fibonacci anyon with topological “charge” 1 fusion rules
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−=
−−
−−
12/1
2/11
ϕϕϕϕ
FF Matrix:
F F
F
F
F
Pentagon
Fusion rules: 1x1 = 0+1; 1x0 = 0x1 = 1; 0x0 = 0
Fibonacci anyon with topological “charge” 1
aR=a a
=
1
0
00R
RR
The R Matrix
a a
=
1
0
00R
RR
The R Matrix
R
1
1 a
1 a
a
Equivalent to
1 a
a
Equivalent to R
a
1 a
a
Equivalent to R
a
R 1 a 1 a
F
F
R
F F
R
F F
R
R
F F
R
F
R
R
F F
F
R
R
F F
R
F F
R
F
R
R
The Hexagon Equation
0 1 0 1
The Hexagon Equation
0 1 0 1
The Hexagon Equation
0 1 0 1
The Hexagon Equation
1
0 1 0 1
The Hexagon Equation
a 1
0 1 0 1
The Hexagon Equation
a 1 F0a
0 1 0 1
The Hexagon Equation
a 1 F0a
0 1 0 1
The Hexagon Equation
a 1 1 a F0a
0 1 0 1
The Hexagon Equation
a 1
δa,0 + R1 δa,1
1 a F0a
0 1 0 1
The Hexagon Equation
a 1
δa,0 + R1 δa,1
1 a F0a
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
0 1
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1 1
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1 1 0
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1 1 0 F00
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1 1 0 F00
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1 1 0 F00
R0
0 1 0 1
The Hexagon Equation
a 1 F0a
δa,0 + R1 δa,1
1 a Fa0
R0
0 1 1 0 F00
R0
000001,10,0 )( RFRFRF aaaa
a =+∑ δδTop path
Bottom path
The Hexagon Equation
Only two solutions:
=
−
5/3
5/4
00π
π
i
i
ee
R
=
− 5/3
5/4
00
π
π
i
i
ee
R
F F
R F
R
R
Basic Content of an Anyon Theory
−=
−−
−−
12/1
2/11
ϕϕϕϕ
F
=
−
5/3
5/4
00π
π
i
i
ee
R
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 =1
=0
Initializing a Qubit
Pull two quasiparticle-quasihole pairs out of the “vacuum”.
0 0 0
These three particles have total q-spin 1
Initializing a Qubit
Pull two quasiparticle-quasihole pairs out of the “vacuum”.
0 1
0
Initializing a Qubit
Pull two quasiparticle-quasihole pairs out of the “vacuum”.
0 1
0
Measuring a Qubit
Try to fuse the leftmost quasiparticle-quasihole pair.
10 βα +
? 1
Measuring a Qubit
If they fuse back into the “vacuum” the result of the measurement is 0.
1 0
0
Measuring a Qubit
If they cannot fuse back into the “vacuum” the result of the measurement is 1
1 1
1
1
Elementary Braid Matrices
a 1
time
?1 =σ
a 1
time
==
−
5/3
5/4
1 00π
π
σ i
i
ee
R
Elementary Braid Matrices
a = 0 1
a 1
time
==
−
5/3
5/4
1 00π
π
σ i
i
ee
R
a 1
?2 =σ
Elementary Braid Matrices
a 1
time
==
−
5/3
5/4
1 00π
π
σ i
i
ee
R
a 1
−−
==−
−−
ττττσ
π
ππ
5/3
5/35/
2 i
ii
eeeFRF
Elementary Braid Matrices
12/)15( −=−= ϕτ
a 1
−−
==−
−−
ττττσσ
π
ππ
5/3
5/35/
12 i
ii
eeeFF
σ1−1 σ2 σ1
−1 σ2 = M
i Ψ
f Ψ M T = i Ψ
=
−
5/3
5/4
1 00π
π
σ i
i
ee
a 1
1 1
Elementary Braid Matrices
A Quantum Bit: A Continuum of States
0
1
θ
φ 2
10 i−2
10 i+
1 2
sin 0 2
cos φ θ θ ψ i e − + =
Single Qubit Operations: Rotations
1 2
sin 0 2
cos φ θ θ ψ i e − + =
0
1
ψα
α = rotation vector
Direction of α is the rotation axis
Magnitude of α is the rotation angle
Single Qubit Operations: Rotations
0
1
ψ
ψαU
α = rotation vector
α 2
exp σαα
⋅=
iU
ψ ψ α U α U
Single Qubit Operations: Rotations
2 π
−2 π
2 π −2 π
The set of all single qubit
rotations lives in a solid sphere of
radius 2π.
