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Quantum error correction and local topological order from Quantum Double model

Diego Arturo Romero Fonseca

Published May 28, 2025• 0 citations
OpenAlex

Abstract

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.

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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.
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.
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.
In this paper, we present a detailed mathematical description of the error correction process for Kitaev's model for finite Abelian groups. The number of errors Kitaev's model can correct depends on the lattice and its topology. Although there is a theoretical maximum number of errors that can be corrected, we prove that correcting this number of errors, in general, is an NP-complete problem. Consequently, we introduce a polynomial-time correction algorithm that corrects a number of errors below the theoretical maximum.
It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by commuting non-Pauli stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.
This thesis addresses whether it is possible to build a robust memory device for quantum information. Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction. In this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square. This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.