# Hopf Algebras in Kitaev's Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics

Conference Date:
Monday, July 31, 2017 (All day) to Friday, August 4, 2017 (All day)
Scientific Areas:
Condensed Matter
Mathematical Physics
Quantum Foundations
Quantum Gravity
Quantum Information

The Kitaev quantum double models are a family of topologically ordered spin models originally proposed to exploit the novel condensed matter phenomenology of topological phases for fault-tolerant quantum computation. Their physics is inherited from topological quantum field theories, while their underlying mathematical structure is based on a class of Hopf algebras. This structure is also seen across diverse fields of physics, and so allows connections to be made between the Kitaev models and topics as varied as quantum gauge theory and modified strong complementarity. This workshop will explore this shared mathematical structure and in so doing develop the connections between the fields of mathematical physics, quantum gravity, quantum information, condensed matter and quantum foundations.

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• Andreas Bauer, Free University of Berlin
• Liang Chang, Chern Institute of Mathematics
• Bianca Dittrich, Perimeter Institute
• Ross Duncan, University of Strathclyde
• Tobias Fritz, Max Planck Institute for Mathematics in the Sciences
• Davide Gaiotto, Perimeter Institute
• Cesar Galindo, Universidad de los Andes
• Robert Koenig, Technical University of Munich
• Catherine Meusburger, Friedrich-Alexander Universitaet
• Michael Mueger, Radboud University Nijmegen
• David Reutter, University of Oxford
• Eric Rowell, Texas A&M University
• Burak Sahinoglu, California Institute of Technology
• Joost Slingerland, National University of Ireland
• Pawel Sobocinski, University of Southampton
• Dominic Williamson, University of Vienna
• Derek Wise, Concordia University

• Andreas Bauer, Free University of Berlin
• Courtney Brell, Perimeter Institute
• Liang Chang, Chern Institute of Mathematics
• Aaron Conlon, Maynooth University
• Bianca Dittrich, Perimeter Institute
• Ross Duncan, University of Strathclyde
• Jason Erbele,  Victor Valley College
• Javier Espiro, Universidade de Sao Paolo
• Tobias Fritz, Max Planck Institute for Mathematics in the Sciences
• Davide Gaiotto, Perimeter Institute
• Cesar Galindo, Universidad de los Andes
• Stefano Gogioso, University of Oxford
• Daniel Gottesman, Perimeter Institute
• Theo Johnson-Freyd, Perimeter Institute
• Martti Karvonen, University of Edinburgh
• Robert Koenig, Technical University of Munich
• Tian Lan, Perimeter Institute
• Claire Levaillant, University of Copehagen
• Catherine Meusburger, Friedrich-Alexander Universitaet
• Michael Mueger, Radboud University Nijmegen
• Domenico Pellegrino, Maynooth University
• Prince Osei, Perimeter Institute
• Abdulmajid Osumanu, University of Waterloo
• David Reutter, University of Oxford
• Eric Rowell, Texas A&M University
• Burak Sahinoglu, California Institute of Technology
• Bernd Schroers, Heriot-Watt University
• Joost Slingerland, National University of Ireland
• Pawel Sobocinski, University of Southampton
• Chenjie Wang, Perimeter Institute
• Alex Weekes, Perimeter Institute
• Carolin Wille, Free University of Berlin
• Dominic Williamson, University of Vienna
• Derek Wise, Concordia University
• Jon Yard, Perimeter Institute
• Shuo Yang, Perimeter Institute
• Theodore Yoder, Massachusetts Institute of Technology
• Beni Yoshida, Perimeter Institute
• Lucy Zhang, University of Toronto

