1. F Matrix: F F F F F Pentagon Fusion rules: 1x1 = 0+1; 1x0 = 0x1 = 1; 0x0 = 0 Fibonacci anyon with topological “charge” 1

2. a a The R Matrix

3. a a R

4. F R R F F R

5. F F R F R R The Hexagon Equation

6. 0 1 0 1

7. 0 1 0 1

8. 0 1 0 1 1

9. 0 1 0 1 a 1

10. 0 1 0 1 a 1 F 0a

11. 0 1 0 1 The Hexagon Equation a 1 F 0a

12. 0 1 0 1 The Hexagon Equation a 1 1 a F 0a

13. 0 1 0 1 The Hexagon Equation a 1  a,0 + R 1  a,1 1 a F 0a

14. 0 1 0 1 The Hexagon Equation a 1  a,0 + R 1  a,1 1 a F 0a

15. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0

16. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 0 1

17. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1

18. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1

19. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1 1

20. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1 1 0

21. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1 1 0 F 00

22. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1 1 0 F 00

23. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1 1 0 F 00 R0R0

24. 0 1 0 1 The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R0 0 1 1 0 F 00 R0R0 Top path Bottom path

25. The Hexagon Equation Only two solutions: F F R F R R

26. Basic Content of an Anyon Theory F Matrix: R Matrix: F F F F F F F R F R R Pentagon Hexagon “Charge” values: 0, 1 Fusion rules: 1x1 = 0+1; 1x0 = 0x1 = 1; 0x0 = 0

27. Qubit Encoding 0 1 Non-Computational State State of qubit is determined by “charge” of two leftmost particles Qubit States 1 0 Transitions to this state are leakage errors 1 1

28. Elementary Braid Matrices a 1 time

29. a 1 Elementary Braid Matrices a = 0 1

30. a 1 time a 1 Elementary Braid Matrices

31. a 1 time a 1 Elementary Braid Matrices

32. a 1  1   2  1   2 = M i  f  M T = i  a 1 1 1 Elementary Braid Matrices

33. A Quantum Bit: A Continuum of States   1 2 sin0 2 cos    i e  

34. Single Qubit Operations: Rotations 1 2 sin0 2 cos    i e     =  rotation vector Direction of  is the rotation axis Magnitude of  is the rotation angle

35. Single Qubit Operations: Rotations  =  rotation vector      U   U

36. Single Qubit Operations: Rotations 2   2  2   2  The set of all single qubit rotations lives in a solid sphere of radius   Qubits are sensitive: Easy to make errors!     U   U

37. 2   2  2   2  1212  1 -2  2 -2  2  2222

38. N = 1

39. N = 2

40. N = 3

41. N = 4

42. N = 5

43. N = 6

44. N = 7

45. N = 8

46. N = 9

47. N = 10

48. N = 11

49. Brute Force Search

50. Braid Length Brute Force Search Braid Length For brute force search: |ln  L. Hormozi, G. Zikos, NEB, S.H. Simon, PRB ‘07  “error”

51. Beyond Brute Force Search Given a sufficiently dense covering of SU(2), braids can be systematically improved by Solovay-Kitaev (see, e.g., Dawson and Nielson, RMP 2002). Braid Length, Using ideas from algebraic number theory can asymptotically achieve: Braid Length Recent Development

52. Single Qubit Operations time U

53. X Two Qubit Gates (CNOT) NEB, L. Hormozi, G. Zikos, & S. Simon, PRL 2005 L. Hormozi, G. Zikos, NEB, and S. H. Simon, PRB 2007

54. Other Two-Qubit Gates L. Hormozi, G. Zikos, NEB, and S.H. Simon, PRB ‘07. Xu and Wan, PRA ‘08. C.Carnahan, D. Zeuch, and NEB, arXiv:1511.00719.

55. X X X X UU Quantum Circuit

56. X X X X UU

57. Braid

58. SU(2) k Nonabelian Particles Quasiparticle excitations of the =12/5 state (if it is a Read- Rezayi state) are described (up to abelian phases) by SU(2) 3 Chern-Simons theory. Universal for quantum computation for k=3 and k > 4. Slingerland and Bais, 2001 Read and Rezyai, 1999 Freedman, Larsen, and Wang, 2001 But before SU(2) k, there was just plain old SU(2)…. In fact, Read and Rezayi proposed an infinite sequence of nonabelian states labeled by an index k with quasiparticle excitations described by SU(2) k Chern-Simons theory.