α
Qubits are sensitive: Easy to make errors!
ψ ψ α U α U
2 π
−2 π
2 π −2 π
σ12
σ1-2
σ2-2
−2 π σ22
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
= )10(0
0 3−+
Oi
i
Braid Length
Brute Force Search
=
Braid Length |ln| εFor brute force search: |ln ε |
L. Hormozi, G. Zikos, NEB, S.H. Simon, PRB ‘07
ε “error”
)10(0
0 3−+
Oi
i
Brute Force Search
=
ε “error”
Brute force searching rapidly becomes infeasible as braids get longer.
Fortunately, a clever algorithm due to Solovay and Kitaev allows for systematic improvement of the braid given a sufficiently dense covering of SU(2).
)10(0
0 3−+
Oi
i
Solovay-Kitaev Construction
Braid Length c|ln| ε 4≈c,
ε “error”
)10(0
0 5−+
Oi
i
Single Qubit Operations
time
U ψ ψU
Two Qubit Gates
U
U
ψ ψ
ψ
0 0
1 1
ψU
X
X
0 0
0 1
0 0
1 1
Controlled-NOT Gate
X
X
1 1
1 0
0 0
1 1
=
0110
X
X
X
X
X U
U
Any unitary operation can be expressed as a quantum circuit
Universal Set of Gates
U X
Quantum Circuit
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
=
0110
X
SK Improved Controlled-NOT Gate
X
X X
X U
U
Quantum Circuit
Quantum Circuit
X
X X
X U
U
Braid
SU(2)k Nonabelian Particles Quasiparticle excitations of the ν=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.
α + β 0 1
1021
21
⊕=⊗For example:
( ) 21212121 1 ssssssss +⊕⊕+−⊕−=⊗
Numbers label total spin of particles inside ovals
N 0
7/2 3
5/2 2
3/2 1
1/2
S
1 2 3 4 5 6 7 8 10 9 11 12
Hilbert Space
N 0
7/2 3
5/2 2
3/2 1
1/2
S
1 2 3 4 5 6 7 8 10 9 11 12
Hilbert Space
1 1/2 1 3/2
Paths are states
N 0
1
1
1
1
2
1
3
2
5
9
5
1
4
5
6
14
14
20
28
14
1
1 7
7/2 3
5/2 2
3/2 1
1/2
S
1 2 3 4 5 6 7 8 10 9 11 12
27
48
42
8
55
110
132
44
20
90
42
35
132
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.
α + β 0 1
1021
21
⊕=⊗For example:
( ) 21212121 1 ssssssss +⊕⊕+−⊕−=⊗
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.
α + β 0 1
1021
21
⊕=⊗For example:
( ) [ ]2,min1 2121212121 kssssssssss −++⊕⊕+−⊕−=⊗
N 0
1
1
1
1
2
1
3
2
5
9
5
1
4
5
6
14
14
20
28
14
1
1 7
7/2 3
5/2 2
3/2 1
1/2
S
1 2 3 4 5 6 7 8 10 9 11 12
27
48
42
8
110
165
132
44
75
90
42
35
132
Dim(N) ~ 2N
N 0
1
1
1
1
2
1
3
2
9
5
4
5
13
14
27
14
7/2 3
5/2 2
3/2 1
1/2
S
1 2 3 4 5 6 7 8 10 9 11 12
40
41
121
122
81
41 122
4 13 40 121
k = 4
Dim(N) ~ 3N/2
N 0
1
1
1
1
2
3
2
8
5
3
5
8
13
21
13
7/2 3
5/2 2
3/2 1
1/2
S
1 2 3 4 5 6 7 8 10 9 11 12
21
34
55
89
55
34 89
k = 3
Dim(N) = Fib(N+1) ~ φΝ
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
1 2 3 4 5 6 7 8 10 9 11 12
16 32
16
16 32
32
k = 2
Dim(N) = 2N/2
The F Matrix
a c = b c
=
100
0]2[
1]2[]3[
0]2[]3[
]2[1
q
q
q
q
−
0 1/2
1 1/2
1 3/2
0 1/2
1 1/2
1 3/2
∑a
cabF
2/12/1
2/2/
][ −
−
−−
=qqqqm
mm
qq-integers: )2/(2 += kieq π;
π4212
++
= kki
e
π42
1+= k
ie
=
+
++
π
π
421
4212
0
0k
i
kki
e
eR0
1
0
1
The R Matrix
R matrix
