1. Anyon and Topological Quantum Computation Beijing Normal university
Su-Peng Kou
Beijing Normal university

2. Outline Part I: Anyons and braiding group
Part II: Quantum computation of topological qubits in Z2 topological orders
Part III : Topological quantum computation by Ising anyons
IV: Topological quantum computation by Fibonacci anyons
Key words: topological string operator, nonAbelian anyon

3. 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 non-Abelian anyons.
2001, Preskill, Freedman and others proposed a universal topological quantum computation.

4. (I) Anyons and braid groups

5. Abelian statistics via non-Abelian statistics

7. Exchange statistics and braid group
Particle Exchange : world lines braiding

8. Braid group

9. 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).

11. 2. Braiding rules: R matrix
3. Associativity relations for fusion: F matrix

12. Pentagon equation:

13. Hexagon equation:

14. 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.

15. (II) Quantum computation of topological qubits in Z2 topological orders

16. SP Kou, M Levin, and XG Wen, PRB 78, 155134 (2008).
1. Z2 topological order
There are four sectors : I (vacuum), ε(fermion), e (Z2 charge), m (Z2 vortex) ;
Z2 gauge theory
U(1)×U(1) mutual Chern-Simons theory
Topological degeneracy : 4 on torus
SP Kou, M Levin, and XG Wen, PRB 78, (2008).

17. Mutual semion statistics between Z2 vortex and Z2 charge
flux
Z2 vortex
Mutual Flux binding
Fermion as the bound state of a Z2 vortex and a Z2 charge

18. Fusion rule
A. Yu. Kitaev, Ann. Phys. 303,

19. Toric-code model
A．Y．Kitaev，Annals Phys. 303, 2 (2003)

20. Wen-plaquette model
X. G. Wen, PRL. 90,
(2003)

21. Solving the Wen-plaquette model
The energy eigenstates are labeled by the eigenstates of
Because of , the eigenvalues are

22. The energy gap For g>0, the ground state is
The ground state energy is E0=Ng
The elementary excitation is
The energy gap for it becomes

23. 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?

24. 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

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

26. The dynamics of the Z2 Vortex and Z2 charge
Z2 vortex (charge) can only move in the same sub-plaquette:
The hopping operators for Z2 vortex (charge) are

27. 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
Fermion as the bound state of a Z2 vortex and a Z2 charge.
X. G. Wen, PRD68, (2003).

28. Controlling the hopping of quasi-particles by external fields
The hopping operators of Z2 vortex and charge are
The hopping operator of fermion is
So one can control the dynamics of different quasi-particles by applying different external.

29. Closed strings
Open strings

30. String net condensation for the ground states
The string operators：
For the ground state, the closed-strings are condensed

31. The toric-code model There are two kinds of Bosonic excitations:
Z2 vortex
Z2 charge
Fermion as the bound state of a Z2 vortex and a Z2 charge.

32. Controlling the hopping of quasi-particles by external fields
The hopping operator of Z2 vortex is
The hopping operator of Z2 charge is
The hopping operator of fermion is
So one can control the dynamics of different quasi-particles by applying different external fields.

33. 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.
Advantage
No local perturbation can introduce decoherence.
Ioffe, &, Nature 415, 503 (2002)

34. Topology of Z2 topological order
Cylinder
Torus
Disc 1 2 4
Hole on a Disc

35. Topological closed string operators on torus – topological qubits

36. Degenerate ground states as eigenstates of topological closed operators
Algebra relationship:
Define pseudo-spin operators:

37. Topological closed string operators
On torus，pseudo-spin representation of topological closed string operators:
S.P. Kou, PHYS. REV. LETT. 102,  (2009).
J. Yu and S. P. Kou, PHYS. REV. B 80, (2009).
S. P. Kou, PHYS. REV. A 80, (2009).

38. Degenerate ground states as eigenstates of topological closed operators

39. 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.

40. 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)

41. 3. Quantum tunneling effect
of topological qubits : topological closed string representation
Tunneling processes are virtual quasi-particle moves around the periodic direction.

42. Topological closed string operator as a virtual particle hopping

43. Topological closed string operators may connect different degenerate ground states
S.P. Kou, PHYS. REV. LETT. 102,  (2009).
J. Yu and S. P. Kou, PHYS. REV. B 80, (2009).
S. P. Kou, PHYS. REV. A 80, (2009).

44. Higher order perturbation approach
Energy splitting : lowest order contribution of topological closed string operators
L0 is the length of topological closed string operator

45. The energy splitting from higher order (degenerate) perturbation approach
L : Hopping steps of quasi-particles
teff : Hopping integral
: Excited energy of quasi-particles
J. Yu and S. P. Kou, PHYS. REV. B 80, (2009).

