Algebraic Aspects of Topological Quantum Computing

Eric RowellIndiana University

Collaborators

Z. Wang (IU and MS Research*)

M. Larsen (IU)

R. Stong (Rice)

* “Project Q” with M. Freedman, A. Kitaev, K. Walker and C. Nayak

What is a Quantum Computer?

Any system for computation based on

quantum mechanical phenomena

Create

Manipulate

Measure

Quantum Systems

Classical vs. Quantum

• Bits {0,1}

• Logical Operationson {0,1}n

• Deterministic: output unique

• Qubits: V=CC22

(superposition)

• Unitary Operationson V n

• Probabilistic: output varies

(Uncertainty principle)

X

1

0

a1

0

Anyons: 2D Electron Gas

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

Algebraic Characterization

Anyonic System Top. Quantum Computer

Modular Categories

Toy Model: Rep(G)

• Irreps: {V1=CC, V2,…,Vk}

• Sum V W, product V W, duals W*

• Semisimple: every W= miVi

• Rep: Sn EndG(V n)

X

X

+

+

Braid Group Bn “Quantum Sn”

Generated by: 1 i i+1 n

Multiplication is by concatenation:

=

bi =i=1,…,n-1

Concept: Modular Category

group G Rep(G) Modular Categorydeform

Sn action Bn action

Axiomatic definition due to Turaev

Modular Category

• Simple objects {X0=CC,X1,…XM-1}

+ Rep(G) properties

• Rep. Bn End(X n) (braid group action)

• Non-degeneracy: S-matrix invertible

X

Dictionary: MCs vs. TQCs

Simple objects Xi Elementary particle types

Bn-action Operations (unitary)

X0 =CC Vacuum state

XX00 Xi Xi* CreationX

Constructions of MCs

Survey: (E.R. Contemp. Math.) (to appear)

g Uqg Rep(Uqg) FF

Lie algebra

quantumgroup

|q|=1

semisimplify

G D(G) Rep(D(GG))

Also,

finite group

quantum double

13

Physical Feasibility

Realizable TQC Bn action Unitary

Uqg Unitarity results: (Wenzl 98), (Xu 98) & (E.R. 05)

Computational Power

Physically realizable {Ui} universal if all

{Ui} = { all unitaries }

TQC universal F(Bn) dense in PkSU(k)

Results: in (Freedman, Larsen, Wang 02) and (Larsen, E.R., Wang 05)

Physical Hurdle: Realizable as Anyonic Systems?

Classify MCs

Recall: distinct particle types Simple objects in MC1-1

Classified for:

M=1, 2 (V. Ostrik), 3 and 4 (E.R., Stong, Wang)

Conjecture (Z. Wang 03): The set { MCs of rank M } is finite.

True for finite groups! (Landau 1903)

Groethendieck Semiring

• Assume X=X*. For a MC DD:

Xi Xj = Nijk Xk

• Semiring Gr(DD):=(Ob(DD), , )

• Encoded in matrices (Ni)jk := Nijk

X +

X+

Modular Group

• Non-dengeneracy S symmetric

• Compatibility T diagonal

•

give a unitary projective rep. of SL(2,ZZ)

1 1

10

0 -1

01T, S

Our Approach

• Study Gr(DD) and reps. of SL(2,Z)

• Ocneanu Rigidity:

MCs {Ni}

• Verlinde Formula:

{Ni} determined by S-matrix

Finite-to-one

Some Number Theory

• Let pi(x) = det(Ni - xI)

and K = Split({pi},QQ).

• Study Gal(K/QQ): always abelian!

• Nijk integers, Sij algebraic, constraints polynomials.

Sketch of Proof (M<5)

1. Show: 1 Gal(K/QQ) SM

2. Use Gal(K/QQ) + constraints to determine (S, {Ni})

3. For each S find T rep. of SL(2,ZZ)

4. Find realizations.

Graphs of MCs

• Simple Xi multigraph Gi :

Vertices labeled by 0,…,M-1

• Question: What graphs possible?

Nijk edges

j k

Example (Lie type G2, q10=-1)

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!

Classification by Graphs

Theorem: (E.R., Stong, Wang)

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

Future Directions

• Classification of all MCs

• Prove Wang’s conjecture

• Images of Bn reps…

• Connections to: link/manifold invariants, Hopf algebras, operator algebras…

Thanks!