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. 2010 NEB, D.P. DiVincenzo, PRB 2012
2D Array of Qubits
Trivalent Lattice
Trivalent Lattice
Trivalent Lattice
∑∑ −−=p
pv
v BQH
“Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005
Trivalent Lattice
∑∑ −−=p
pv
v BQH
“Fibonacci” Levin-Wen Model
Vertex Operator
v
Qv = 0,1
Levin & Wen, PRB 2005
Trivalent Lattice
∑∑ −−=p
pv
v BQH
“Fibonacci” Levin-Wen Model
Vertex Operator
Plaquette Operator
v p
Qv = 0,1 Bp = 0,1
Levin & Wen, PRB 2005
Trivalent Lattice
∑∑ −−=p
pv
v BQH
Vertex Operator
Plaquette Operator
v p
Ground State Qv = 1 on each vertex Bp = 1 on each plaquette
Qv = 0,1 Bp = 0,1
“Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005
Trivalent Lattice
∑∑ −−=p
pv
v BQH
Vertex Operator
Plaquette Operator
v p
Ground State Qv = 1 on each vertex Bp = 1 on each plaquette
Excited States are Fibonacci Anyons Qv = 0,1 Bp = 0,1
“Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005
Vertex Operator: Qv
i
j
k v vQ ijkδ= i
j
k v 0001010100 === δδδ
All other 1=ijkδ
∑∑ −−=p
pv
v BQH
“Fibonacci” Levin-Wen Model
∑∑ −−=p
pv
v BQHVertex Operator: Qv
i
j
k v vQ ijkδ= i
j
k v
0=
1=
1v =Qon each vertex “branching” loop states.
0001010100 === δδδAll other 1=ijkδ
“Fibonacci” Levin-Wen Model
Plaquette Operator: Bp
Very Complicated 12-qubit Interaction!
∑∑ −−=p
pv
v BQH
1p =B on each plaquette superposition of loop states
( ) fmnmns
elmlms
dklkls
cjkjks
bijijs
aninmlkjisijklmn FFFFFFabcdefB ′′′′′′′′′′′′
′′′′′′ = nis,,p
sBp f
a
b
d
c
m
j
k
l
n
i
e
p ( )∑′′′′′′
′′′′′′=nmlkji
nmlkjisijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1 ϕϕ
+
+= pp
p
BBB
Plaquette Operator: Bp ∑∑ −−=p
pv
v BQH
1p =B on each plaquette superposition of loop states
Very Complicated 12-qubit Interaction!
( ) fmnmns
elmlms
dklkls
cjkjks
bijijs
aninmlkjisijklmn FFFFFFabcdefB ′′′′′′′′′′′′
′′′′′′ = nis,,p
sBp f
a
b
d
c
m
j
k
l
n
i
e
p ( )∑′′′′′′
′′′′′′=nmlkji
nmlkjisijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1 ϕϕ
+
+= pp
p
BBB
Plaquette Operator: Bp ∑∑ −−=p
pv
v BQH
1p =B on each plaquette superposition of loop states
Very Complicated 12-qubit Interaction!
( ) fmnmns
elmlms
dklkls
cjkjks
bijijs
aninmlkjisijklmn FFFFFFabcdefB ′′′′′′′′′′′′
′′′′′′ = nis,,p
sBp f
a
b
d
c
m
j
k
l
n
i
e
p ( )∑′′′′′′
′′′′′′=nmlkji
nmlkjisijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1 ϕϕ
+
+= pp
p
BBB
Plaquette Operator: Bp ∑∑ −−=p
pv
v BQH
1p =B on each plaquette superposition of loop states
Very Complicated 12-qubit Interaction!
( ) fmnmns
elmlms
dklkls
cjkjks
bijijs
aninmlkjisijklmn FFFFFFabcdefB ′′′′′′′′′′′′
′′′′′′ = nis,,p
sBp f
a
b
d
c
m
j
k
l
n
i
e
p ( )∑′′′′′′
′′′′′′=nmlkji
nmlkjisijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1 ϕϕ
+
+= pp
p
BBB
“Surface Code” Approach
v p
Use quantum computer to repeatedly measure Qv and Bp.
