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arXiv:1908.02829 · Quantum

Kitaev's quantum double model as an error correcting code

Shawn Xingshan Cui, Dawei Ding, Xizhi Han, Geoff Penington, Daniel Ranard, Brandon C. Rayhaun, Zhou Shangnan

Published Sep 24, 2020• 63 citations
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Abstract

Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.

DOI
10.22331/q-2020-09-24-331

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Kitaev's quantum double models in 2D provide some of the most commonly studied examples of topological quantum order. In particular, the ground space is thought to yield a quantum error-correcting code. We offer an explicit proof that this is the case for arbitrary finite groups. Actually a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region. Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. We contrast this result with the fact that local properties of gauge-invariant states are not generally determined by specifying all of their non-Abelian fluxes --- that is, the Wilson loops of lattice gauge theory do not form a complete commuting set of observables. We also note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of the error correcting property of Kitaev's model. Finally, we compute the topological entanglement entropy in Kitaev's model, and show, contrary to previous claims in the literature, that it does not depend on whether the ``log dim R'' term is included in the definition of entanglement entropy.
This thesis presents a detailed analysis of the quantum double model and its twisted version, focusing on their properties as quantum error-correcting codes. In the case of Kitaev's quantum double model for finite Abelian groups, the error correction process is explicitly described. The number of correctable errors depends on the lattice and the topology of the underlying surface. Although there exists a theoretical maximum number of errors that can be corrected, it is proven that correcting this number of errors is, in general, an NP-complete problem. As an alternative, a polynomial-time correction algorithm is proposed that corrects a number of errors below the theoretical maximum. In regard to the twisted quantum double model, it is studied within the framework of Local Topological Order (LTO). Originally formulated for square lattices, the definition of LTO is extended here to arbitrary two-dimensional lattices, enabling an explicit characterization of the ground states space through invariant spaces of monomial representations. By reformulating the LTO conditions to include such general lattices, it is proven that the twisted model satisfies all four axioms of LTO on any 2D lattice. As a result, the ground states space of the model is shown to define a quantum error-correcting code, highlighting the role of topological order in fault-tolerant quantum computation.
We study the Twisted Kitaev Quantum Double model within the framework of Local Topological Order (LTO). We extend its definition to arbitrary 2D lattices, enabling an explicit characterization of the ground state space through the invariant spaces of monomial representations. We reformulate the LTO conditions to include general lattices and prove that the twisted model satisfies all four LTO axioms on any 2D lattice. As a corollary, we show that its ground state space is a quantum error-correcting code.
Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
Abstract The Kitaev honeycomb model is an approximate topological quantum error correcting code in the same phase as the toric code, but requiring only a 2-body Hamiltonian. As a frustrated spin model, it is well outside the commuting models of topological quantum codes that are typically studied, but its exact solubility makes it more amenable to analysis of effects arising in this noncommutative setting than a generic topologically ordered Hamiltonian. Here we study quantum error correction in the honeycomb model using both analytic and numerical techniques. We first prove explicit exponential bounds on the approximate degeneracy, local indistinguishability, and correctability of the code space. These bounds are tighter than can be achieved using known general properties of topological phases. Our proofs are specialized to the honeycomb model, but some of the methods may nonetheless be of broader interest. Following this, we numerically study noise caused by thermalization processes in the perturbative regime close to the toric code renormalization group fixed point. The appearance of non-topological excitations in this setting has no significant effect on the error correction properties of the honeycomb model in the regimes we study. Although the behavior of this model is found to be qualitatively similar to that of the standard toric code in most regimes, we find numerical evidence of an interesting effect in the low-temperature, finite-size regime where a preferred lattice direction emerges and anyon diffusion is geometrically constrained. We expect this effect to yield an improvement in the scaling of the lifetime with system size as compared to the standard toric code.
A rigorous analysis is presented for the entanglement spectrum of quantum many-body states possessing a higher-form group-representation symmetry generated by topological Wilson loops, which is generally non-invertible. A general framework based on elementary algebraic topology and category theory is developed to determine the block structure of reduced density matrices for arbitrary bipartite manifolds on which the states are defined. Within this framework, we scrutinize the impact of topology on the entanglement structure for low-dimensional manifolds, including especially the torus, the Klein bottle, and lens spaces. By further incorporating gauge invariance, we refine our framework to determine the entanglement structure for topological gauge theories in arbitrary dimensions. In particular, in two dimensions, it is shown for the Kitaev quantum double model that not only the topological entanglement entropy can be reproduced, but also the Li-Haldane conjecture concerning the full entanglement spectrum holds exactly.