Kitaev quantum double models and their
generalizations
Liang Chang
University of California, Santa Barbara
Texas A&M University
December 5, 2012
1 / 25
Background
Local errors, thermic noise are considered the obstacles in the
realization of a quantum computer.
Topological properties of physical systems probably the best
answer to overcome those problems.
Qubits encoded in topological states can be insensitive to local
perturbations
2 / 25
Overview
In 1997, Kitaev constructed a lattice model with the following features:
There is a finite energy gap between the ground state and excited
states.
The ground state degeneracy is protected by topology from local
perturbation.
The excited states are anyons.
3 / 25
Lattice models
Choose a lattice on a closed oriented surface.
Place spins/qubits at the edges of a lattice.
N
Hilbert space H =
V.
edges
Physical observable: Hermitian operators on H.
4 / 25
Lattice models
V
V
V
V
V
V
V
V
V
V
V
V
Choose a lattice on a closed oriented surface.
Place spins/qubits at the edges of a lattice.
N
Hilbert space H =
V.
edges
Physical observable: Hermitian operators on H.
4 / 25
Kitaev models
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
G is a finite group.
Group algebra C[G] is a vector space spanned by group elements.
N
Hilbert space H =
C[G].
edges
5 / 25
Kitaev models
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
G is a finite group.
Group algebra C[G] is a vector space spanned by group elements.
N
Hilbert space H =
C[G].
edges
5 / 25
Kitaev models
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
C[G]
G is a finite group.
Group algebra C[G] is a vector space spanned by group elements.
N
Hilbert space H =
C[G].
edges
5 / 25
Vertex and plaquette operators
Plaquette operators Bp :
g3
Bp g2
g3
+
p
g4
= δg1 g2 g3 g4 ,e g2
g1
+
p
g4
g1
Vertex operators Av :
g2
+
g
g1
3
=
Av v
g4
hg2 +
hg
P
hg1
3
1
|G|
v
h∈G
hg4
6 / 25
Hamiltonian
Hamiltonian H = −
P
v
Av −
P
Bp .
p
All [Av , Av0 ] = 0, [Bp , Bp0 ] = 0, [Av , Bp ] = 0. They can be
diagonalized simultaneously.
All Av ’s and Bp ’s are projectors, i.e., A2v = Av and Bp2 = Bp . As a
consequence, the eigenvalues are -1 and 1.
In particular, the ground state |Ψ > is the common eigenspace of
all Av ’s and Bp ’s for eigenvalue 1, i.e., Av |Ψ >= |Ψ > and
Bp |Ψ >= |Ψ > for all v and p.
7 / 25
Hamiltonian
Hamiltonian H = −
P
v
Av −
P
Bp .
p
All [Av , Av0 ] = 0, [Bp , Bp0 ] = 0, [Av , Bp ] = 0. They can be
diagonalized simultaneously.
All Av ’s and Bp ’s are projectors, i.e., A2v = Av and Bp2 = Bp . As a
consequence, the eigenvalues are -1 and 1.
In particular, the ground state |Ψ > is the common eigenspace of
all Av ’s and Bp ’s for eigenvalue 1, i.e., Av |Ψ >= |Ψ > and
Bp |Ψ >= |Ψ > for all v and p.
7 / 25
Hamiltonian
Hamiltonian H = −
P
v
Av −
P
Bp .
p
All [Av , Av0 ] = 0, [Bp , Bp0 ] = 0, [Av , Bp ] = 0. They can be
diagonalized simultaneously.
All Av ’s and Bp ’s are projectors, i.e., A2v = Av and Bp2 = Bp . As a
consequence, the eigenvalues are -1 and 1.
In particular, the ground state |Ψ > is the common eigenspace of
all Av ’s and Bp ’s for eigenvalue 1, i.e., Av |Ψ >= |Ψ > and
Bp |Ψ >= |Ψ > for all v and p.
7 / 25
Hamiltonian
Hamiltonian H = −
P
v
Av −
P
Bp .
p
All [Av , Av0 ] = 0, [Bp , Bp0 ] = 0, [Av , Bp ] = 0. They can be
diagonalized simultaneously.
All Av ’s and Bp ’s are projectors, i.e., A2v = Av and Bp2 = Bp . As a
consequence, the eigenvalues are -1 and 1.