59. 1.Particles have spin s = 0, 1/2, 1, 3/2, … spin = ½ 2. “Triangle Rule” for adding angular momentum: Particles with Ordinary Spin: SU(2) Two particles can have total spin 0 or 1.   01 For example: Numbers label total spin of particles inside ovals

60. Hilbert Space

61. N 0 7/2 3 5/2 2 3/2 1 1/2 S 12345678109 1112 Hilbert Space 1 1/2 1 3/2 Paths are states

62. N 0 1 1 1 1 2 1 3 2 5 9 5 1 4 5 6 14 20 28 14 1 1 7 7/2 3 5/2 2 3/2 1 1/2 S 12345678109 1112 27 48 42 8 55 110 132 44 20 90 42 35 132 Hilbert Space

63. 1.Particles have spin s = 0, 1/2, 1, 3/2, … spin = ½ 2. “Triangle Rule” for adding angular momentum: Particles with Ordinary Spin: SU(2) Two particles can have total spin 0 or 1.   01 For example:

64. 1.Particles have topological charge s = 0, 1/2, 1, 3/2, …, k/2 topological charge = ½ 2. “Fusion Rule” for adding topological charge: Nonabelian Particles: SU(2) k Two particles can have total topological charge 0 or 1.   01 For example:

65. N 0 1 1 1 1 2 1 3 2 5 9 5 1 4 5 6 14 20 28 14 1 1 7 7/2 3 5/2 2 3/2 1 1/2 S 12345678109 1112 27 48 42 8 110 165 132 44 75 90 42 35 132 Dim(N) ~ 2 N

66. k = 4 Dim(N) ~ 3 N/2

67. k = 3 Dim(N) = Fib(N+1) ~  

68. N 0 1 1 1 2 2 2 4 4 4 8 8 8 7/2 3 5/2 2 3/2 1 1/2 S 12345678109 1112 16 32 16 32 k = 2 Dim(N) = 2 N/2

69. The F Matrix a c b c = 0 1/2 1 1 3/2 0 1/2 1 1 3/2 q-integers: ;

70. 0 1 0 1 The R Matrix R matrix

71. k=2 k=4 k=3 k=5 N = 1

72. k=2 k=4 k=3 k=5 N = 2

73. k=2 k=4 k=3 k=5 N = 3

74. k=2 k=4 k=3 k=5 N = 4

75. k=2 k=4 k=3 k=5 N = 5

76. k=2 k=4 k=3 k=5 N = 6

77. k=2 k=4 k=3 k=5 N = 7

78. k=3 k=5 k=2 k=4 N = 8

79. k=2 k=4 k=3 k=5 N = 9

80. k=6 k=8 k=7 k=9 N = 1

81. k=6 k=8 k=7 k=9 N = 2

82. k=6 k=8 k=7 k=9 N = 3

83. k=6 k=8 k=7 k=9 N = 4

84. k=6 k=8 k=7 k=9 N = 5

85. k=6 k=8 k=7 k=9 N = 6

86. k=6 k=8 k=7 k=9 N = 7

87. k=6 k=8 k=7 k=9 N = 8

88. k=6 k=8 k=7 k=9 N = 9

89. X CNOT gate for SU(2) 5 L. Hormozi, NEB, & S.H. Simon, PRL 2009

90. “Surface Code” Approach Key Idea: Use a quantum computer to simulate a theory of anyons. More speculative idea: simulate Fibonacci anyons. The simulation “inherits” the fault-tolerance of the anyon theory. Most promising approach is based on “Abelian anyons.” High error thresholds, but can’t compute purely by braiding. Bravyi & Kitaev, 2003 Raussendorf & Harrington, PRL 2007 Fowler, Stephens, and Groszkowski, PRA 2009 Konig, Kuperberg, Reichardt, Ann. Phys. 2010 NEB, D.P. DiVincenzo, PRB 2012

91. 2D Array of Qubits

92. Trivalent Lattice

94. “Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005

95. Trivalent Lattice “Fibonacci” Levin-Wen Model Vertex Operator v Q v = 0,1 Levin & Wen, PRB 2005

96. Trivalent Lattice “Fibonacci” Levin-Wen Model Vertex Operator Plaquette Operator v p Q v = 0,1 B p = 0,1 Levin & Wen, PRB 2005

97. Trivalent Lattice “Fibonacci” Levin-Wen Model Vertex Operator Plaquette Operator v p Q v = 0,1 B p = 0,1 Levin & Wen, PRB 2005

98. Trivalent Lattice Vertex Operator Plaquette Operator v p Ground State Q v = 1 on each vertex B p = 1 on each plaquette Q v = 0,1 B p = 0,1 “Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005

99. Trivalent Lattice Vertex Operator Plaquette Operator v p Ground State Q v = 1 on each vertex B p = 1 on each plaquette Excited States are Fibonacci Anyons Q v = 0,1 B p = 0,1 “Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005