SCHEDULE SUBJECT TO CHANGES

Monday, July 31st, 2017

 Time Event Location 9:00 – 9:30am Registration Reception 9:30 – 9:35am Welcome and Opening Remarks Bob Room 9:35 – 10:35am Cesar Galindo, Universidad de los AndesSemisimple Hopf algebras and fusion categories Bob Room 10:35 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Andreas Bauer, Free University of BerlinThe Hopf C*-algebraic quantum double models - symmetries beyond group theory Bob Room 12:00 – 2:00pm Lunch Bistro – 1st Floor 2:00 – 3:00pm Michael Mueger, Radboud University NijmegenModular categories and the Witt group Bob Room 3:00 – 4:00pm Eric Rowell, Texas A&M UniversityTopological Quantum Computation Bob Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Davide Gaiotto, Perimeter InstituteGapped phases of matter vs. Topological field theories Bob Room

Tuesday, August 1st, 2017

 Time Event Location 9:30 – 10:30am Derek Wise, Concordia UniversityAn Introduction to Hopf Algebra Gauge Theory Bob Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Catherine Meusburger, Friedrich-Alexander UniversitaetKitaev lattice models as a Hopf algebra gauge theory Bob Room 12:00 - 12:10pm Conference Photo TBA 12:10 – 2:00pm Lunch Bistro – 1st Floor 2:00 – 3:00pm Free Discussion 1 Bob Room 3:00 – 4:00pm Jurgen Fuchs, Karlstad UniversityTopological defects and higher-categorical structures Bob Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Dominic Williamson, University of ViennaSymmetry-enriched topological order in tensor networks: Gauging and anyon condensation Bob Room

Wednesday, August 2nd, 2017

 Time Event Location 9:30 – 10:30am Liang Chang, Chern Institute of MathematicsKitaev models based on  unitary quantum groupoids Bob Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Ross Duncan, University of StrathclydeIntroduction to CQM Bob Room 12:00 – 2:00pm Lunch Bistro – 1st Floor 2:00 – 3:00pm Free Discussion 2 Bob Room 3:00 – 4:00pm Pawel Sobocinski, University of SouthamptonInteracting Hopf monoids and Graphical Linear Algebra Bob Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Poster Session Bob Room 5:30pm onwards Banquet Bistro – 2nd Floor

Thursday, August 3rd, 2017

 Time Event Location 9:30 – 10:30am Burak Sahinoglu, California Institute of TechnologyA tensor network framework for topological phases of quantum matter Bob Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Tobias Fritz, Max Planck Institute for Mathematics in the SciencesThe Kitaev model and aspects of semisimple Hopf algebras via the graphical calculus Bob Room 12:00 – 2:00pm Lunch Bistro – 1st Floor 2:00 – 3:00pm Free Discussion 3 Bob Room 3:00 – 4:00pm Robert Koenig, Technical University of MunichQuantum computation with Turaev-Viro codes Bob Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm David Reutter, University of OxfordFrobenius algebras, Hopf algebras and 3-categories Bob Room

Friday, August 4th, 2017

 Time Event Location 9:30 – 10:30am Joost Slingerland, National University of IrelandHopf algebras and parafermionic lattice models Bob Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00 – 12:00pm Bianca Dittrich, Perimeter InstituteFrom 3D TQFTs to 4D models with defects Bob Room 12:00 – 2:00pm Lunch Bistro – 1st Floor 2:00 – 4:00pm Chaired Discussion Bob Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor

Andreas Bauer, Free University of Berlin

The Hopf C*-algebraic quantum double models - symmetries beyond group theory

Liang Chang, Chern Institute of Mathematics

Kitaev models based on  unitary quantum groupoids

Kitaev originally constructed his quantum double model based on finite groups and anticipated the extension based on Hopf algebras, which was achieved later by Buerschaper, etc. In this talk, we will present the work on the generalization of Kitaev model for quantum groupoids and discuss its ground states.

Bianca Dittrich, Perimeter Institute

From 3D TQFTs to 4D models with defects

I will explain a general strategy to lift (2+1)D topological phases, in particular string nets, to (3+1)D models with line defects. This allows a systematic construction of (3+1)D topological theories with defects, including an improved version of the Walker-Wang Model. It has also an interesting application to quantum gravity as it leads to quantum geometry realizations for which all geometric operators have discrete and bounded spectra. I will furthermore comment on some interesting (self-) duality relations that emerge in these constructions.

