Anyon and Topological Anyon and Topological Quantum ComputationQuantum Computation

Su-Peng Kou

Beijing Normal university

Outline1. 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 topological string operator, nonAbelian

anyonanyon

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 non-Abelian anyons.

2001, Preskill, Freedman and others proposed a universal topological quantum computation.

Milestone for topological quantum computation

(I) Anyons and braid groups

1221 ffff

Fermion

,, 1221 bbbb

Boson

,,

,,,, 01221 ie

anyonAbelian

matrixaisMM

anyonAbeliannon

,,, 1221

Abelian statistics via non-Abelian statistics

. angle lstatistica the,2 ,for E.g.

phase Bohm-Aharonov The

π/n/neΦeq

qΦ

'82) (Wilczek, composites flux - charge

:AnyonsAbelian of model)(toy Example

Φq

Exchange statistics and braid group

Particle Exchange : world lines braiding

Braid group

i

k

0],[ ki

| i k | 2

i1 i

i i1

i i1 i

i1 i i1

12 i

Baxter-Yang

General anyon theory

1. A finite set of quasi-particles or anyonic “charges.”

2. Fusion rulesFusion rules (specifying how charges can combine or split).

3. Braiding rulesBraiding rules (specifying behavior under particle exchange).

matrix N :rules Fusion1.

cNbac

cab

f

efabc

dF ][

b a b a

abcR

3. Associativity relations for fusion: F matrix

2. Braiding rules: R matrix

Pentagon equation:

Hexagon equation:

Non-Abelian statisticsExchanging particles 1 and 2:

• Matrices M12 and M23 don’t commute;• Matrices M form a higher-dimensional representation

of the braid-group.

Exchanging particles 2 and 3:

(II) Quantum computation of (II) Quantum computation of topological qubitstopological qubits

in Z2 topological ordersin Z2 topological orders

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).

1. Z2 topological order

flux

Mutual semion statistics between Z2 vortex and Z2 chargeMutual semion statistics between Z2 vortex and Z2 charge

Z2 vortex

Z2 charge

Fermion as the bound state of a Z2 vortex and a Z2 charge

Mutual Flux binding

Fusion rule

A. Yu. Kitaev, Ann. Phys. 303, 2 2003.

Toric-code model

A ． Y ． Kitaev ， Annals Phys. 303, 2 (2003)

BPA

S

Wen-plaquette model

yei

xeei

yei

xii

ii

yyxxF

FgH

ˆˆˆˆˆ

,ˆ

X. G. Wen, PRL. 90,

016803 (2003)

y

y

x

x

• The energy eigenstates are labeled by the eigenstates of

• Because of , the eigenvalues are

0ˆ,ˆ0,ˆ jii FFHF

Solving the Wen-plaquette model

1ˆ 2 iF

iF̂

11 ii FandF

yei

xeei

yei

xii

ii yyxx

FFgH ˆˆˆˆˆ,ˆ

• For g>0, the ground state is

The ground state energy is E0=Ng

The elementary excitation is

The energy gap for it becomes

1iF

1iF

1,201 iFforgEE

The energy gapy

eix

eeiy

eixii

ii yyxx

FFgH ˆˆˆˆˆ,ˆ

The statistics for the elementary excitations

• There are two kinds of Bosonic excitations: • Z2 vortex

• Z2 charge

• Each kind of excitations moves on each sub-plaquette:

• Why?

1 eveniii yxF

-1

1

1 1

1 1 1

1 1

-1

1

1 11

1 1

1 1

1 oddiii yxF

• There are two constraints (the even-by-even lattice): One for the even plaquettes, the other for the odd plaquettes

• The hopping from even plaquette to odd violates the constraints :

You cannot change

a Z2 vortex into a Z2 charge -1

1

1 1

1 1 1

1 1

1

eveniiieveniii

yx

yx

F 1

oddiiioddiii

yx

yx

F

-1

1

1 1

1 1 1

1 1

• On an even-by-even lattice, there are totally

states

• Under the constaints,

the number of states are only

• For the ground state , it must be four-fold degeneracy.

Topological degeneracy on a torus (even-by-even lattice) :

11 22

ioddii

ievenii

FandFyxyx

N2

4

2N

1iF

yx LLN

• Z2 vortex (charge) can only move in the same sub-plaquette:

• The hopping operators for Z2 vortex (charge) are

The dynamics of the Z2 Vortex and Z2 charge

yi

xi and

ieix

ieix FF yy

ˆˆ ˆˆ

iiy

iiy FF ˆˆ

X. G. Wen, PRD68, 024501 (2003).

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 odd-plaquette, the operator is

• it is -1 with an excitation on it

• This is the character for mutual mutual semion statistics

yei

xeei

yei

xii yyxx

F ˆˆˆˆ

1iF

Fermion as the bound state of a Z2 vortex and a Z2 charge.

• The hopping operators of Z2 vortex and charge are

Controlling the hopping of quasi-particles by external fields

yi

xi and

• The hopping operator of fermion is

zi

So one can control the dynamics of different quasi-particles by applying different external.

