Topological Quantum Computation:
Localizable Anyons and Property F
Eric Rowell
Supported by USA NSF grant DMS1108725
May 2015
Encuentro Colombiano de Computación Cuántica
Quantum Circuit Model
Fix d∈ Z and let V = Cd .
Definition
The n-qudit state space is the n-fold tensor product:
Mn = V ⊗ V ⊗ · · · ⊗ V .
A quantum gate set is a collection S = {Ui } of unitary operators
Ui ∈ U(Mni ) (ni -local) usually ni ≤ 4.

1
0

0
0
0
1
0
0
0
0
0
1

0
0

1
0
Quantum Circuits
Definition
A quantum circuit on S = {Ui } is:
I
G = G1 · G2 · · · Gm ∈ U(Mn )
I
where Gj = P(IV⊗a ⊗ Ui ⊗ IV⊗b )P −1 .
Here P permutes the n qudit spaces.
Given U ∈ U(Mn ) can only reasonably hope to approximate U as
G1 · G2 · · · Gm
Definition
In the quantum circuit model, a gate set S is Universal if for any
U ∈ U(V ⊗n ) and > 0 there is a quantum circuit G such that
||U − G|| < .
Remark
Sometimes we can get S strictly universal, with circuits
Hj = (IV⊗a ⊗ Ui ⊗ IV⊗b ): no qudit swaps.
Topological 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 associated
Y -sector Hilbert space V (X n ; Y ).
Example
X0 = 1, X1 . V 06 := V (X16 ; X0 ) is
V06=H
1
1
1
1
1
1
0
fusion rules→ dim(V 06 ).
Ising: dim(V 06 ) = 4,
Fibonacci: dim(V 06 ) = 5
Braid Action
Bn acts on state spaces: V (X n ; Y ) = Hom(Y , X ⊗n )
I Fix anyons X , Y
I Braid group acts linearly: ρ : Bn → U(V (X n ; Y ))
ρ(σi )h = IX⊗i−1 ⊗ cX ,X ⊗ IXn−i−2 ◦ h
Physically: particle exchange
i
i+1
i
Mathematical Formulation
I
Sum over the Y -sectors: V (X n ) :=
I
n
ρX
n : Bn → U(V (X )) by particle exchange is a unitary
representation.
L
Y
V (X n ; Y )
Definition
An anyon X is braiding universal if ρX
n (Bn ) is dense on each
irreducible component.
In Wang’s talk: QCM can be simulated on a (universal) TQC. For
example X =Fibonacci anyon.
Questions
1. Can we simulate TQC with a QCM?
2. How can we detect universality?
3. Are these two questions related?
Hidden Locality
Freedman, Kitaev and Wang showed that anyons have a hidden
locality: Let U(β) ∈ U(V (X n )) be a unitary braiding matrix.
Goal: simulate U on V ⊗k(n) for some v.s. V .
L
⊗(n−1)
I Set V =
(A,B,C ) V (A, B; C ) and W = V
I
TQFT axioms imply:
V (X n ) ,→ W
Remark
V can be quite large and U(β) only acts on the subspace V (X n ).
Bratteli Diagrams
Consider Bn -rep. V (X n ; Y ).
Bn−1 ⊂ Bn also acts.
How does V (X n ; Y ) decompose under Bn−1 action?
M (n−1)
V (X n ; Y ) ∼
mYZ V (X n−1 ; Z )
=
Z
(j−1)
Recorded in graph: vertices: {V (X j ; Z )}, mYZ
V (X j−1 ; Z )
(j−1)
mYZ
Bn−1 :
V (X j ; Y )
Bratteli diagram
···
1)
V (X n−1 ; Z
(n−1)
1
mYZ
Bn :
edges
V (X n ; Y )
(n−1)
mYZ
k
V (X n−1 ; Zk )
Example: Fibonacci anyons
Bratteli diagrams determine
dim V (X n ; Y ):
Example
Fibonacci anyons X = f :
f ⊗2 = 1 ⊕ f .
count (downward) paths for
dim V (f n ; Y )
V (f ; f )
V (f 2 ; 1)
V (f 2 ; f )
V (f 3 ; f )
V (f 3 ; 1)
V (f 4 ; 1)
V (f 4 ; f )
V (f 5 ; f )
V (f 5 ; 1)
V (f 6 ; 1)
V (f 6 ; f )
Example: Fibonacci anyons
Bratteli diagrams determine
dim V (f n ; Y ):
1
Example
1
1
2
1
2
3
5
3
5
8
Fibonacci anyons X = f :
f ⊗2 = 1 ⊕ f .
dim V (f n ; 1) = Fib(n − 2),
dim V (f n ; f ) = Fib(n − 1)
Example: Metaplectic anyons
1
Example
1
1
2
1
2
3
5
4
5
9
SO(3)2 -Metaplectic anyons
X : {1, X , X 0 , Y , Z }
X ⊗2 = 1 ⊕ Y ,
X ⊗ Y = X ⊕ X 0,
X ⊗ X0 = Y ⊕ Z,
X ⊗ Z = X0
1
n
dim V (X 2n+1 , X ) = (3 2+1) ,
n
dim V (X 2n+1 , X 0 ) = (3 2−1) .
4
Algebraic Locality
Definition
M ∈ End(V ⊗n ) is k-local if there is a A ∈ End(V ⊗k ) such that
M = I ⊗i ⊗ A ⊗ I ⊗(n−i−k) .
Remark
I
Note this is strict locality: no permuting the factors V .
I
n
⊗k(n) .
ρX
n (σi ) almost never local: usually V (X ) 6= V
Local Braiding Gates: 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)
Get 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
Square Peg, Round Hole?
Definition (R,Wang)
Anyon X is localizable if there is a braided vector space (R, W )
⊗n ) such that
and injective algebra maps τn : CρX
n (Bn ) → End(W
the following diagram commutes:
CBn
ρn
CρX
n (Bn )
ρR
τn
&
/ End(W ⊗n )
L
Combinatorially: W ⊗n ∼
= Z mZ V (X n ; Z ) for multiplicities
mZ > 0.
Example: Fibonacci
If (R, V ) localizes f
1
1
1
2
1
2
3
5
3
Example: Fibonacci
If (R, V ) localizes f
with mult. vectors (an , bn )
1
1 a2
1 b2
2 a3
1 b3
2 a4
3 b4
5 a5
3 b5
Example: Fibonacci
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 f
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 , a2 , b2 ∈ Z.
Impossible!
Example: SO(3)2 -Metaplectic
1
1 a2
1 b2
2 a3
1 b3
2 a4
3 b4
5 a5
4 b5
5 a6
9 b6
If (R, V ) localizes X
with dim(V ) = k then
 