46. Topological closed string operators of four degenerate ground states for the Wen-plaquette model under x- and z-component external fields

47. Effective model of four degenerate ground states for the Wen-plaquette model under x- and z-component external fields

48. External field along z direction
In anisotropy limit, the four degenerate ground states split two groups,
2×6 lattice on the Wen-plaquette model under z direction field

49. External field along z direction
Isotropy limit， the four degenerate ground states split three groups
4×4 lattice on Wen-plaquette model under z-direction

50. External field along x direction
Under x-direction field, the four degenerate ground states split three groups:
4×4 lattice on Wen-plaquette model under x-direction

51. Ground states energy splitting of Wen-plaqutte model on torus under a magnetic field along x-direction

52. Ground states energy splitting of Wen-plaqutte model on torus under a magnetic field along z-direction

53. Planar codes : topological qubits on surface with holes
Fermionic based
L. B. Ioffe, et al., Nature 415, 503 (2002).

54. Effective model of the degenerate ground states of multi-hole
S.P. Kou, PHYS. REV. LETT. 102,  (2009).
S. P. Kou, PHYS. REV. A 80, (2009).
The four parameters Jz, Jx, hx, hz are determined by the quantum effects of different quasi-particles.

55. Unitary operations A general operator becomes :
For example , Hadamard gate is

56. CNOT gate and quantum entangled state of topological qubits
S. P. Kou, PHYS. REV. A 80, (2009).

57. III. Topological quantum computation by braiding Ising anyons

58. Topological Quantum Computation
Physics
measure
output
operation
braid
create
particles
initialize
Eric Rowell

59. (I) Ising anyons
Fusion rules:

60. σ:π-Flux binding a Majorana Fermion
Ising anyons
Majorana fermion
Another anyon
σ:π-Flux binding a Majorana Fermion
flux
SU(2)2 non-Abelian statistics between π-flux with a trapped majorana fermion.

61. µ>0, non-Abelian Topologial state µ<0, Abelian Topologial state
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.

62. Winding number in momentum space

63. BdG equation of px+ipy superconductor
Bogoliubov deGennes Hamiltonian:
Eigenstates in +/- E pairs
Spectrum with a gap
Excitations: Fermionic quasiparticles above the gap

64. BdG equation of vortex in px+ipy superconductor

65. 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

66. Chiral edge state Edge state y x p+ip superconductor
Edge Majorana fermion
p+ip superconductor
Chiral fermion propagates along edge
Edge state encircling a droplet
Spinor rotates by 2π
encircling sample
Antiperiodic boundary condition

67. 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

68. E = nω

69. “5/2” FQHE states Pan et al. PRL 83,1999 Xia et al. PRL 93, 2004
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

70. Moore-Read wavefunction for 5/2 FQHE state
Greiter, Wen, Wilczek (1992)
“Paired” Hall state
Pfaffian:

71. Moore/Read = Laughlin × BCS

72. Ising anyons in the generalized Kitaev model
Gapped B phase are SU(2)2‏ non- Abelian 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 )‏

73. px+ipy SC for generalized Kitaev model by Jordan-Wigner transformation
Y. Yue and Z. Q. Wang, Europhys. Lett. 84, (2008)

74. 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

75. Braiding operator for two-anyons
The braiding matrices are (Ivanov, 2001) :

76. Braiding matrices for the degenerate states of four Ising anyons
Two- qubit

77. N matrices
F matrices
R matrices

78. Topological Quantum Computationf Y
time i Y = f i Y

79. Topological quantum computation by Ising anyons
Two pairs of Ising anyons
R matrices of two pairs of anyons : braiding operators

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

81. L.S.Georgiev, PRB74,235112(2006)
Hadmard gate

82. L.S.Georgiev, PRB74,235112(2006)
CNOT gate

83. No π/8 gate
Toffoli gate ?
L. S. Georgiev, PRB74,235112(2006)

84. IV. Topological quantum computation by braiding Fibonacci anyons

85. (2) Fibonacci anyon There are two sectors : I and τ.
Two anyons (τ) can “fuse” two ways.
Fusion rules

86. Fibonacci anyon
Fib(n) = Fib(n–1) + Fib(n–2)

87. Fibonacci anyons
=1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

88. Fibonacci anyon
N matrices
F matrices
R matrices

89. Other examples of Fibonacci anyon

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

91. Read-Rezayi wave-function
Para-fermion state :
bound state of three fermions
N. Read and E. H. Rezayi, Phys. Rev. B 59, 8084 (1999).

92. Topological Qubit of Fibonacci anyons
1 × 1 = 0 + 1
Two Fibonacci span a 2-dimensional Hilbert space 1
To do non-trivial operation we need three Fibonacci anyons 1 1 1

94. P. Bonderson et. al

95. P. Bonderson et. al

96. P. Bonderson et. al

97. Single qubit rotation Fibonacci anyons Universal computation
Ising anyons

98. P. Bonderson et. al

99. P. Bonderson et. al

100. P. Bonderson et. al

101. P. Bonderson et. al

102. P. Bonderson et. al

103. P. Bonderson et. al

104. Topological Quantum Computation (Kitaev, Preskill, Freedman, Larsen, Wang)1
(Bonesteel, et. al.)

105. P. Bonderson et. al

106. Thank you!