Transcript Algebraic Aspects of Topological Quantum Computing

Algebraic Aspects of Topological
Quantum Computing
Eric Rowell
Indiana 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}
1
• Qubits:
V=C2
1
(superposition)
0
• Logical Operations
on {0,1}n
• Unitary Operations
on V X n
• Deterministic: output
unique
• Probabilistic: output
varies
(Uncertainty principle)
a
0
Anyons: 2D Electron Gas
1011 electrons/cm2
particle exchange
9 mK
fusion
defects=quasi-particles
10 Tesla
Topological Computation
Computation
Physics
output
measure
apply operators
braid
initialize
create
particles
Algebraic Characterization
Anyonic System
Top. Quantum Computer
Modular Categories
Toy Model: Rep(G)
•
Irreps: {V1=C, V2,…,Vk}
•
Sum V + W, product V X W, duals W*
•
Semisimple: every W= + miVi
•
Rep: Sn
EndG(V X n)
Braid Group Bn “Quantum Sn”
Generated by:
1
i
i+1
n
i=1,…,n-1
bi =
Multiplication is by concatenation:
=
Concept: Modular Category
deform
group G
Rep(G)
Sn action
Modular Category
Bn action
Axiomatic definition due to Turaev
Modular Category
• Simple objects {X0=C,X1,…XM-1}
+ Rep(G) properties
• Rep. Bn
End(X X n) (braid group action)
• Non-degeneracy: S-matrix invertible
Dictionary: MCs vs. TQCs
Simple objects Xi
Bn-action
Elementary particle
types
Operations (unitary)
X0 =C
Vacuum state
X0
Xi X Xi*
Creation
Constructions of MCs
quantum
group
g
Uqg
Lie algebra
Rep(Uqg)
semisimplify
F
|q|=1
Survey: (E.R. Contemp. Math.) (to appear)
Also,
quantum double
G
D(G)
Rep(D(G))
finite group
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
1-1
Recall: distinct particle types
Simple objects in MC
Conjecture (Z. Wang 03):
The set { MCs of rank M } is finite.
Classified for:
M=1, 2 (V. Ostrik), 3 and 4 (E.R., Stong, Wang)
True for finite groups! (Landau 1903)
Groethendieck Semiring
•
Assume X=X*. For a MC D:
Xi X Xj = + Nijk Xk
•
Semiring Gr(D):=(Ob(D), + , X )
•
Encoded in matrices (Ni)jk := Nijk
Modular Group
• Non-dengeneracy
S symmetric
• Compatibility
T diagonal
•
1
1
0
1
T,
0
-1
1
0
S
give a unitary projective rep. of SL(2,Z)
Our Approach
• Study Gr(D) and reps. of SL(2,Z)
• Ocneanu Rigidity:
Finite-to-one
MCs
{Ni}
• Verlinde Formula:
{Ni} determined by S-matrix
Some Number Theory
• Let pi(x) = det(Ni - xI)
and K = Split({pi},Q).
• Study Gal(K/Q): always abelian!
• Nijk integers, Sij algebraic, constraints
polynomials.
Sketch of Proof (M<5)
1. Show: 1
Gal(K/Q)
SM
2. Use Gal(K/Q) + constraints to determine
(S, {Ni})
3. For each S find T
rep. of SL(2,Z)
4. Find realizations.
Graphs of MCs
• Simple Xi
multigraph Gi :
Vertices labeled by 0,…,M-1
Nijk edges
j
k
• Question: What graphs possible?
Example (Lie type G2, q10=-1)
Rank 4 MC with fusion rules:
N111=N113=N123=N222=N233=N333=1;
N112=N122= N223=0
G1:
G2:
G 3:
0
1
0
2
2
0
3
1
3
3
2
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!