Ross Duncan, University of Strathclyde

Introduction to CQM

Categorical quantum mechanics is a research programme which aims to axiomatise (finite dimensional) quantum theory as an algebraic theory inside an abstract symmetric monoidal category.  The central idea is that quantum observables can be axiomatised as certain Frobenius algebras, and that two observables are (strongly) complementary when their Frobenius algebras jointly form a Hopf algebra.  The resulting theory is surprisingly powerful, especially when combined with its graphical notation.  In this talk I'll introduce the main concepts and present some applications to quantum computation.

Tobias Fritz, Max Planck Institute for Mathematics in the Sciences

The Kitaev model and aspects of semisimple Hopf algebras via the graphical calculus

The quantum double models are parametrized by a finite-dimensional semisimple Hopf algebra (over $\mathbb{C}$). I will introduce the graphical calculus of these Hopf algebras and sketch how it is equivalent to the calculus of two interacting symmetric Frobenius algebras. Since symmetric Frobenius algebras are extended 2D TQFTs, this suggests that there is a canonical way to 'lift' a compatible pair of 2D TQFTs to a 3D TQFT. Time permitting, I will also showcase how to rederive graphically parts of the Larson-Sweedler theorem, giving various equivalent characterizations of semisimplicity, thereby generalizing these results to arbitrary Hopf monoids in traced symmetric monoidal categories.

Topological defects and higher-categorical structures

I will discuss some (higher-)categorical structures present in three-dimensional topological field theories that include topological defects of any codimension. The emphasis will be on two topics:
(1) For Reshetikhin-Turaev type theories, regarded as 3-2-1-extended TFTs, I will explain why codimension-1 boundaries and defects form bicategories of module categories over suitable fusion categories.
In the case of defects separating three-dimensional regions supporting the same theory, the relevant fusion category $A$ is the modular tensor category underlying that theory, while for defects separating two theories of Turaec-Viro type with underlying fusion categories $A_1$ and $A_2$, respectively, $A$ is the the Deligne product $A_1 \boxtimes A_2^{op}$.
(2) I will indicate the building blocks of a generalization of the TV-BW state-sum construction to theories with defects. Making use of ends and coends, various aspects of this construction can be formulated without requiring semi simplicity.

Davide Gaiotto, Perimeter Institute

Gapped phases of matter vs. Topological field theories

I will discuss the relation between topological field theories and gapped phases of matter. I will propose a general formalism to define a class of TFTs which can be realized by commuting projector Hamiltonians.  This allows one to apply rigorous mathematical theorems about TFTs to gapped phases of matter. I will also discuss the role of generalized cohomology theories and spectra in the classification of SPT phases.

Cesar Galindo, Universidad de los Andes

Semisimple Hopf algebras and fusion categories

This talk will be a short introduction to the semisimple Hopf algebras over an algebraically closed field of characteristic 0 and their representation theories. It is intended to outline the main basic results about structure and known methods for the construction of semisimple Hopf algebras: extensions, twisting, Tannakian reconstruction. Basic notions concerning tensor categories will be introduced: braided structures, center construction, fiber functors. Special emphasis is given to the notion of fusion category. At the same time, the relations between these notions and those of the Hopf algebras are studied.

Robert Koenig, Technical University of Munich

Quantum computation with Turaev-Viro codes

The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For a manifold with boundary, the tensor network has free indices that can be associated to qudits, and its contraction gives the coefficients of a quantum error-correcting code. The code has local stabilizers determined by Levin and Wen. By studying braid group representations acting on equivalence classes of colored ribbon graphs embedded in a punctured sphere, we identify the anyons, and give a simple recipe for mapping fusion basis states of the doubled category to ribbon graphs. Combined with known universality results for anyonic systems, this provides a large family of schemes for quantum computation based on local deformations of stabilizer codes. These schemes may serve as a starting point for developing fault-tolerance schemes using continuous stabilizer measurements and active error-correction.

This is joint work with Greg Kuperberg and Ben Reichardt.

