Topological Quantum Computation II
Eric Rowell
Supported by USA NSF grant DMS1108725
Joint with C. Galindo, D. Naidu, S.-M. Hong and Z. Wang
July 2014
Quantum Topology
Pocket Dictionary
Unitary Modular Category
Simple objects Xi
1
dual objects X ∗
Hom(X , Y )
cX ,Y
det(S) 6= 0
Anyonic System
anyon types
vacuum type
antiparticles
state spaces
particle exchange
anyons distinguishable
State Spaces
Definition
Let X and Y be particle-types in an anyonic system. The state
space for n anyons of type X with total charge Y is the Hilbert
space Hom(Y , X ⊗n ).
Example
Two particle-types X0 = 1, X1 .
V 06 := Hom(1, X ⊗6 ) is
V06=H
1
1
1
1
1
1
0
fusion rules→ dim(V 06 ).
Z2 : dim(V 06 ) = 1,
Fibonacci: dim(V 06 ) = 5
Definition
A (gate set in a) quantum computational model is Universal if any
unitary operator can be approximately achieved.
In the Quantum Circuit Model:
Definition
A gate set G = {Ui } is Universal if quantum circuits are dense in
U(V ⊗m ) for m >> 0.
That is hIVa ⊗ Ui ⊗ IVb ⊂ U(V ⊗m )i = U(V ⊗m ).
Example

1
0
1
0
1 1
,
,
{
πi/4
0
1 −1
0 e
0
0
1
0
0
0
0
0
1

0
0
} is universal.
1
0
Fix X L
∈ C modular category. Bn acts linearly on
Hn = i Hom(Xi , X ⊗n ):
⊗(i−1)
⊗(n−i−1)
I
ϕX : Bn → Aut(X ⊗n ) via σi 7→ IX
I
ϕX (CBn ) ⊂ End(X ⊗n ) acts on Hom(Xi , X ⊗n ) via f ◦ h.
I
ϕX (CBn−1 ) ⊂ End(X ⊗(n−1) ) acts via (f ⊗ IX ) ◦ h
L
Hn | B ∼
mi Hom(Xi , X ⊗(n−1) ) by naturality.
=
I
n−1
i
⊗ cX ,X ⊗ IX
Suppose:
∼
(ϕX
n , Hn ) =
M
(ϕjn , Vnj )
j
(irreps.)
For Top. Quant.
Comp. to be universal, Bn image must be dense:
Q
j
ϕX
(B
)
=
SU(V
n)
n
n
j
Question (1)
For which X is |ϕX
n (Bn )| = ∞?
Sources of Modular Categories
Example
I
C(g, `): semisimple subquotient of quantum group Rep(Uq g)
at q = e πi/` . Chern-Simons TQFT.
I
Rep(DG ): quantum double DG of finite group G
Dijkgraaf-Witten TQFT.
Definition
We say X ∈ C has Property F if |ϕX
n (Bn )| < ∞ for all n.
Theorem (Jones ’86)
Fund. rep. X ∈ SU(2)`−2 = C(sl2 , `) has property F if and only if
` ∈ {3, 4, 6}.
Theorem (Etingof,R,Witherspoon ’08)
X ∈ Rep(DG ) always has property F .
Algebraic Locality
Definition
M ∈ End(V ⊗n ) is k-local if there is a A ∈ End(V ⊗k ) such that
⊗(n−i−k)
M = IV⊗i ⊗ A ⊗ IV
.
Remark
Quantum Circuit Model: Gates Ui ∈ U(V ⊗ni ) are ni -local.
Topological Model: Gates ϕX
n (σi ) not typically local.
Example: PSU(2)3
Input: modular category 21 C(sl2 , 5) = PSU(2)3 :
I
L = {0, 1}
I
Define: Vki := H(D 2 \ {zj }kj=1 ; i, 1, · · · , 1)
(
Fib(n − 2) i = 0
dim Vni =
Fib(n − 1) i = 1
I
dim(Vn0 ⊕ Vn1 ) 6= mf (n) , not local.
Local Bn representations: Yang-Baxter eqn.
Definition
(R, V ) is a braided vector space if R ∈ Aut(V ⊗ V ) satisfies
(R ⊗ IV )(IV ⊗ R)(R ⊗ IV ) = (IV ⊗ R)(R ⊗ IV )(IV ⊗ R)
Induces a sequence of local Bn -reps (ρR , V ⊗n ) by
ρR (σi ) = IV⊗i−1 ⊗ R ⊗ IV⊗n−i−1
ρR (σi )
v1 ⊗ · · · ⊗ vi ⊗ vi+1 ⊗ · · · ⊗ vn −→ v1 ⊗ · · · ⊗ R(vi ⊗ vi+1 ) ⊗ · · · ⊗ vn
Remark
Turaev: enhanced braided vector spaces (R, V , µ, α, β) give link
invariants.
Square Peg, Round Hole?
Definition (R,Wang)
A localization of a sequence of Bn -reps. (ρn , Vn ) is a braided
vector space (R, W ) and injective algebra maps
τn : Cρn (Bn ) → End(W ⊗n ) such that the following diagram
commutes:
CBn
ρn
Cρn (Bn )
ρR
τn
&
/ End(W ⊗n )
X ∈ C is localizable if (ϕX
n , Hn ) is localizable.
Combinatorial Consequence
L
Suppose (ρn , Vn ) ∼
= i (ρin , Vni ) with ρin irreducible and (R, W )
localizes (ρn , VL
n ).
Then W ⊗n ∼
= i µi Vni with multiplicities µi > 0.
Idea: Add redundancy to Vn to make it 2-local.
Question
If (R, W ) localizes (ϕX , Hn ), does (R, W , µ, α, β) give the same
invariant as InvL (X ) = trC (ϕX (γL )) where γbL = L?
Weak Integrality
P
Z Z where
fusion rules: X ⊗ Y = Z NXY
Z
NXY := dim Hom(X ⊗ Y , Z )
Definition
Let dim(X) ∈ R>0 be the largest eigenvalue of the matrix
Z ]
NX := [NXY
YZ .
X is weakly integral if dim(X )2 ∈ Z.
Example
Fibonacci and SO(3)2
:
0
1

