Anyon and Topological
Quantum Computation
Su-Peng Kou
Beijing Normal university
Outline
1. Part I: Anyons and braiding group
2. Part II: Quantum computation of topological
qubits in Z2 topological orders
3. Part III : Topological quantum computation by
Ising anyons
4. IV: Topological quantum computation by
Fibonacci anyons
Key words: topological string operator, nonAbelian
anyon
Milestone for topological
quantum computation
1997, Kitaev proposed the idea of
topological quantum bit and fault
torrent quantum computation in an
Abelian state.
2001, Kitaev proposed the topological
quantum compuation by braiding nonAbelian anyons.
 2001, Preskill, Freedman and others
proposed a universal topological
quantum computation.
(I) Anyons and braid groups
Boson
b1 , b2  b2 , b1
Fermion
f1 , f 2   f 2 , f1
Abelian statistics via non-Abelian
statistics
Abelian anyon
 1 , 2  e
i
 2 , 1 ,   0, 
non  Abelian anyon
 1 , 2  M  2 , 1 , M is a matrix
Example (toy model) of Abelian Anyons :
charge q - flux Φ composites (Wilczek, '82)
The Aharonov - Bohm phase   qΦ
E.g. for q  e, Φ  2/ne, the statistica l angle   π/n.
Exchange statistics and braid group
Particle Exchange : world lines braiding


Braid group
[ i ,  k ]  0
i
|i  k |  2
 i2  1
k


 i
i 1



Yang - Baxter


i 1
i

 i i 1 i

 i 1 i i 1
General anyon theory
1. A finite set of quasi-particles or anyonic
“charges.”
2. Fusion rules (specifying how charges
can combine or split).
3. Braiding rules (specifying behavior under
particle exchange).
1. Fusion rules : N matrix
ab   N c
c
ab
c
2. Braiding rules: R matrix
b
a
a
b
 R
ab
c
3. Associativity relations for fusion: F matrix

 [F
f
abc
d
] ef
Pentagon equation:
Hexagon equation:
Non-Abelian statistics
Exchanging particles 1 and 2:
Exchanging particles 2 and 3:
• Matrices M12 and M23 don’t commute;
• Matrices M form a higher-dimensional representation
of the braid-group.
(II) Quantum computation of
topological qubits
in Z2 topological orders
1. Z2 topological order
1. There are four sectors : I (vacuum), ε(fermion),
e (Z2 charge), m (Z2 vortex) ;
2. Z2 gauge theory
3. U(1)×U(1) mutual Chern-Simons theory
4. Topological degeneracy : 4 on torus
SP Kou, M Levin, and XG Wen, PRB 78, 155134 (2008).
Mutual semion statistics between Z2 vortex and Z2 charge
Z2 charge
Z2 vortex
  flux
Mutual Flux binding
Fermion as the bound state of a Z2 vortex and
a Z2 charge
Fusion rule
A. Yu. Kitaev, Ann. Phys. 303, 2 2003.
Toric-code model
AS
A．Y．Kitaev，Annals
Phys. 303, 2 (2003)
BP
Wen-plaquette model
H   g  Fˆ i ,
i
x
y
x
y
Fˆ i   i  i  eˆ x  i  eˆ x  eˆ y  i  eˆ y
X. G. Wen, PRL. 90,
016803 (2003)