100. Vertex Operator: Q v i j k v i j k v All other “Fibonacci” Levin-Wen Model

101. Vertex Operator: Q v i j k v i j k v on each vertex “branching” loop states. All other “Fibonacci” Levin-Wen Model

102. Plaquette Operator: B p Very Complicated 12-qubit Interaction! on each plaquette superposition of loop states f a b d c m j k l n i e p f a b d c m’m’ j’j’ k’k’ l’l’ n’n’ i’i’ e p

103. Plaquette Operator: B p on each plaquette superposition of loop states Very Complicated 12-qubit Interaction! f a b d c m j k l n i e p f a b d c m’m’ j’j’ k’k’ l’l’ n’n’ i’i’ e p

104. Plaquette Operator: B p on each plaquette superposition of loop states Very Complicated 12-qubit Interaction! f a b d c m j k l n i e p f a b d c m’m’ j’j’ k’k’ l’l’ n’n’ i’i’ e p

105. Plaquette Operator: B p on each plaquette superposition of loop states Very Complicated 12-qubit Interaction! f a b d c m j k l n i e p f a b d c m’m’ j’j’ k’k’ l’l’ n’n’ i’i’ e p

106. “Surface Code” Approach v p Use quantum computer to repeatedly measure Q v and B p. How hard is it to measure Q v and B p with a quantum computer? If Q v =1 on each vertex and B p =1 on each plaquette, good. If not, treat as an “error.” Konig, Kuperberg, Reichardt, Ann. Phys. 2010

107. Quantum Circuit for Measuring Q v 1 2 3 v 1 2 3 XXXX NEB, D.P. DiVincenzo, PRB 2012

108. Quantum Circuit for Measuring Q v 1 2 3 v 1 2 3 XXXX Syndrome qubit NEB, D.P. DiVincenzo, PRB 2012

109. Quantum Circuit for Measuring Q v That was easy! What about B p ? 1 2 3 v 1 2 3 XXXX Syndrome qubit NEB, D.P. DiVincenzo, PRB 2012

110. 6-sided Plaquette: B p is a 12-qubit Operator

111. Physical qubits are fixed in space 6-sided Plaquette: B p is a 12-qubit Operator

112. Physical qubits are fixed in space Trivalent lattice is abstract and can be “redrawn” 6-sided Plaquette: B p is a 12-qubit Operator

113. Physical qubits are fixed in space Trivalent lattice is abstract and can be “redrawn”

114. Physical qubits are fixed in space Trivalent lattice is abstract and can be “redrawn” 6-sided Plaquette: B p is a 12-qubit Operator

115. Physical qubits are fixed in space Trivalent lattice is abstract and can be “redrawn” 6-sided Plaquette: B p is a 12-qubit Operator 6-sided plaquette becomes 5-sided plaquette!

116. The F–Move Locally redraw the lattice five qubits at a time.

117. The F–Move ad bc e e’e’ ad bc Apply a unitary operation on these five qubits to stay in the Levin-Wen ground state.

118. a c e b d = F X X X X X ad bc e e’e’ ad bc F Quantum Circuit NEB, D.P. DiVincenzo, PRB 2012

119. a c e b d 1 2 3 4 5 ad bc e e’e’ ad bc F Quantum Circuit 1 5 4 3 2 1 5 4 3 2 NEB, D.P. DiVincenzo, PRB 2012

120. Plaquette Reduction

145. 2-qubit “Tadpole”

146. Measuring B p for a Tadpole is Easy! S Quantum Circuit a b a b a b X = a b S XX

147. Plaquette Restoration

159. 6 1 2 4 3 12 9 10 11 7 8 5 p Quantum Circuit for Measuring B p NEB, D.P. DiVincenzo, PRB 2012

160. Gate Count 371 CNOT Gates 392 Single Qubit Rotations 8 5-qubit Toffoli Gates 2 4-qubit Toffoli Gates 10 3-qubit Toffoli Gates 43 CNOT Gates 24 Single Qubit Gates 6 1 2 4 3 12 9 10 11 7 8 5 p NEB, D.P. DiVincenzo, PRB 2012