 
a4
1 1 0
a4
1 2 1 b4  = λ b4 
c4
0 1 1
c4
1 c4
and 2a4 + 3b4 + c4 = k 4
k = λ = 3, a4 = b4 /2 = c4 = 9
works!
4 c6
Example: SO(3)2 -Metaplectic
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 


ω 0 0
0 ω 0 0 0 1


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
ω = e 2πi/3 Localizes X .
Property F
Definition
An anyon X has property F if ρX
n (Bn ) is a finite group for all n.
Detecting Locality and Property F
Question
I
Fibonacci anyon f : not localizable, not property F
I
Metaplectic anyon X : localizable, property F
I
WHY?
Weak Integrality
fusion rules: X ⊗ Y =
P
Z
Z Z Recall:
NXY
Definition
Let FPdim(X) be the largest eigenvalue of the matrix
Z ]
NX := [NXY
YZ .
X is weakly integral if FPdim(X )2 ∈ Z.
Conjecture
The following are equivalent for an anyon X :
I
X has property F ,
I
X is localizable,
I
X is weakly-integral
Evidence
Results
I
I
I
Conjecture true for all quantum group anyons.
L
n
R-Wang: If ρX
n (Bn ) generates
Z End(V (X ; Z )) then
localizable ⇒weakly integral.
R-Galindo-Hong: group-type Yang-Baxter operators: ρR
n (Bn )
virtually abelian.
Conclusions
If conjecture holds, weakly integral anyons:
I
generate only finitely many n-circuits from braiding gates
I
can be directly simulated with qudit models (localizable).
Thank you!