y

x

x

y
Solving the Wen-plaquette model
x
y
x
y
H   g  Fˆi , Fˆi   i  i  eˆx  i  eˆx  eˆ y  i  eˆ y
i
• The energy eigenstates are labeled by the
eigenstates of Fˆi
Fˆ , H   0 Fˆ , Fˆ   0
i
• Because of
2
Fˆi  1 ,
i
j
the eigenvalues are
Fi  1 and
Fi   1
The energy gap
x
y
x
y
H   g  Fˆi , Fˆi   i  i  eˆx  i  eˆx  eˆ y  i  eˆ y
i
• For g>0, the ground state is
Fi  1
1
1
1
1
1
1
1
1
1
-1
1
1
1
1
1
1
1
1
The ground state energy is E0=Ng
The elementary excitation is
Fi   1
The energy gap for it becomes
E1  E 0  2 g ,
for
Fi   1
The statistics for the elementary excitations
• There are two kinds of Bosonic excitations:
• Z2 vortex
F
 1
ii x i y even
• Z2 charge
Fiix i y odd  1
• Each kind of excitations moves on each subplaquette:
• Why?
-1
1
1
1
1
1
1
1
1
-1
1
1
1
1
1
1
1
1
• There are two constraints (the even-by-even lattice):
One for the even plaquettes, the other for the odd
plaquettes

ii x  i y  even
Fiix i y even  1

Fiix i y odd  1
ii x  i y  odd
• The hopping from even plaquette to odd violates
the constraints :
You cannot change
-1
1
1
1
-1
a Z2 vortex into a Z2 charge
1
1
1
1
1
1
Topological degeneracy on a torus
(even-by-even lattice) :
• On an even-by-even lattice, there are totally
N
2 states N  Lx  Ly
2
2
F

1
and
F
 i 1
• Under the constaints,  i
i  i  even
i  i  odd
N
2
the number of states are only
4
• For the ground state Fi  1 , it must be four-fold
degeneracy.
x
y
x
y
The dynamics of the Z2 Vortex and Z2 charge
• Z2 vortex (charge) can only move in the same subplaquette:
• The hopping operators for Z2 vortex (charge) are


x
x
i
and 
y
i
x
ˆ
ˆ
F

i  eˆ y i
i  eˆ y   Fi
y
ˆ
 i Fi i   Fˆi
y
The mutual semion statistics between the Z2
Vortex and Z2 charge
• When an excitation (Z2 vortex) in even-plaquette
move around an excitation (Z2 charge) in oddplaquette, the operator is F   x y  x
y

i
i
i  eˆx
i  eˆ x  eˆ y
i  eˆ y
• it is -1 with an excitation on it
Fi  1
• This is the character for
mutual semion statistics
Fermion as the bound state of a Z2
vortex and a Z2 charge.
X. G. Wen, PRD68, 024501 (2003).
mutual
Controlling the hopping of quasi-particles by
external fields
• The hopping operators of Z2
vortex and charge are
i
x
and  i
y
• The hopping operator of
fermion is  iz
So one can control the
dynamics of different quasiparticles by applying
different external.
• Closed strings
• Open strings
String net condensation for the ground states
The string operators：
Wc
C
， Wv 
C和 W f 
C
，
For the ground state, the closed-strings are condensed
The toric-code model
H  g
 Zi  g
ieven
 X i , Z i   i  i eˆx  i eˆx eˆy  i eˆy ,
z
iodd
X i   i  i  eˆx  i  eˆx  eˆ y  i  eˆ y
x
x
x
x
• There are two kinds of
Bosonic excitations:
• Z2 vortex Z ii  i even  1
• Z2 charge X ii  i odd  1
x
y
x
y
Fermion as the bound state of
a Z2 vortex and a Z2 charge.
z
z
z
Controlling the hopping of quasi-particles by
external fields
• The hopping operator of Z2 vortex is
 ix
• The hopping operator of Z2 charge is  iz
• The hopping operator of fermion is  iy
So one can control the dynamics of
different quasi-particles by applying different
external fields.
2. Topological qubit
A. Yu. Kitaev, Annals Phys. 303, 2 (2003)
|0> and |1> are the degenerate ground-states of a (Z2)
topological order due to the (non-trivial) topology.
  0  1
Advantage
No local perturbation can introduce decoherence.
Ioffe, &, Nature 415, 503 (2002)
Topology of Z2 topological order
E
E
1
Disc
E
2
Cylinder
4
Torus
Hole on a Disc
Topological closed string operators
on torus – topological qubits
Degenerate ground states as eigenstates of
topological closed operators
• Algebra relationship:
• Define pseudo-spin operators:
Topological closed string
operators
• On torus，pseudo-spin representation of
topological closed string operators:
S.P. Kou, PHYS. REV. LETT. 102, 120402 (2009).
J. Yu and S. P. Kou, PHYS. REV. B 80, 075107 (2009).
S. P. Kou, PHYS. REV. A 80, 052317 (2009).
Degenerate ground states as eigenstates of
topological closed operators
 m1 , m2   m1  m2 
zl  ml  ml,
ml  0
zl  ml    ml. ml  1
zl (l  1, 2)
Toric codes : topological qubits on
torus
There are four degenerate ground states for the
Z2 topological order on a torus: m, n = 0, 1 label the
flux into the holes of the torus.
How to control the
topological qubits?
A. Y. Kitaev :
“Unfortunately, I do not know any way this
quantum information can get in or out. Too few
things can be done by moving abelian anyons.
All other imaginable ways of accessing the
ground state are uncontrollable.”
A．Y．Kitaev，Annals Phys. 303, 2 (2003)
3. Quantum tunneling effect
of topological qubits : topological closed
E representation
E
string
Tunneling processes are
virtual quasi-particle
moves around the
1
periodic direction.
Disc
Cylinder
Topological closed string operator as
a virtual particle hopping
Topological closed string operators may
connect different degenerate ground states
S.P. Kou, PHYS. REV. LETT. 102, 120402 (2009).
J. Yu and S. P. Kou, PHYS. REV. B 80, 075107 (2009).
S. P. Kou, PHYS. REV. A 80, 052317 (2009).
Higher order perturbation approach
0