161. B p = ? 5 4 3 2 1 6 12 7 8 9 10 11 Plaquette Swapping

162. Freshly initialized “tadpole” in LW ground state B p = ? 5 4 3 2 1 6 12 7 8 9 10 11 13 14 15 Plaquette Swapping

163. B p =1 B p = ? Plaquette Swapping

164. B p =1 B p = ? Plaquette Swapping

165. B p =1 B p = ? Plaquette Swapping

166. B p =1 B p = ? Plaquette Swapping

167. B p =1 B p = ? Plaquette Swapping

168. B p =1 B p = ? Plaquette Swapping

169. B p =1 B p = ? Plaquette Swapping

170. B p =1 B p = ? Plaquette Swapping

171. B p =1 B p = ? Plaquette Swapping

172. B p =1 B p = ? Plaquette Swapping

173. B p =1 B p = ? Plaquette Swapping

174. B p =1 B p = ? Plaquette Swapping

175. B p =1 B p = ? Plaquette Swapping

176. B p =1 B p = ? Plaquette Swapping

177. B p =1 B p = ? Plaquette Swapping

178. B p =1 B p = ? Plaquette Swapping

179. B p =1 B p = ? Plaquette Swapping

180. B p =1 Measure B p Plaquette Swapping

181. Plaquette Swapping Circuit = SWAP W. Feng, NEB, D.P. DiVincenzo, Unpublished

182. B p =1 Measure B p After Swapping

183. B p =1 No Error After Swapping

184. B p =1 Simply Remove Swapped Plaquette After Swapping

185. B p =1 B p =0 Error After Swapping

186. B p =1 B p =0 Error Move “Packaged” Error Using F-Moves After Swapping

187. B p =1 Move “Packaged” Error Using F-Moves After Swapping

188. B p =1 Move “Packaged” Error Using F-Moves After Swapping

189. B p =1 Move “Packaged” Error Using F-Moves After Swapping

190. B p =1 Move “Packaged” Error Using F-Moves After Swapping

191. B p =1 Move “Packaged” Error Using F-Moves After Swapping

192. B p =1 Move “Packaged” Error Using F-Moves After Swapping

193. B p =1 Move “Packaged” Error Using F-Moves After Swapping

194. B p =1 Move “Packaged” Error Using F-Moves After Swapping

195. B p =1 Move “Packaged” Error Using F-Moves After Swapping

196. 1234 5 6 7

197. 1234 5 6 7 5 6 7 1234

198. 5 5 6 7 1234 6 7 1234 1234 5 6 7

199. 6 5 6 7 1234 5 6 7 1234 5 7 1234 1234 5 6 7

200. 5 6 1234 5 6 7 1234 7 5 6 7 1234 5 6 7 1234 1234 5 6 7

201. 6 1234 7 1234 5 7 1234 5 6 7 5 6 7 1234 5 6 7 1234 1234 5 6 7 6 5

202. 6 6 1234 6 5 7 1234 5 7 1234 5 7 5 6 7 1234 5 6 7 1234 1234 5 6 7

203. 6 6 1234 6 5 7 SWAP 1234 5 7 1234 5 7 5 6 7 1234 5 6 7 1234 1234 5 6 7

204. 3 4 5 6 1 2 7 b e a d c d e c b a c e b a d a e d c b c e b a d = NEB, D.P. DiVincenzo, PRB 2012 Pentagon Equation as a Quantum Circuit

205. 3 4 5 6 1 2 7 b e a d c d e c b a c e b a d a e d c b c e b a d = SWAP NEB, D.P. DiVincenzo, PRB 2012 Pentagon Equation as a Quantum Circuit

206. Simplified Pentagon Circuit = SWAP F F F F F NEB, D.P. DiVincenzo, PRB 2012

207. Creating Excited States: Fibonacci Anyons Konig, Kuperberg, Reichardt, Ann. Phys. 2010

208. Vertex Errors = String Ends

209. String Ends = Fibonacci Anyons Fibonacci anyons

210. Konig, Kuperberg, Reichardt, Ann. Phys. 2010

211. Konig, Kuperberg, Reichardt, Ann. Phys. 2010

213. 1 F-moves

215. 4 F-moves

217. 8 F-moves

220. 32 F-moves

221. 48 F-moves

223. 51 F-moves

225. 56 F-moves

229. Braid Dehn Twist Konig, Kuperberg, Reichardt, Ann. Phys. 2010

245. Non-Abelian Anyons and Topological Quantum Computation, C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008). (arXiv:0707.1889v2) Lectures on Topological Quantum Computation, J. Preskill, Available online at: www.theory.caltech.edu/~preskill/ph219/topological.pdf Some excellent reviews: Support: US DOE NEB, L. Hormozi, G. Zikos, S.H. Simon, Phys. Rev. Lett. 95 140503 (2005). S.H. Simon, NEB, M.Freedman, N, Petrovic, L. Hormozi, Phys. Rev. Lett. 96, 070503 (2006). L. Hormozi, G. Zikos, NEB, and S.H. Simon, Phys. Rev. B 75, 165310 (2007). L. Hormozi, NEB, and S.H. Simon, Phys. Rev. Lett. 103, 160501 (2009). M. Baraban, NEB, and S.H. Simon, Phys. Rev. A 81, 062317 (2010). NEB and D.P. DiVincenzo, Phys. Rev. B 86, 165113 (2012).