Catherine Meusburger, Friedrich-Alexander Universitaet

Kitaev lattice models as a Hopf algebra gauge theory

We show that Kitaev's lattice model for a finite-dimensional semisimple Hopf algebra H is equivalent to the combinatorial quantisation of Chern-Simons theory for the Drinfeld double D(H). As a result, Kitaev models are a special case of combinatorial quantization of Chern-Simons theory by Alekseev, Grosse and Schomerus.  This equivalence is an analogue of the relation between Turaev-Viro and Reshetikhin-Turaev TQFTs and relates them to the quantisation of moduli spaces of flat connections.
We show that the topological invariants of the two models, the algebra of operators acting on the protected space of the Kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism, are isomorphic. This is established in a gauge theoretical picture, in which both models appear as Hopf algebra valued lattice gauge theories.

Modular categories and the Witt group

The aim of this talk is to give an introduction to modular categories, touching both basics and recent developments. I will begin with a quick reminder concerning tensor categories, in particular braided and symmetric ones, and notions like duality, fusion and spherical categories. We'll meet algebras in tensor categories, categories of modules, module categories and their connection. I will then focus on modular categories and some basic structure theory. We will consider two ways of obtaining modular categories: modularization and the Drinfeld center. (The important third one, quantum groups at root of unity, is too complicated to be discussed in any depth.) The Drinfeld center will be used to define Witt equivalence of modular categories and the Witt group. Several equivalent characterization of Witt equivalence, using module categories, will be discussed. The Witt group will be crucial for any (future) classification of modular categories, as well as for application to physics in condensed matter physics and conformal field theory.

David Reutter, University of Oxford

Frobenius algebras, Hopf algebras and 3-categories

It is well known that commutative Frobenius algebras can be represented as topological surfaces, using the graphical calculus of dualizable objects in monoidal 2-categories. We build on related ideas to show that the interacting Frobenius algebras of Duncan and Dunne, which have a Hopf algebra structure, arise naturally in a similar way, by requiring a single 3-morphism in a 3-category to be invertible. We show that this gives a purely geometrical proof of Mueger's version of Tannakian reconstruction of Hopf algebras from fusion categories equipped with a fibre functor. We also relate our results to the theory of lattice code surgery.

Eric Rowell, Texas A&M University

Topological Quantum Computation

The (Freedman-Kitaev) topological model for quantum computation is an inherently fault-tolerant computation scheme, storing information in topological (rather than local) degrees of freedom with quantum gates typically realized by braiding quasi-particles in two dimensional media. I will give an overview of this model, emphasizing the mathematical aspects.

Burak Sahinoglu, California Institute of Technology

A tensor network framework for topological phases of quantum matter

We present a general scheme for constructing topological lattice models in any space dimension using tensor networks. Our approach relies on finding "simplex tensors" that satisfy a finite set of tensor equations. Given any such tensor, we construct a discrete topological quantum field theory (TQFT) and local commuting projector Hamiltonians on any lattice. The ground space degeneracy of these models is a topological invariant that can be computed via the TQFT, and the ground states are locally indistinguishable when the ground space is nondegenerate on the sphere. Any ground state can be realized by a tensor network obtained by contracting simplex tensors. Our models are exact renormalization fixed points, covering a broad range of models in the literature. We identify symmetries on the virtual level of the tensor networks of our models that generalize the topological invariance properties beyond fixed point models. This framework combined with recent tensor network techniques is convenient for studying excitations, their statistics, phase transitions, and ultimately for classification of gapped phases of many-body theories in 3+1 and higher dimensions.

Joost Slingerland, National University of Ireland

Hopf algebras and parafermionic lattice models

Ground state degeneracy is an important characteristic of topological order. It is a natural question under what conditions such topological degeneracy extends to higher energy states or even to the full energy spectrum of a model, in such a way that the degeneracy is preserved when the Hamiltonian of the system is perturbed. It appears that Ising/Majorana wires have this property due to the presence of robust edge zero modes. Generalized wire models based on parafermions also have edge zero modes and topological degeneracy at special points in parameter space, but the  stability of these modes is a much more intricate question. These models are related to Hopf algebras or tensor categories in several ways. In particular they are "golden chain" type models based on fusion categories for boundary defects of Abelian TQFTs. As such they are part of a much larger class of Hopf algebra based chain models with edge modes. It is natural to ask which of these have stable edge zero modes and/or full spectrum degeneracy.