U

0,



T
exp

i
H

t

dt

，

I
I

iH
iH


0t H
0t .
H

t

e
e
I
I

j

U
0, 
0, 
|m   U
|m
I
I 
j0
j0

U

0, 
|m  
I
1

E 0 H
0
j
H
.
I |m
L1
E   m,n 
n  H
0, 
|m.
I UI
• Energy splitting : lowest order contribution of
topological closed string operators
E 
n  H

I
E
m,n
0
1
L 1
H
.
I 0 |m

 H0
L0 is the length of topological closed string operator
#
The energy splitting from higher order
(degenerate) perturbation approach
E 
n  H

I
E
m,n
0
1
L 1
H
.
I 0 |m
0
H
 t eff
E   
 
L : Hopping steps of quasi-particles
teff : Hopping integral
 : Excited energy of quasi-particles




L
J. Yu and S. P. Kou, PHYS.
REV. B 80, 075107 (2009).
#
Topological closed string operators of four
degenerate ground states for the Wen-plaquette
model under x- and z-component external fields
Effective model of four degenerate ground
states for the Wen-plaquette model under xand z-component external fields
External field along z direction
• In anisotropy limit, the four degenerate
ground states split two groups, Lx  Ly ,
z
z
z
z

E 1  h
h
h
2, E2  
2, E3  h2 和 E4  
2。
z
z

E 1  E 3  2h
1 ， E 1  E 2  2h 2 .
2×6 lattice on the Wenplaquette model under z
direction field
External field along z direction
• Isotropy limit， Lx  Ly , the four
degenerate ground states split three groups
z
z

E 1  E 2  0, E 3  2h
和
E


2h
4
1
1.
4×4 lattice on Wenplaquette model under zdirection
External field along x direction
• Under x-direction field, the four degenerate
ground states split three groups:
E 1  2J xx , E 2  2J xx ,
E 3  E 4  J zz  0,
4×4 lattice on Wenplaquette model under xdirection
#
Ground states energy splitting of Wen-plaqutte model on
torus under a magnetic field along x-direction
Ground states energy splitting of Wen-plaqutte model on
torus under a magnetic field along z-direction
Planar codes : topological qubits
on surface with holes
  flux


Fermionic based
L. B. Ioffe, et al., Nature 415, 503 (2002).
  