In particular, the ground state |Ψ > is the common eigenspace of
all Av ’s and Bp ’s for eigenvalue 1, i.e., Av |Ψ >= |Ψ > and
Bp |Ψ >= |Ψ > for all v and p.
7 / 25
Toric code model
The surface is a torus T .
G = Z2 = {1, −1}.
Plaquette operators Bp :
g3
Bp g2
g3
+
p
g4
g1
= δg1 g2 g3 g4 ,1 g2
+
p
g4
g1
Vertex operators Av :
g2
+
g
g1
3
Av =
v
g4
1
2
g2 + −g2 +
−g
g
g
−g1 1
3
3
+
v
v
g4
−g4
8 / 25
Eigenstates of all Bp
c
A common eigenstate of all Bp ’s can be represented by closed
curves in the dual lattice.
9 / 25
Eigenstates of all Bp
c
A common eigenstate of all Bp ’s can be represented by closed
curves in the dual lattice.
9 / 25
Eigenstates of all Av
v
c
c ∼ c 0 : c is homologous to c 0 .
P
A common eigenstate of all Av ’s is of the form |ci > where
ci ∼ cj .
10 / 25
Eigenstates of all Av
v
c0
c ∼ c 0 : c is homologous to c 0 .
P
A common eigenstate of all Av ’s is of the form |ci > where
ci ∼ cj .
10 / 25
Eigenstates of all Av
v
c0
c ∼ c 0 : c is homologous to c 0 .
P
A common eigenstate of all Av ’s is of the form |ci > where
ci ∼ cj .
10 / 25
Ground states of toric code
The ground state space has 4 basis vetors: [c0 ], [c1 ], [c2 ], [c1 ∪ c2 ].
The ground state space has a topological interpretation as the
homology group H1 (T ) with two generators c1 and c2 .
c0
c2
c1
11 / 25
Generalized Kitaev models
To generalize the group algebra models, we expect to construct a
Hamiltonian
X
X
H =−
Av −
Bp
v
p
such that all Av ’s and Bp ’s commute with each other.
It was anticipated by Kitaev to be possible for finite dimensional C ∗
Hopf algebras and was achieved by Buerschaper, et al.
It can be generalized in a general quantum groupoid setting.
12 / 25
Generalized Kitaev models
To generalize the group algebra models, we expect to construct a
Hamiltonian
X
X
H =−
Av −
Bp
v
p
such that all Av ’s and Bp ’s commute with each other.
It was anticipated by Kitaev to be possible for finite dimensional C ∗
Hopf algebras and was achieved by Buerschaper, et al.
It can be generalized in a general quantum groupoid setting.
12 / 25
Generalized Kitaev models
To generalize the group algebra models, we expect to construct a
Hamiltonian
X
X
H =−
Av −
Bp
v
p
such that all Av ’s and Bp ’s commute with each other.
It was anticipated by Kitaev to be possible for finite dimensional C ∗
Hopf algebras and was achieved by Buerschaper, et al.
It can be generalized in a general quantum groupoid setting.
12 / 25
Quantum groupoids
Quantum groupoid H(µ, ∆, S, 1, ε) over C is a complex vector space
equipped with the following compatible maps:
Multiplication µ : H ⊗ H → H;
Unit i : C → H.
i(1) = 1H ;
Comultiplication ∆ : H → H ⊗ H.
P
∆(x) = (x) x(1) ⊗ x(2) (Sweedler notation);
Counit ε : H → C;
Antipode S : H → H.
13 / 25
Example of quantum groupoids
For a finite group G, its group algebra C(G) is Hopf algebra.
For g, h ∈ G,
µ(g ⊗ h) = gh,
i(1) = e,
∆(g) = g ⊗ g,
ε(g) = 1,
S(g) = g −1
14 / 25
Exactly solvable Hamiltonian
It turns out vertex and plaquette operators can be built from two
cocommutative elements Λ ∈ H and λ ∈ H ∗ .
1 P
For group algebra C[G], λ (g) = δg,e and Λ = |G|
g.
g∈G
For general quantum groupoids, λ and Λ play a similar role as
trace on H and H ∗ .