• Closed strings

• Open strings

String net condensation for the ground states

The string operators：

For the ground state, the closed-strings are condensed

WcC， WvC 和 WfC，

xei

xeei

xei

xii

zei

zeei

zei

zii

oddii

evenii

yyxx

yyxx

X

ZXgZgH

ˆˆˆˆ

ˆˆˆˆ ,,

The toric-code model

• There are two kinds of Bosonic excitations:

• Z2 vortex • Z2 charge

1 eveniii yxZ

1 oddiii yxX

Fermion as the bound state of a Z2 vortex and a Z2 charge.

• The hopping operator of Z2 vortex is

Controlling the hopping of quasi-particles by external fields

xi

• The hopping operator of fermion is

zi

So one can control the dynamics of different quasi-particles by applying different external fields.

• The hopping operator of Z2 charge is

yi

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.

Advantage

No local perturbation can introduce decoherence.

10

2. Topological qubit

Ioffe, &, Nature 415, 503 (2002)

Topology of Z2 topological order

E

Cylinder Torus

E

Disc

E

1 2 4

Hole on a Disc

Topological closed string operators on torus – topological qubits

Degenerate ground states as eigenstates of topological closed operators

• Define pseudo-spin operators:

• Algebra relationship:

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

lz (l 1, 2)

m1, m2 m1 m2

ml 0 lz ml ml,

ml 1 lz ml ml.

Toric codes : topological qubits topological qubits on toruson 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 How to control the topological qubits? topological qubits?

A. Y. Kitaev :

“Unfortunately, I do not know any way this Unfortunately, I do not know any way this

quantum information can get in or out. Too quantum information can get in or out. Too

few things can be done by moving abelian few things can be done by moving abelian

anyons. All other imaginable ways of accessing anyons. All other imaginable ways of accessing

the ground state are uncontrollablethe ground state are uncontrollable.”

A ． Y ． Kitaev ， Annals Phys. 303, 2 (2003)

3. Quantum tunneling effectof topological qubits : topological closed

string representation

Tunneling processes are

virtual quasi-particle

moves around the

periodic direction.

E

Cylinder Torus

E

Disc

E

1 2 4

E

Cylinder Torus

E

Disc

E

1 2 4

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

• Energy splitting : lowest order contribution of topological closed string operators

UI0, T exp i

0H I

tdt， H It e i H0tH Ie i H0t.

UI0, |m j 0

U I

j0, |m

U Ij 00, |m 1

E0 H 0

HI j |m.

E m,nn HIU I

L 10, |m.

E m,n

n HI 1E0 H 0

HIL0 1|m. #

L0 is the length of topological closed string operator

The energy splitting from higher order (degenerate) perturbation approach

L

efftE

L : Hopping steps of quasi-particlesteff : Hopping integral : Excited energy of quasi-particles

J. Yu and S. P. Kou, PHYS. REV. B 80, 075107 (2009).

E m,n

n HI 1E0 H 0

HIL0 1|m. #

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

x- and z-component external fields

External field along z direction

• In anisotropy limit, the four degenerate ground states split two groups, Lx Ly,

E1 h 2z , E2 h 2

z , E3 h 2z 和 E4 h 2

z。E1 E2 2h 2

z .E1 E3 2h 1z，

2×6 lattice on the Wen-plaquette model under z direction field

External field along z direction

• Isotropy limit ， the four degenerate ground states split three groups

Lx Ly,

E1 E2 0, E3 2h 1z 和 E4 2h 1

z .

4×4 lattice on Wen-plaquette model under z-direction

External field along x direction

• Under x-direction field, the four degenerate ground states split three groups:

E1 2Jxx , E2 2Jxx ,

E3 E4 Jzz 0, #

4×4 lattice on Wen-plaquette model under x-direction

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

flux

Planar codes : topological qubits topological qubits on surface with holeson surface with holes

L. B. Ioffe, et al., Nature 415, 503 (2002).

Fermionic based

Effective model of the degenerate ground states of multi-hole

i

xi

xi

i

zi

zi

ij

xj

xi

xij

ij

zj

zi

zijeff hhJJH

The four parameters Jz, Jx, hx, hz are determined by the quantum effects of different quasi-particles.

S.P. Kou, PHYS. REV. LETT. 102, 120402 (2009).S. P. Kou, PHYS. REV. A 80, 052317 (2009).

Unitary operations

• A general operator becomes :

zxziii

eeeU

For example , Hadamard gate is

i

xi

xi

i

zi

zi

ij

xj

xi

xij

ij

zj

zi

zijeff hhJJH

CNOT gate and quantum entangled state of topological qubits

S. P. Kou, PHYS. REV. A 80, 052317 (2009).

III. Topological quantum computation III. Topological quantum computation by braiding Ising anyonsby braiding Ising anyons

initialize create particles

operation braid

output measure

ComputationComputation PhysicsPhysics

Eric Rowell

Topological Quantum Computation

(I) Ising anyons

1

1

1

1

Fusion rules:

flux

Ising anyons

SU(2)2 non-Abelian statistics between π-flux with a trapped majorana fermion.