Pawel Sobocinski, University of Southampton

Interacting Hopf monoids and Graphical Linear Algebra

The interaction of Hopf monoids and Frobenius monoids is the productive nucleus of the ZX calculus, where famously each Frobenius monoid-comonoid pair corresponds to a complementary basis and the Hopf structure describes the interaction between the bases. The theory of Interacting Hopf monoids (IH), introduced by Bonchi, Sobocinski and Zanasi, features essentially the same Hopf-Frobenius interaction pattern. The free symmetric monoidal category generated by IH is isomorphic to the category of linear relations over the field of rationals: thus the string diagrams of IH are an alternative graphical language for elementary concepts of linear algebra. IH has a modular construction via distributive laws of props, and has been applied as a compositional language of signal flow graphs. In this talk I will outline the equational theory, its construction and applications, as well as report on ongoing and future work.

Dominic Williamson, University of Vienna

Symmetry-enriched topological order in tensor networks: Gauging and anyon condensation

I will describe a framework for the study of symmetry-enriched topological order using graded matrix product operator algebras. The approach is based upon an explicit construction of the extrinsic symmetry defects, which facilitates the extraction of their physical properties. This allows for a simple analysis of dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory.

Derek Wise, Concordia University

An Introduction to Hopf Algebra Gauge Theory

A variety of models, especially Kitev models, quantum Chern-Simons theory, and models from 3d quantum gravity, hint at a kind of lattice gauge theory in which the gauge group is generalized to a Hopf algebra.  However, until recently, no general notion of Hopf algebra gauge theory was available.  In this self-contained introduction, I will cover background on lattice gauge theory and Hopf algebras, and explain our recent construction of Hopf algebra gauge theory on a ribbon graph (arXiv:1512.03966).  The resulting theory parallels ordinary lattice gauge theory, generalizing its structure only as necessary to accommodate more general Hopf algebras.  All of the key features of gauge theory, including gauge transformations, connections, holonomy and curvature, and observables, have Hopf algebra analogues, but with a richer structure arising from non-cocommuntativity, the key property distinguishing Hopf algebras from groups. Main results include topological invariance of algebras of observables, and a gauge theoretic derivation of algebras previously obtained in the combinatorial quantization of Chern-Simons theory.

## From 3D TQFTs to 4D models with defects

Friday Aug 04, 2017
Speaker(s):

I will explain a general strategy to lift (2+1)D topological phases, in particular string nets, to (3+1)D models with line defects. This allows a systematic construction of (3+1)D topological theories with defects, including an improved version of the Walker-Wang Model. It has also an interesting application to quantum gravity as it leads to quantum geometry realizations for which all geometric operators have discrete and bounded spectra. I will furthermore comment on some interesting (self-) duality relations that emerge in these constructions.

Scientific Areas:

## Hopf algebras and parafermionic lattice models

Friday Aug 04, 2017
Speaker(s):

Ground state degeneracy is an important characteristic of topological order. It is a natural question under what conditions such topological degeneracy extends to higher energy states or even to the full energy spectrum of a model, in such a way that the degeneracy is preserved when the Hamiltonian of the system is perturbed. It appears that Ising/Majorana wires have this property due to the presence of robust edge zero modes.

Scientific Areas:

## Frobenius algebras, Hopf algebras and 3-categories

Thursday Aug 03, 2017
Speaker(s):

It is well known that commutative Frobenius algebras can be represented as topological surfaces, using the graphical calculus of dualizable objects in monoidal 2-categories. We build on related ideas to show that the interacting Frobenius algebras of Duncan and Dunne, which have a Hopf algebra structure, arise naturally in a similar way, by requiring a single 3-morphism in a 3-category to be invertible. We show that this gives a purely geometrical proof of Mueger's version of Tannakian reconstruction of Hopf algebras from fusion categories equipped with a fibre functor.

