deep in the $D_4$ TO for any decoherence strength, while the $n\to \infty$ limit becomes critical upon maximally decohering a particular anyon type, similar to our wavefunction deformation result. The information-theoretic threshold ($n\to 1$) appears to be controlled by a disordered version of these stat-mech models, akin to the toric code case although significantly more robust. We furthermore use Monte Carlo simulations to explore the phase diagrams when multiple anyon types proliferate at the same time, leading to a continued stability of the $D_4$ TO in addition to critical phases with emergent $U(1)$ symmetry. Instead of loop models, these are now described by net models corresponding to different anyon types coupled together according to fusion rules.This opens up the exploration of statistical mechanical models for decohered non-Abelian TO, which can inform optimal decoders, and which in an ungauged formulation examples of non-Abelian strong-to-weak symmetry breaking.

27 + 19 pages. Close to published version

References in corpus (71)

• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Topological quantum memory
• String-net condensation: A physical mechanism for topological phases
• Quantum Error Correction for Quantum Memories
• Preparation of Entangled States by Quantum Markov Processes
• Criticality, the area law, and the computational power of PEPS
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Non-stationary coherent quantum many-body dynamics through dissipation
• Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Quantum memories at finite temperature
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Two-dimensional periodic frustrated Ising models in a transverse field
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• Anyon condensation and its applications
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• Topological Order at Non-zero Temperature
• Quantum criticality under decoherence or weak measurement
• Topological phases with generalized global symmetries
• Gauging quantum states: from global to local symmetries in many-body systems
• Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Towards Universal Topological Quantum Computation in the $\nu =5/2$ Fractional Quantum Hall State
• Average Symmetry-Protected Topological Phases
• Nishimori point in the 2D +/- J random-bond Ising model
• Nishimori's cat: stable long-range entanglement from finite-depth unitaries and weak measurements
• Random-bond Ising model in two dimensions, the Nishimori line, and supersymmetry
• Diagnostics of mixed-state topological order and breakdown of quantum memory
• Stable Quantum-Correlated Many Body States through Engineered Dissipation
• Detecting Topological Order at Finite Temperature Using Entanglement Negativity
• Shortest Route to Non-Abelian Topological Order on a Quantum Processor
• Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States
• Intrinsic Mixed-state Topological Order
• Gapless Coulomb state emerging from a self-dual topological tensor-network state
• Local topological order inhibits thermal stability in 2D
• Mixed-state long-range order and criticality from measurement and feedback
• Symmetry protected topological phases under decoherence
• Classification of phases for mixed states via fast dissipative evolution
• Symmetry Fractionalization in Two Dimensional Topological Phases
• A Noisy Approach to Intrinsically Mixed-State Topological Order
• Topological order from quantum loops and nets
• Three-dimensional color code thresholds via statistical-mechanical mapping
• Towards a classification of mixed-state topological orders in two dimensions
• Decoding non-Abelian topological quantum memories
• Separability transitions in topological states induced by local decoherence
• Hilbert Space Fragmentation in Open Quantum Systems
• Multicritical point of Ising spin glasses on triangular and honeycomb lattices
• Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids
• Topological Phases with Average Symmetries: the Decohered, the Disordered, and the Intrinsic
• Subsystem symmetry enriched topological order in three dimensions
• Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
• Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Fault-Tolerant Quantum Error Correction for non-Abelian Anyons
• Symmetry-enforced many-body separability transitions
• Mixed-state quantum anomaly and multipartite entanglement
• Condensation-Driven Phase Transitions in Perturbed String Nets
• Mapping between Morita equivalent string-net states with a constant depth quantum circuit
• Tensor-network approach to phase transitions in string-net models
• Classical Simulation of Quantum Error Correction in a Fibonacci Anyon Code
• Intersecting Loop Models on Z^D: Rigorous Results
• Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states
• Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
• Tapestry of dualities in decohered quantum error correction codes
• System-environment entanglement phase transitions
• Active error correction for Abelian and non-Abelian anyons
• Worldsheet matter for electric flux strings
• Conformal Field Theories generated by Chern Insulators under Quantum Decoherence
• Highly entangled stationary states from strong symmetries
• Symmetry-induced decoherence-free subspaces
• Quasi-long-range order in the 2D XY model with random phase shifts
• Phase diagram and the strong-coupling fixed point in the disordered O(n) loop model

Cited by in corpus (7)

• Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions
• Stabilizer Rényi Entropy and Conformal Field Theory
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Viewing protected superconducting qubits through the lens of the cat qubit
• Intrinsic Heralding and Optimal Decoders for Non-Abelian Topological Order
• Phases of Floquet code under local decoherence
• Universal Decay of Mutual Information and Conditional Mutual Information in Gapped Pure- and Mixed-State Quantum Matter