Algebraic Aspects of Topological Quantum Computing Eric Rowell Indiana University.
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Transcript of Algebraic Aspects of Topological Quantum Computing Eric Rowell Indiana University.
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!