Scientific Areas:

## Quantum computation with Turaev-Viro codes

Thursday Aug 03, 2017
Speaker(s):

The Turaev-Viro invariant for a closed 3-manifold is defined as the contraction of a certain tensor network. The tensors correspond to tetrahedra in a triangulation of the manifold, with values determined by a fixed spherical category. For a manifold with boundary, the tensor network has free indices that can be associated to qudits, and its contraction gives the coefficients of a quantum error-correcting code. The code has local stabilizers determined by Levin and Wen.

Scientific Areas:

## The Kitaev model and aspects of semisimple Hopf algebras via the graphical calculus

Thursday Aug 03, 2017
Speaker(s):

The quantum double models are parametrized by a finite-dimensional semisimple Hopf algebra (over $\mathbb{C}$). I will introduce the graphical calculus of these Hopf algebras and sketch how it is equivalent to the calculus of two interacting symmetric Frobenius algebras. Since symmetric Frobenius algebras are extended 2D TQFTs, this suggests that there is a canonical way to 'lift' a compatible pair of 2D TQFTs to a 3D TQFT.

Scientific Areas:

## A tensor network framework for topological phases of quantum matter

Thursday Aug 03, 2017
Speaker(s):

We present a general scheme for constructing topological lattice models in any space dimension using tensor networks. Our approach relies on finding "simplex tensors" that satisfy a finite set of tensor equations. Given any such tensor, we construct a discrete topological quantum field theory (TQFT) and local commuting projector Hamiltonians on any lattice. The ground space degeneracy of these models is a topological invariant that can be computed via the TQFT, and the ground states are locally indistinguishable when the ground space is nondegenerate on the sphere.

Scientific Areas:

## Interacting Hopf monoids and Graphical Linear Algebra

Wednesday Aug 02, 2017
Speaker(s):

The interaction of Hopf monoids and Frobenius monoids is the productive nucleus of the ZX calculus, where famously each Frobenius monoid-comonoid pair corresponds to a complementary basis and the Hopf structure describes the interaction between the bases. The theory of Interacting Hopf monoids (IH), introduced by Bonchi, Sobocinski and Zanasi, features essentially the same Hopf-Frobenius interaction pattern.

Scientific Areas:

## Introduction to CQM

Wednesday Aug 02, 2017
Speaker(s):

Categorical quantum mechanics is a research programme which aims to axiomatise (finite dimensional) quantum theory as an algebraic theory inside an abstract symmetric monoidal category. The central idea is that quantum observables can be
axiomatised as certain Frobenius algebras, and that two observables are (strongly) complementary when their Frobenius algebras jointly form a Hopf algebra. The resulting theory is surprisingly powerful, especially when combined with its graphical notation. In this talk

Scientific Areas:

## Kitaev models based on unitary quantum groupoids

Wednesday Aug 02, 2017
Speaker(s):

Kitaev originally constructed his quantum double model based on finite groups and anticipated the extension based on Hopf algebras, which was achieved later by Buerschaper, etc. In this talk, we will present the work on the generalization of Kitaev model for quantum groupoids and discuss its ground states.

Scientific Areas:

## Symmetry-enriched topological order in tensor networks: Gauging and anyon condensation

Tuesday Aug 01, 2017
Speaker(s):

I will describe a framework for the study of symmetry-enriched topological order using graded matrix product operator algebras. The approach is based upon an explicit construction of the extrinsic symmetry defects, which facilitates the extraction of their physical properties. This allows for a simple analysis of dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory.

Scientific Areas:

## Pages

Scientific Organizers:

• Courtney Brell, Perimeter Institute
• Ross Duncan, University of Strathclyde
• Daniel Gottesman, Perimeter Institute
• Prince Osei, Perimeter Institute
• Lucy Zhang, University of Toronto