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arXiv:1609.02037

Topological Quantum Computation with Gapped Boundaries

Iris Cong, Meng Cheng, Zhenghan Wang

Published Sep 7, 2016• 24 citations
arXivHTMLPDF

Abstract

This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protected operations and obtain quantum gates. To demonstrate a potential physical realization, we provide quantum circuits for surface codes that can perform all basic operations on gapped boundaries. Finally, we show how gapped boundaries of the abelian theory $\mathfrak{D}(\mathbb{Z}_3)$ can be used to perform universal quantum computation.

DOI
10.48550/arxiv.1609.02037

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This paper studies fault-tolerant quantum computation with gapped boundaries. We first introduce gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories using their Hamiltonian realizations. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also provide a commuting Hamiltonian to realize defects between boundaries in any quantum double model. Next, we present the algebraic/categorical structure of gapped boundaries and boundary defects, which will be used to describe topologically protected operations and obtain quantum gates. To demonstrate a potential physical realization, we provide quantum circuits for surface codes that can perform all basic operations on gapped boundaries. Finally, we show how gapped boundaries of the abelian theory D(Z3)\mathfrak{D}(\mathbb{Z}_3)D(Z3​) can be used to perform universal quantum computation.
Defects between gapped boundaries provide a possible physical realization of projective non-Abelian braid statistics. A notable example is the projective Majorana/parafermion braid statistics of boundary defects in fractional quantum Hall/topological insulator and superconductor heterostructures. In this paper, we develop general theories to analyze the topological properties and projective braiding of boundary defects of topological phases of matter in two spatial dimensions. We present commuting Hamiltonians to realize defects between gapped boundaries in any (2+1)D(2+1)D(2+1)D untwisted Dijkgraaf-Witten theory, and use these to describe their topological properties such as their quantum dimension. By modeling the algebraic structure of boundary defects through multifusion categories, we establish a bulk-edge correspondence between certain boundary defects and symmetry defects in the bulk. Even though it is not clear how to physically braid the defects, this correspondence elucidates the projective braid statistics for many classes of boundary defects, both amongst themselves and with bulk anyons. Specifically, three such classes of importance to condensed matter physics/topological quantum computation are studied in detail: (1) a boundary defect version of Majorana and parafermion zero modes, (2) a similar version of genons in bilayer theories, and (3) boundary defects in D(S3)\mathfrak{D}({S}_{3})D(S3​).
We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. We give explicit formulas for the monomial matrices and the ground state degeneracy of the Kitaev models that are Hamiltonian realizations of Dijkgraaf-Witten theories. Our results imply that braiding gapped boundaries alone cannot provide universal gate sets for topological quantum computing with gapped boundaries.
We study representations of the braid groups from braiding gapped boundaries of Dijkgraaf-Witten theories and their twisted generalizations, which are (twisted) quantum doubled topological orders in two spatial dimensions. We show that the resulting braid (pure braid) representations are all monomial with respect to some specific bases, hence all such representation images of the braid groups are finite groups. We give explicit formulas for the monomial matrices and the ground state degeneracy of the Kitaev models that are Hamiltonian realizations of Dijkgraaf-Witten theories. Our results imply that braiding gapped boundaries alone cannot provide universal gate sets for topological quantum computing with gapped boundaries.
The macroscopic theory of anyon condensation, rooted in the categorical structure of topological excitations, provides a complete classification of gapped boundaries in topologically ordered systems, where distinct boundaries correspond to the condensation of different Lagrangian algebras. However, an intrinsic and direct understanding of anyon condensation in lattice models, grounded in the framework of Lagrangian algebras, remains undeveloped. In this paper, we propose a systematic framework for constructing all gapped boundaries of Kitaev's quantum double models directly from the data of Lagrangian algebras. Central to our approach is the observation that bulk interactions in the quantum double models admit two complementary interpretations: the anyon-creating picture and the anyon-probing picture. Generalizing this insight to the boundary, we derive the consistency condition for boundary ribbon operators that respect the mathematical axiomatic structure of Lagrangian algebras. Solving these conditions yields explicit expressions for the local boundary interactions required to realize gapped boundaries. We also provide three families of solutions that cover a broad range of cases. Our construction provides a microscopic characterization of the bulk-to-boundary anyon condensation dynamics via the action of ribbon operators. Moreover, all these boundary terms are supported within a common effective Hilbert space, making further studies on pure boundary phase transitions natural and convenient. Given the broad applicability of anyon condensation theory, we believe that our approach can be generalized to planar topological codes, extended string-net models, or higher-dimensional topologically ordered systems.
This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the Z_{3} toric code [D(Z_{3})], is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of D(Z_{3}) can be realized in bilayer fractional quantum Hall 1/3 systems. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely Abelian topological phase.
This is a comprehensive review on fault-tolerant topological quantum computation with the surface codes. The basic concepts and useful tools underlying fault-tolerant quantum computation, such as universal quantum computation, stabilizer formalism, and measurement-based quantum computation, are also provided in a pedagogical way. Topological quantum computation by brading the defects on the surface code is explained in both circuit-based and measurement-based models in such a way that their relation is clear. The interdisciplinary connections between quantum error correction codes and subjects in other fields such as topological order in condensed matter physics and spin glass models in statistical physics are also discussed. This manuscript will be appeared in SpringerBriefs.