Another anyon

Majorana fermion

σ:π-Flux binding a 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

y

x

p+ip superconductor

Edge state

Edge Majorana fermion

Chiral fermion propagates along edge

Edge state encircling a droplet

Antiperiodic boundary conditionSpinor rotates by 2π encircling sample

Vortex (πflux) in px+ipy superconductor

Single vortex

Fermion picks up π phase around vortex: Changes to periodic boundary condition

E=0 Majorana fermion encircling sample : an encircling vortex - a “vortex zero mode”

E = nω

“5/2” FQHE states

Pan et al. PRL 83,1999Gap at 5/2 is 0.11 K

Xia et al. PRL 93, 2004Gap at 5/2 is 0.5K, at 12/5: 0.07K

Moore-Read wavefunction for 5/2 FQHE state

Moore, Read (1991)Greiter, Wen, Wilczek (1992)“Paired” Hall state

Pfaffian:

Moore/Read = Laughlin × BCS

Ising anyons in the generalized Kitaev model

Gapped B phaseGapped B phase are SU(2) are SU(2)22 non- non-Abelian topological order for Abelian topological order for KK>0.>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 modelby 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 .

A qubit

)(

)(

21

21

2

12

1

id

id

00

11

dd

dd ,

01

Braiding operator for two-anyons

ii

ii

iT

1

1

:

iiddi

T

z

iiiii

0

01

412

4

12

1

4 11

)exp(exp

)()exp(

The braiding matrices are (Ivanov, 2001) :

Braiding matrices for the degenerate states of four Ising anyons

)4

exp(

),1(2

1

),4

exp(

)2(34

)2()1(23

)1(12

z

xy

z

iT

T

iT

22

0000

432211

2121

/)(,/)(

,,,:

ididwhere

ddddbasethechooseWe

Two- qubit

N matrices

100

010

001

1N

N 0 1 0

1 0 1

0 1 0

N 0 0 1

0 1 0

1 0 0

11

11

2

1F

10

01F

R1 e

i

8

R e

3 i

8R matrices

F matrices

MMM

M

aa

aa

1

111

i=f

i

f

time

Topological Quantum Computation

Topological quantum computation by Ising anyons

• Two pairs of Ising anyons

R matrices of two pairs of anyons :

braiding operators)(

),exp(

),exp(

)()(

)(

)(

212323

23434

11212

12

14

4

xy

z

z

TR

iTR

iTR

X gate and Z gate L.S.Georgiev, PRB74,235112(2006)

Hadmard gate

L.S.Georgiev, PRB74,235112(2006)

CNOT gateL.S.Georgiev, PRB74,235112(2006)

No π/8 gate

Toffoli gate ?

a|

b|b|

c|

a|

cab|

01000000

10000000

00100000

00010000

00001000

00000100

00000010

00000001

L. S. Georgiev, PRB74,235112(2006)

IV. Topological quantum computation IV. Topological quantum computation by braiding Fibonacciby braiding Fibonacci anyonsanyons

(2) Fibonacci anyon

There are two sectors : I and τ.

Two anyons (τ) can “fuse” two ways.

I

II

III

Fusion rules

Fibonacci anyon

• Fib(n) = Fib(n–1) + Fib(n–2)

538

25

3

2

1

1

I

I

I

I

I

)(

Fibonacci anyons

)()()( 21 nFibnFibnFib

121 )()( FibFib

)(nFib =1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

N matrices

1 1

Fibonacci anyon

10

011N

11

01N

121

211

/

/

F 2

51

R1 e

4 i

5

R e

3 i

5R matrices

F matrices

Other examples of Fibonacci anyon

Possible example of Fibonacci anyon in “12/5” FQHE state

Read-Rezayi wave-function

Para-fermion state : bound state of three fermions

N. Read and E. H. Rezayi, Phys. Rev. B 59, 8084 (1999).

))(...())((

))((,

13222

2111

skkkrkskkkrskkrskkr

skkrskkrsr

zzzzzzzz

zzzz

Topological Qubit of Fibonacci anyons

1 × 1 = 0 + 1 1 × 1 = 0 + 1

Two Fibonacci span a 2-dimensional Hilbert spaceTwo Fibonacci span a 2-dimensional Hilbert space

0 1

010 1 1 1

To do non-trivial operation we need three Fibonacci anyonsTo do non-trivial operation we need three Fibonacci anyons

P. Bonderson et. al

P. Bonderson et. al

P. Bonderson et. al

Single qubit rotation

Universal Universal computationcomputation

Ising anyons

Fibonacci 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

eterInterferom

(Bonesteel, et. al.)

Topological Quantum Computation(Kitaev, Preskill, Freedman, Larsen, Wang)

0 1

P. Bonderson et. al

Thank you!