Algebraic Aspects of Topological Quantum Computing

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Algebraic Aspects of Topological Quantum Computing
Eric RowellIndiana University
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Collaborators
Project Q with M. Freedman, A. Kitaev, K.
Walker and C. Nayak
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What is a Quantum Computer?

• Any system for computation based on
• quantum mechanical phenomena

Create Manipulate
Measure
Quantum Systems
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Classical vs. Quantum
1
a

• Bits 0,1
• Logical Operations
• on 0,1n
• Deterministic output unique

• Qubits VC2
• (superposition)
• Unitary Operations
• on V n
• Probabilistic output varies
• (Uncertainty principle)

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0
0
X
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Anyons 2D Electron Gas
1011 electrons/cm2
particle exchange
9 mK
fusion
defectsquasi-particles
10 Tesla
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Topological Computation
Computation
Physics
output
measure
apply operators
braid
create particles
initialize
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Algebraic Characterization
Anyonic System
Top. Quantum Computer
Modular Categories
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Toy Model Rep(G)

• Irreps V1C, V2,,Vk
• Sum V W, product V W, duals W
• Semisimple every W miVi
• Rep Sn EndG(V n)

X


X
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Braid Group Bn Quantum Sn
Generated by
i1,,n-1
bi
Multiplication is by concatenation

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Concept Modular Category
deform
group G
Rep(G)
Modular Category
Sn action
Bn action
Axiomatic definition due to Turaev
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Modular Category

• Simple objects X0C,X1,XM-1
• Rep(G) properties
• Rep. Bn End(X n) (braid group action)
• Non-degeneracy S-matrix invertible

X
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Dictionary MCs vs. TQCs
X
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Constructions of MCs
quantum group
semisimplify
g Uqg Rep(Uqg)
F
Lie algebra
q1

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

Also,
quantum double
G D(G) Rep(D(G))

finite group
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Physical Feasibility
Realizable TQC
Bn action Unitary
Uqg Unitarity results (Wenzl 98), (Xu 98)
(E.R. 05)
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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?
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General Problem

• Rep. Y G U(N), G discrete
• Y(G) ?
• SU(N)
• Finite group
• SO(N), E7
• For G Bn see (Franko, E.R., Wang) and (Larsen,
  E.R.)

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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 M1, 2 (V. Ostrik), 3 and 4
(E.R., Stong, Wang)
True for finite groups! (Landau 1903)
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Groethendieck Semiring

• Assume XX. For a MC D
• Xi Xj Nijk Xk
• Semiring Gr(D)(Ob(D), , )
• Encoded in matrices (Ni)jk Nijk

X
X

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Modular Group

• Non-dengeneracy S symmetric
• Compatibility T diagonal
• give a unitary projective rep. of SL(2,Z)

T,
S
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Our Approach

• Study Gr(D) and reps. of SL(2,Z)
• Ocneanu Rigidity
• MCs Ni
• Verlinde Formula
• Ni determined by S-matrix

Finite-to-one
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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.

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Sketch of Proof (Mlt5)

• Show 1 Gal(K/Q) SM
• Use Gal(K/Q) constraints to determine
  (S, Ni)
  
• For each S find T rep. of SL(2,Z)
• 4. Find realizations.

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Graphs of MCs

• Simple Xi multigraph Gi
• Vertices labeled by 0,,M-1
• Question What graphs possible?

Nijk edges
j
k
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Example (Lie type G2, q10-1)

• Rank 4 MC with fusion rules
• N111N113N123N222N233N3331
• N112N122 N2230

G1
Tensor Decomposable!
G2
0
2
1
3
G3
2
0
3
1
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Classification by Graphs

• Theorem (E.R., Stong, Wang)
• Indecomposable, self-dual MCs of ranklt5 are
  classified by

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Future Directions

• Classification of all MCs
• Prove Wangs conjecture
• Images of Bn reps
• Connections to link/manifold invariants, Hopf
  algebras, operator algebras

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Thanks!
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Bn Images vs. Link Invariants

• Roughly, we expect
• Finite image Link invariant classical
  (easy).
• Dense image Link invariant P-hard
• Universal TQC

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Improvements over Classical Computation

• Integer factorization in polyl time
• Simulation of Quantum Mechanics
• Quantum database search

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Analogy

• Theorem (E. Landau 1903)
• The set G Rep(G)N is finite.
• Proof Exercise (Hint Use class equation)

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More Graphs
Lie type B2 q9-1
D(S3)
Lie type B3 q12-1
Lie type G2 q21-1