15 / 25
Exactly solvable Hamiltonian
Plaquette operators Bp :
c(1)
c
Bp b
+
p
d
P
=
λ (d(2) c(2) b(2) a(2) ) b(1)
(a,b,c,d)
+
p
d(1)
a(1)
a
Vertex operators Av :
b
+
a
c
Av v
d
=
P
(Λ)
Λ(2) b +
Λ c
Λ(1) a
(3)
v
Λ(4) d
16 / 25
Fibonacci category F
Simple objects: 1 and τ;
Self dual: 1∗ = 1 and τ ∗ = τ;
Quantum dimension: d1 = 1 and dτ = φ =
Fusion rule:
τ2
√
1+ 5
2
(golden ratio);
= 1 + τ.
F-moves:
τ
τ
τ
τ
= φ −1
1
τ
τ
τ
+φ
τ
τ
τ
1
τ
τ
τ
τ
τ
1
φ−2
τ
− 21
τ
=
τ
τ
1
τ
τ
τ
τ
τ
−φ −1
τ
τ
τ
17 / 25
Fibonacci algebra
In 2010, Kitaev and Kong constructed a C ∗ -quantum groupoid HC from
a unitary fusion category C. Here we illustrate their construction by
using the example of Fibonacci category F.
As vector space, HF is spanned by
a
ab
ei;cd
c
i
=
b
d
where a, b, c, d, i ∈ {1, τ}.
Multiplication
a
b
a0
c
i
·
d
b0
i0
c0
=
d0
a
i0
δc,a0 δd,b0 δi,i 0
√
c0
di
b
d0
18 / 25
Unit
P√
η=
di
a
a
i
a,b,i
b
b
Comultiplication
a
i
∆
b
c !
=
P
j,k
p,q
d
a
√
di dp dq iap idq
√
Fk ,jk Fj,kc
p
c
j
⊗
db dc
p
q
q
k
b
d
Counit
a
i
ε
b
c !
= δa,b δc,d δi,1
d
19 / 25
Antipode
a
c !
i
S
d
√
db dc
=√
b
i
da dd
b
c
d
a
Star
a
i
b
c !∗
c
=
d
a
i
d
b
20 / 25
Examples of quantum groupoid structure
1
τ
τ
τ
∆
τ
τ
1 !
1
1
τ
⊗
τ
τ
− 12
τ
1
τ
+
1
τ
1
⊗
τ
τ
τ
τ
τ
τ
τ
⊗
τ
τ
τ
τ
1
τ
−φ
τ
τ
1
τ
τ
√1
φ
τ
1
1
τ
=
τ
τ
=
τ
1
τ
·
τ
τ
1
τ
τ
21 / 25
Solvable Hamiltonian for HF
By choosing
a
Λ=
1
1+φ 2
P
da db
a,b
a
a
λ=
1
1+φ 2
P p
a,b,µ
b
1
b
∧
a
µ
dµ
b
We get an exactly solvable Hamiltonian HK = −
b
P
v
AKv −
P
p
BpK .
22 / 25
Ground states
The ground state is
k1
a2
c1
a2
6
P Q
dbn
kn in µ
k
c
b
n
n
n
√ Fi ,i a Fj ,j i djn n+1 n n+1 n+1 n n+1 j,k ,µ n=1
a,b,c
c2
k2
a3
c2
b1
i2
c1
i1
a1
b6
b1
a1
c6
c6
b6
b2
b2
a3
i3
i6
a6
b5
b5
b3
c3
c3
k3
k6
a4
b3
a4
b4
i4
c4
b4
i5
a5
c5
a6
k5
+
c5
a5
c4
k4
23 / 25
Main result and future work
Theorem
Given a trivalent lattice Γ on a closed oriented surface Σ, the ground
space LK (Σ, Γ) of the Kitaev model based on HF is canonically
isomorphic to the ground space LLW (Σ, Γ) of Levin-Wen Models based
on F. As a consequence, LK (Σ, Γ) is canonically isomorphic to the
target space ZTV (Σ) of the Turaev-Viro TQFT based on F.
Excitation(Anyons).
Non-semisimple generalization.
24 / 25
Main result and future work
Theorem
Given a trivalent lattice Γ on a closed oriented surface Σ, the ground
space LK (Σ, Γ) of the Kitaev model based on HF is canonically
isomorphic to the ground space LLW (Σ, Γ) of Levin-Wen Models based
on F. As a consequence, LK (Σ, Γ) is canonically isomorphic to the
target space ZTV (Σ) of the Turaev-Viro TQFT based on F.
To do:
Excitation(Anyons).
Non-semisimple generalization.
24 / 25
Thank You!
25 / 25