  


Effective model of the degenerate ground
states of multi-hole
S.P. Kou, PHYS. REV. LETT.
102, 120402 (2009).
S. P. Kou, PHYS. REV. A 80,
052317 (2009).
H eff   J ij i  j   J ij  i  j   hi  i   hi  i
z
ij
z
z
x
ij
x
x
z
i
z
x
x
i
The four parameters Jz, Jx, hx, hz are determined by the
quantum effects of different quasi-particles.
Unitary operations
• A general operator becomes :
U e

i

 z

e
i


x

e
i

 z
H eff   J ij i  j   J ij i  j   hi  i   hi  i
z
ij
z
z
x
ij
x
x
z
i
For example , Hadamard gate is
z
x
i
x
CNOT gate and quantum entangled state
of topological qubits
S. P. Kou, PHYS. REV.
A 80, 052317 (2009).
III. Topological quantum computation
by braiding Ising anyons
Topological Quantum Computation
Computation
Physics
output
measure
operation
braid
initialize
create
particles
Eric Rowell
(I) Ising anyons
Particle type : I ,  , 
    1 
Fusion rules:
   
   1
Ising anyons
Majorana fermion
Another anyon
  flux
σ:π-Flux binding a
Majorana Fermion
SU(2)2 non-Abelian statistics between π-flux with a
trapped majorana fermion.
px+ipy-wave superconductor : an example of
symmetry protected topological order
• µ>0, non-Abelian Topologial state
• µ<0, Abelian Topologial state
Read, Green, 2000.
S. P. Kou and X.G. Wen, 2009.
Winding number in momentum space
BdG equation of px+ipy superconductor
Bogoliubov deGennes Hamiltonian:
Eigenstates in +/- E pairs
Spectrum with a gap
Excitations: Fermionic
quasiparticles above the gap
BdG equation of vortex in px+ipy superconductor
E=0
Whyπ vortex in px+ipy wave superconductors
traps majorona fermion?
• The existence of zero mode in πflux
for chiral superconducting state :
cancelation between the π flux of
vortex and edge chiral angle (winding
numer in momentum space)
• Majorana fermion in chiral p-wave –
mixed annihilation operator and
generation operation
Chiral edge state
Edge state
y
x
Edge Majorana fermion
p+ip superconductor
Chiral fermion propagates along edge
Edge state encircling a droplet
Antiperiodic boundary condition
Spinor rotates by 2π
encircling sample
Vortex (πflux) in px+ipy superconductor
Single vortex
E=0 Majorana fermion encircling sample :
an encircling vortex - a “vortex zero mode”
Fermion picks up π phase around vortex:
Changes to periodic boundary condition
E = nω
“5/2” FQHE
states
Pan et al. PRL 83,1999
Gap at 5/2 is 0.11 K
Xia et al. PRL 93, 2004
Gap at 5/2 is 0.5K, at 12/5: 0.07K
Moore-Read wavefunction for 5/2 FQHE state
“Paired” Hall state
Pfaffian:
Moore, Read (1991)
Greiter, Wen, Wilczek (1992)
Moore/Read = Laughlin × BCS
Ising anyons in the generalized Kitaev model
Gapped B phase are SU(2)2 nonAbelian topological order for K>0.
Boundaries:
•
•
•
Vortex-free: J=1/2
Full-vortex: J=1/√2
Sparse: 1/2 ≤ J ≤ 1/√2
(Jz = 1 and J = Jx = Jy )
px+ipy SC for generalized Kitaev model
by Jordan-Wigner transformation
Y. Yue and Z. Q. Wang, Europhys. Lett. 84, 57002 (2008)
Topological qubits of Ising anyons
• Pairs of Ising anyons : each
anyon binds to a Majorana
fermion, the fermion state of two
anyons is described by a regular
fermion which is a qubit .

