Properties of Modular Categoriesand their

Computation Consequences

Eric C. Rowell, Texas A&M U.UT Tyler, 21 Sept. 2007

A Few Collaborators

Z. Wang (Microsoft)

M. Larsen (Indiana)

S. Witherspoon (TAMU)

P. Etingof (MIT)

Y. Zhang (Utah, Physics)

Publications/Preprints

• [Franko,ER,Wang] JKTR 15, no. 4, 2006

• [Larsen,ER,Wang] IMRN 2005, no. 64

• [ER] Contemp. Math. 413, 2006

• [Larsen, ER] MP Camb. Phil. Soc.

• [ER] Math. Z 250, no. 4, 2005

• [Etingof,ER,Witherspoon] preprint

• [Zhang,ER,et al] preprint

Motivation

Top. Quantum Computer

Modular Categories

Top. States (anyons)

3-D TQFT(Turaev)

definition

(Kitaev)

(Freedman)

What is a Topological Phase?

[Das Sarma, Freedman, Nayak, Simon, Stern]

“…a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory…”

Working definition…

Topological States: FQHE

1011 electrons/cm2

10 Tesla

defects=quasi-particles

particle exchange

fusion

9 mK

Topological Computation

initialize create particles

apply operators braid

output measure

Computation Physics

MC Toy Model: Rep(G)

• Irreps: {V1=CC, V2,…,Vk}

• Sums VW, tens. prod. VW, duals W*

• Semisimple: each W=imiVi

• Rep: Sn EndG(V n)

Modular Categories

group G Rep(G) Modular Categorydeform

axioms

Sn action

(Schur-Weyl)

Bn action

(braiding)

9

Braid Group Bn “Quantum Sn”

Generated by: 1 i i+1 n

Multiplication is by concatenation:

=

bi =

Modular Category

• Simple objects {X0=CC,X1,…XM-1}

+ Rep(G) properties

• Rep. Bn End(Xn) (braid group action)

• Non-degeneracy: S-matrix invertible

Uses of Modular Categories

• Link, knot and 3-manifold invariants

• Representations of mapping class groups

• Study of (special) Hopf algebras

• “Symmetries” of topological states of matter. (analogy: 3D crystals and space groups)

Partial Dictionary

Simple objects Xi Indecomposable particle types

Bn-action Particle exchange

X0 =CC Vacuum state

Xi* Antiparticle

X0 Xi Xi* Creation

In Pictures

Simple objects Xi Quasi-particles

Braiding Particleexchange

Unit object X0 Vacuum

X0 Xi Xi* Create

Two Hopf Algebra Constructions

g Uqg Rep(Uqg) F(g,q,L)Lie algebra

quantumgroup

qL=-1

semisimplify

G DG Rep(DGG) finite group

twistedquantum double

Finite dimensional quasi-Hopf algebra

Other Constructions

• Direct Products of Modular Categories

• Doubles of Spherical Categories

• Minimal Models, RCFT, VOAs, affine Kac-Moody, Temperley-Lieb, and von Neumann algebras…

Groethendieck Semiring

• Assume self-dual: X=X*. For a MC DD:

Xi Xj = k Nijk Xk (fusion rules)

• Semiring Gr(DD):=(Ob(DD),,)

• Encoded in matrices (Ni)jk = Nijk

17

Generalized Ocneanu Rigidity

Theorem: (see [Etingof, Nikshych, Ostrik])

For fixed fusion rules { Nijk } there are finitely many inequivalent modular categories with these fusion rules.

• Simple Xi multigraph Gi :

Vertices labeled by 0,…,M-1

Graphs of Fusion Rules

Nijk edges

j k

Example: F(g2,q,10)

Rank 4 MC with fusion rules:

N111=N113=N123=N222=N233=N333=1;

N112=N122= N223=0

G1: 0 1 2 3

G2: 0 2 1 3

G3: 20 3

1

Tensor Decomposable, 2 copies of Fibbonaci!

More Graphs

D(S3)Lie type B2 q9=-1

Lie type B3 q12=-1

Extra colors for different objects…

Classify Modular Categories

Verified for:

M=1, 2 [Ostrik], 3 and 4 [ER, Stong, Wang]

Conjecture (Z. Wang 2003): The set { MCs of rank M } is finite.

Rank of an MC: # of simple objects

Analogy

Theorem (E. Landau 1903):

The set { G : |Rep(G)|=N } is finite.

Proof: Exercise (Hint: Use class equation)

Classification by Graphs

Theorem: (ER, Stong, Wang)

Indecomposable, self-dual MCs of rank<5 are determined and classified by:

Physical Feasibility

Realizable TQC Bn action Unitary

i.e. Unitary Modular Category

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Two Examples

Unitary, for some q Never Unitary, for any qLie type G2 q21=-1 “even part” for

Lie type B2 q9=-1

For quantum group categories, can be complicated…

General Problem

G discrete,

(G) U(N) unitary irrep.What is the closure of (G)? (modulo center)

• SU(N)• Finite group

• SO(N), E7, other compact groups…

Key example: i(Bn) U(Hom(Xn,Xi))

Braid Group Reps.

• Let X be any object in a unitary MC

• Bn acts on Hilbert spaces End(X n)

as unitary operators: a braid.

• The gate set: {bi)}, bi braid generators.

Computational Power

{Ui} universal if

{promotions of Ui} U(kn)

Topological Quantum Computer universal

i(Bn) dense in SU(Ni)

qubits: k=2

Dense Image Paradigm

UniversalTop. Quantum Computer

Class #P-hardLink invariant

(Bn) dense

Eg. FQHE at =12/5?

Property F

A modular category DD has property F

if the subgroup:

(Bn) GL(End(Vn))

is finite for all objects V in DD.

Example 1

Theorem:

F(sl2, q , L) has property F if and only if L=2,3,4 or 6.

(Jones ‘86, Freedman-Larsen-Wang ‘02)

Example 2

Theorem: [Etingof,ER,Witherspoon]

Rep(DG) has property F for any finite group G and 3-cocycle .

More generally, true for braided group-theoretical fusion categories.

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Finite Group Paradigm

Non-Universal Top. Quantum Computer

Modular Cat. with prop. F

Abelian anyons,FQHE at =5/2?

Poly-timeLink invariant

quantum errorcorrection?

Categorical Dimensions

For modular category DD define

dim(X) = TrDD(IdX) = RR

dim(DD)=i(dim(Xi))2

IdX

dim[End(Xn)] [dim(X)]n

Examples

• In Rep(DG) all dim(V)

• In F(sl2, q , L),

dim(Xi) =

For L=4 or 6, dim(Xi) [L/2],

for L=2 or 3, dim(Xi)

sin((i+1)/L)sin(/L)

Property F Conjecture

Conjecture: (ER)

Let DD be a modular category. Then DD has property F dim(CC).

Equivalent to: dim(Xi)2 for all simple Xi.

Observations

• Wang’s Conjecture is true for modular categories with dim(DD)

(Etingof,Nikshych,Ostrik)

• My Conjecture would imply Wang’s for modular categories with property F.

Current Problems

• Construct more modular categories (explicitly!)

• Prove Wang’s Conj. for more cases

• Explore Density Paradigm

• Explore Finite Image Paradigm

• Prove Property F Conjecture

Thanks!