0 1
Nf =
, NX = 
0
1 1
0
0
dim(f ) =
√
1+ 5
2
dim(X ) =
1
0
0
1
0
√
0
0
0
1
1
3.
0
1
1
0
0

0
0

0

0
0
Example C(sl2 , 4) = SU(2)2

1

0
Let R = √12 
0
−1

0 0 1
1 −1 0

1 1 0
0 0 1
Theorem (Franko,R,Wang ’06)
(R, C2 ) localizes (ρX
n , Hn ) for X = X1 ∈ C(sl2 , 4) = SU(2)2
Indeed, (ρR , (C2 )⊗n ) ∼
= 2a H0 ⊕ 2b H1 for n even...
n
n
Remark
Notice: X is not a vector space! (dim(X ) =
√
2)
Example C(sl2 , 5) = SU(2)3
If (R, V ) localizes ϕX
n
1
1
1
2
1
2
3
5
3
Example C(sl2 , 5) = SU(2)3
If (R, V ) localizes ϕX
n
with mult. vectors (an , bn )
1
1 a2
1 b2
2 a3
1 b3
2 a4
3 b4
5 a5
3 b5
Example C(sl2 , 5) = SU(2)3
1
1 a2
1 b2
G(2)
2 a3
1 b3
2 a4
3 b4
5 a5
3 b5
G(3)
G(2)
If (R, V ) localizes (ϕX
n , Hn )
with mult. vectors (an , bn )
then by Perron-Frobenius
Theorem
a2
a
(3)
(2)
G G
=λ 2
b2
b
2 1
1
where G (3) G (2) =
1 2
√ 2
λ = 1+2 5 = FPdim(X )2 ,
a2 , b2 ∈ Z.
Impossible!
Example: C(sl2 , 6) = SU(2)4
If (R, V ) localizes (ϕX
n , Hn )
with dim(V ) = k then
 

 
a4
1 1 0
a4
1 2 1 b4  = λ b4 
c4
0 1 1
c4
1
1 a2
1 b2
2 a3
1 b3
2 a4
3 b4
5 a5
4 b5
5 a6
9 b6
1 c4
and 2a4 + 3b4 + c4 = k 4
k = λ = 3, a4 = b4 /2 = c4 = 9
works!
4 c6
Example: C(sl2 , 6) = SU(2)4
Is there a 9 × 9 R-matrix?
1

ω 0 0
0 1 0 0 0 ω
0 ω 0
0 0 ω 1 0 0


 0 0 ω ω2 0 0 0 ω2 0 


 0 0 ω2 ω 0 0 0 ω2 0 


0 ω 0 0 0 1
γ
ω 0 0

0 1 0
0 0 ω ω 0 0


0 ω 0

0
0
1
ω
0
0


 0 0 ω2 ω2 0 0 0 ω 0 
1 0 0
0 ω 0 0 0 ω

1 3
1 6
2 9
1 9
2 9
3 18
5 33
4 33
5 33
9 54
1 9
4 33
Localizes (ϕX
n , Hn ).
Conjecture
The following are equivalent for X ∈ C:
I
X has property F ,
I
X is localizable,
I
X is weakly-integral
Case Study
Theorem (R-Wenzl,R-Wang,Galindo-R.)
X ∈ SO(N)2 , N odd is:
I
weakly integral: dim(X ) =
I
localizable:
R :=
N−1
X
2
ζNj U j ;
√
N.
U(ei ⊗ej ) = ζNj−i (ei+1 ⊗ej+1 );
ζN = e 2πi/N
j=0
on CN ⊗ CN (indices modulo N).
I
I
Property F : hR1 , · · · , Rn−1 i a finite group.
√
For N prime, InvL (X ) ∼ ±( N)dim H1 (ML ,ZN ) ML double cyclic
cover of S 3 branched over L. Classical.
Evidence
Results
I
X ∈ Rep(DG ) is weakly integral, localizable and property F .
Invariant: | Hom(π1 (S 3 \ L), G )| Classical.
I
Conjecture true for all quantum groups.
Property F Conjecture suggests:
Conjecture
If (R, V ) is a unitary, finite order solution to the Yang-Baxter eqn.
then |ρR (Bn )| < ∞
Theorem (Galindo,R)
If R(xi ⊗ xj ) = (gi (xj ) ⊗ xi ) for some gi ∈ GL(V ), then ρR (Bn ) is
virtually abelian.
Thank you!