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Anyon and Topological Quantum Computation Su-Peng Kou Beijing Normal university.
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Transcript of Anyon and Topological Quantum Computation Su-Peng Kou Beijing Normal university.
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!