Properties of Modular Categories and their Computation Consequences Eric C. Rowell, Texas A&M U. UT Tyler, 21 Sept. 2007.
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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
25
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.
33
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!
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