d 
1
2
d
1
2
( 1  i 2 )
( 1  i 2 )

d d1  1,

d d 0 0
A qubit
 1  0
Braiding operator for two-anyons
  i   i 1

Ti : 
  
i
 i 1
The braiding matrices are (Ivanov, 2001) :
Ti  exp(

4
 i 1 i ) 
1
2
(1   i 1 i )
1

 


 exp i
2d d  1   exp(i
z)  
0
4
4





0

i

Braiding matrices for the degenerate
states of four Ising anyons




We choose the base : 0 , d1 0 , d 2 0 , d1 d 2 0


where d1  ( 1  i 2 ) / 2, d 2  ( 3  i 4 ) / 2
T12  exp( i
T23 
1
2

4
 z ),
(1)
Two- qubit
(1   y  x ),
T34  exp( i
(1)

4
 z( 2 ) )
( 2)
N matrices
1

N1   0
0

0
1
0
0
0

0  N  1


0
1

F matrices
R matrices

F
1
R  e
0
1
0
0
1 N  0
  
0
1

1 1


2 1

1
1 

 1

F
 1
 
 0
i
8
3 i

R
e
8
0
1
0
1
0

0
0 

 1
Topological Quantum Computation

f
time
f

=
 a11

 
a
 M1
i



a1M 

 
aMM 

i
Topological quantum computation
by Ising anyons
• Two pairs of Ising anyons
R12  T12  exp(i
R matrices of two
pairs of anyons :
braiding operators
R34  T34  exp(i
R23  T23 
1
2

4

4
 z(1) ),
 z( 2 ) ),
(1   y  x )
(1)
( 2)
X gate and Z gate
L.S.Georgiev,
PRB74,235112(2006)
L.S.Georgiev,
PRB74,235112(2006)
Hadmard gate
CNOT gate
L.S.Georgiev,
PRB74,235112(2006)
No π/8 gate
|a
Toffoli gate ?
|b
|c
L. S. Georgiev, PRB74,235112(2006)
|a
|b
| ab  c
1

0

0

0
0

0
0


0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0

0

0

0
0

0
1

0

IV. Topological quantum computation
by braiding Fibonacci anyons
(2) Fibonacci anyon
There are two sectors : I and τ.
Two anyons (τ) can “fuse” two ways.
II  I
Fusion rules
I     I  
   I  
Fibonacci anyon
1 I
1
2     I  
3      I    
5           2( I   )
8            3 I  5
• Fib(n) = Fib(n–1) + Fib(n–2)
Fibonacci anyons
Fib( n)  Fib( n  1)  Fib( n  2)
Fib(1)  Fib( 2)  1
Fib(n) =1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…
Fibonacci anyon
   1  
N matrices
F matrices
R matrices
1
N1  
0

F
1
  1
  1/ 2


R  e
1 0 

N   
1 1 
0

1

4 i
5
1 / 2


1 
 

 
1 5
2
3 i

R
e
5
Other examples of Fibonacci anyon
Possible example of Fibonacci anyon in
“12/5” FQHE state
Read-Rezayi wave-function
 r , s  ( z kr 1  z sk 1 )( z kr 1  z sk  2 ) 
( z kr  2  z sk  2 )( z kr  2  z sk 3 )  ...( z kr  k  z sk  k )( z kr  k  z sk 1 )
Para-fermion state :
bound state of three fermions
N. Read and E. H. Rezayi, Phys. Rev. B 59, 8084 (1999).
Topological Qubit of Fibonacci anyons
1×1=0+1
Two Fibonacci span a 2-dimensional Hilbert space



1
0
To do non-trivial operation we need three Fibonacci anyons




 
 
0
1
1
1
1
0
P. Bonderson et. al
P. Bonderson et. al
P. Bonderson et. al
Single qubit rotation
Fibonacci anyons
Universal
computation
Ising anyons
P. Bonderson et. al
P. Bonderson et. al
P. Bonderson et. al
P. Bonderson et. al
P. Bonderson et. al
P. Bonderson et. al

Topological Quantum Computation
(Kitaev, Preskill, Freedman, Larsen, Wang)



1
0
 

(Bonesteel, et. al.)

Interferom
eter
P. Bonderson et. al
Thank you!