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.

arXiv:1707.03884Jul 12, 20172 citations

Topological Quantum Computation with Gapped Boundaries

Iris Cong, Meng Cheng, Zhenghan Wang

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.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.

arXiv:1609.02037Sep 7, 201624 citations

Defects between gapped boundaries in two-dimensional topological phases of matter

Iris Cong, Meng Cheng, Zhenghan Wang

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 $(2+1)D$ 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 $\mathfrak{D}({S}_{3})$ .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 $(2+1)D$ 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 $\mathfrak{D}({S}_{3})$.

arXiv:1703.03564Physical review. B./Physical review. BNov 14, 201736 citations

Twisted quantum double model of topological phases in two dimensions

Yuting Hu, Yidun Wan, Yong-Shi Wu

We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group $G$ and a 3-cocycle $\ensuremath{\alpha}$ over $G$ . The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of $G$ . When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group $G$ , in which the elementary excitations are known to be classified by the quantum double $D(G)$ of $G$ . Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net $6j$ symbols to the corresponding 3-cocycles.We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group $G$ and a 3-cocycle $\ensuremath{\alpha}$ over $G$. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of $G$. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group $G$, in which the elementary excitations are known to be classified by the quantum double $D(G)$ of $G$. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net $6j$ symbols to the corresponding 3-cocycles.

arXiv:1211.3695Physical Review BMar 11, 2013213 citations

Twisted quantum double model of topological order with boundaries

Alex Bullivant, Yuting Hu, Yidun Wan

We generalize the twisted quantum double model of topological orders in two dimensions to the case with boundaries by systematically constructing the boundary Hamiltonians. Given the bulk Hamiltonian defined by a gauge group $G$ and a 3-cocycle in the third cohomology group of $G$ over $U(1)$ , a boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a 2-cochain in the second cochain group of $K$ over $U(1)$ . The consistency between the bulk and boundary Hamiltonians is dictated by what we call the Frobenius condition that constrains the 2-cochain given the 3-cocyle. We offer a closed-form formula computing the ground-state degeneracy of the model on a cylinder in terms of the input data only, which can be naturally generalized to surfaces with more boundaries. We also explicitly write down the ground-state wave function of the model on a disk also in terms of the input data only.We generalize the twisted quantum double model of topological orders in two dimensions to the case with boundaries by systematically constructing the boundary Hamiltonians. Given the bulk Hamiltonian defined by a gauge group $G$ and a 3-cocycle in the third cohomology group of $G$ over $U(1)$, a boundary Hamiltonian can be defined by a subgroup $K$ of $G$ and a 2-cochain in the second cochain group of $K$ over $U(1)$. The consistency between the bulk and boundary Hamiltonians is dictated by what we call the Frobenius condition that constrains the 2-cochain given the 3-cocyle. We offer a closed-form formula computing the ground-state degeneracy of the model on a cylinder in terms of the input data only, which can be naturally generalized to surfaces with more boundaries. We also explicitly write down the ground-state wave function of the model on a disk also in terms of the input data only.

arXiv:1706.03611Physical review. B./Physical review. BOct 20, 201735 citations

Drinfel’d double symmetry of the 4d Kitaev model

Hank Chen

Abstract Following the general theory of categorified quantum groups developed by the author previously, we construct the Drinfel’d double 2-bialgebra associated to a finite group N = G0. For N = ℤ2, we explicitly compute the braided 2-categories of 2-representations of certain version of this Drinfel’d double 2-bialgebra, and prove that they characterize precisely the 4d toric code and its spin-ℤ2 variant. This result relates the two descriptions (categorical vs. field theoretical) of 4d gapped topological phases in existing literature and displays an instance of higher Tannakian duality for braided 2-categories. In particular, we show that particular twists of the underlying Drinfel’d double 2-bialgebra is responsible for much of the higher-structural properties that arise in 4d topological orders.Abstract Following the general theory of categorified quantum groups developed by the author previously, we construct the Drinfel’d double 2-bialgebra associated to a finite group N = G0. For N = ℤ2, we explicitly compute the braided 2-categories of 2-representations of certain version of this Drinfel’d double 2-bialgebra, and prove that they characterize precisely the 4d toric code and its spin-ℤ2 variant. This result relates the two descriptions (categorical vs. field theoretical) of 4d gapped topological phases in existing literature and displays an instance of higher Tannakian duality for braided 2-categories. In particular, we show that particular twists of the underlying Drinfel’d double 2-bialgebra is responsible for much of the higher-structural properties that arise in 4d topological orders.

arXiv:2305.04729Journal of High Energy PhysicsSep 21, 20232 citations

\mathfrak{D}(\mathbb{Z}_3)

can be used to perform universal quantum computation.

(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

\mathfrak{D}({S}_{3})

We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group

G

and a 3-cocycle $\ensuremath{\alpha}$ over

G

. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double ${D}^{\ensuremath{\alpha}}(G)$ of

G

. When $\ensuremath{\alpha}$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group

G

, in which the elementary excitations are known to be classified by the quantum double

D(G)

G

. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net

6j

symbols to the corresponding 3-cocycles.

G

and a 3-cocycle in the third cohomology group of

G

over

U(1)

, a boundary Hamiltonian can be defined by a subgroup

K

G

and a 2-cochain in the second cochain group of

K

over

U(1)

. The consistency between the bulk and boundary Hamiltonians is dictated by what we call the Frobenius condition that constrains the 2-cochain given the 3-cocyle. We offer a closed-form formula computing the ground-state degeneracy of the model on a cylinder in terms of the input data only, which can be naturally generalized to surfaces with more boundaries. We also explicitly write down the ground-state wave function of the model on a disk also in terms of the input data only.