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

a a The R Matrix

a a R

F R R F F R

F F R F R R The Hexagon Equation

a 1

a 1 F 0a

The Hexagon Equation a 1 F 0a

The Hexagon Equation a 1 1 a F 0a

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

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

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

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

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

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

The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R

The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R

The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R F 00

The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R F 00

The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R F 00 R0R0

The Hexagon Equation a 1 F 0a  a,0 + R 1  a,1 1 a F a0 R0R F 00 R0R0 Top path Bottom path

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

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

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

Elementary Braid Matrices a 1 time

a 1 Elementary Braid Matrices a = 0 1

a 1 time a 1 Elementary Braid Matrices

a 1 time a 1 Elementary Braid Matrices

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

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

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

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

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

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

N = 1

N = 2

N = 3

N = 4

N = 5

N = 6

N = 7

N = 8

N = 9

N = 10

N = 11

Brute Force Search

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

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

Single Qubit Operations time U

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

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:

X X X X UU Quantum Circuit

X X X X UU

Braid

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.

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

Hilbert Space

N 0 7/2 3 5/2 2 3/2 1 1/2 S Hilbert Space 1 1/2 1 3/2 Paths are states

N /2 3 5/2 2 3/2 1 1/2 S Hilbert Space

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:

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:

N /2 3 5/2 2 3/2 1 1/2 S Dim(N) ~ 2 N

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

k = 3 Dim(N) = Fib(N+1) ~  

N /2 3 5/2 2 3/2 1 1/2 S k = 2 Dim(N) = 2 N/2

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

The R Matrix R matrix

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

k=2 k=4 k=3 k=5 N = 2

k=2 k=4 k=3 k=5 N = 3

k=2 k=4 k=3 k=5 N = 4

k=2 k=4 k=3 k=5 N = 5

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

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

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

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

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

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

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

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

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

k=6 k=8 k=7 k=9 N = 6

k=6 k=8 k=7 k=9 N = 7

k=6 k=8 k=7 k=9 N = 8

k=6 k=8 k=7 k=9 N = 9

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

“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 NEB, D.P. DiVincenzo, PRB 2012

2D Array of Qubits

Trivalent Lattice

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

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

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

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

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

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

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

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

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

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

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

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

“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

Quantum Circuit for Measuring Q v v XXXX NEB, D.P. DiVincenzo, PRB 2012

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

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

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

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

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

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

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

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!

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

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.

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

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

Plaquette Reduction

2-qubit “Tadpole”

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

Plaquette Restoration

p Quantum Circuit for Measuring B p NEB, D.P. DiVincenzo, PRB 2012

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 p NEB, D.P. DiVincenzo, PRB 2012

B p = ? Plaquette Swapping

Freshly initialized “tadpole” in LW ground state B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 B p = ? Plaquette Swapping

B p =1 Measure B p Plaquette Swapping

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

B p =1 Measure B p After Swapping

B p =1 No Error After Swapping

B p =1 Simply Remove Swapped Plaquette After Swapping

B p =1 B p =0 Error After Swapping

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

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

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

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

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

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

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

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

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

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

SWAP

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

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

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

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

Vertex Errors = String Ends

String Ends = Fibonacci Anyons Fibonacci anyons

Konig, Kuperberg, Reichardt, Ann. Phys. 2010

Konig, Kuperberg, Reichardt, Ann. Phys. 2010

1 F-moves

4 F-moves

8 F-moves

32 F-moves

48 F-moves

51 F-moves

56 F-moves

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

Non-Abelian Anyons and Topological Quantum Computation, C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008). (arXiv: v2) Lectures on Topological Quantum Computation, J. Preskill, Available online at: Some excellent reviews: Support: US DOE NEB, L. Hormozi, G. Zikos, S.H. Simon, Phys. Rev. Lett (2005). S.H. Simon, NEB, M.Freedman, N, Petrovic, L. Hormozi, Phys. Rev. Lett. 96, (2006). L. Hormozi, G. Zikos, NEB, and S.H. Simon, Phys. Rev. B 75, (2007). L. Hormozi, NEB, and S.H. Simon, Phys. Rev. Lett. 103, (2009). M. Baraban, NEB, and S.H. Simon, Phys. Rev. A 81, (2010). NEB and D.P. DiVincenzo, Phys. Rev. B 86, (2012).