How hard is it to measure Qv and Bp with a quantum computer?
If Qv=1 on each vertex and Bp=1 on each plaquette, good. If not, treat as an “error.”
Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Quantum Circuit for Measuring Qv
1
2
3
v
0
1
2
3
v1 Q−X X X X
NEB, D.P. DiVincenzo, PRB 2012
Quantum Circuit for Measuring Qv
1
2
3
v
0
1
2
3
v1 Q−X X X X
Syndrome qubit
NEB, D.P. DiVincenzo, PRB 2012
Quantum Circuit for Measuring Qv
That was easy! What about Bp?
1
2
3
v
0
1
2
3
v1 Q−X X X X
Syndrome qubit
NEB, D.P. DiVincenzo, PRB 2012
6-sided Plaquette: Bp is a 12-qubit Operator
Physical qubits are fixed in space
6-sided Plaquette: Bp is a 12-qubit Operator
Physical qubits are fixed in space
Trivalent lattice is abstract and can be “redrawn”
6-sided Plaquette: Bp is a 12-qubit Operator
6-sided Plaquette: Bp 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: Bp is a 12-qubit Operator
Physical qubits are fixed in space
Trivalent lattice is abstract and can be “redrawn”
6-sided Plaquette: Bp 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
a d
b c
e ∑′
′e
ebaedcF e’
a d
b c
Apply a unitary operation on these five qubits to stay in the Levin-Wen ground state.
−=
−−
−−
12/1
2/11
ϕϕϕϕ
F2
15 +=ϕ
a
c
e
b
d = F
X X
X
X X
a d
b c
e ∑′
′e
ebaedcF e’
a d
b c
F Quantum Circuit
NEB, D.P. DiVincenzo, PRB 2012
a
c
e
b
d
1 2 3 4 5
a d
b c
e ∑′
′e
ebaedcF e’
a d
b c
F Quantum Circuit
1 5 4
3 2
1 5 4 3 2
NEB, D.P. DiVincenzo, PRB 2012
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
Plaquette Reduction
2-qubit “Tadpole”
−+
=1
1
11
2 ϕϕ
ϕS
Measuring Bp for a Tadpole is Easy!
S Quantum Circuit
a
b a
b
0p1 B−
a
b
X
= a b S
X X
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
Plaquette Restoration
0
3 4 5 6
1 2
9 10 11 12
7 8
b
a e c
d
b
e c
d a
a e
b
c a b
b
c
d
a
b
a e c
d
b
e c
d a
a e
b
c a b
b
e c
d
a
b
e c
d
a
p1 B−
a
b
c
d
e e
X
6
1
2
4
3
12
9 10
11 7
8
5
p
Quantum Circuit for Measuring Bp
NEB, D.P. DiVincenzo, PRB 2012
Gate Count
0
3 4 5 6
1 2
9 10 11 12
7 8
b
a e c
d
b
e c
d a
a e
b
c a b
b
c
d
a
b
a e c
d
b
e c
d a
a e
b
c a b
b
e c
d
a
b
e c
d
a
p1 B−
a
b
c
d
e e
X
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
Creating Excited States: Fibonacci Anyons Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Introduce “holes” (Stop measuring Qv and Bp
inside hole)
Creating Excited States: Fibonacci Anyons Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Fix boundary condition by modifying Qv operators
0=
1=
Creating Excited States: Fibonacci Anyons Konig, Kuperberg, Reichardt, Ann. Phys. 2010
0=
1=
Fibonacci anyons
Creating Excited States: Fibonacci Anyons Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Konig, Kuperberg, Reichardt, Ann. Phys. 2010
1 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
1 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
4 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
4 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
8 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
8 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
8 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
32 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
48 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
48 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
51 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
51 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
56 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
56 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
56 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
56 F-moves Konig, Kuperberg, Reichardt, Ann. Phys. 2010
56 F-moves Braid
Konig, Kuperberg, Reichardt, Ann. Phys. 2010
1 2 3 4
5 6
7
1 2 3 4
5 6
7
5 6
7
1 2 3 4
5
5 6
7 1 2 3 4
6
7
1 2 3 4
1 2 3 4
5 6
7
6
5 6
7 1 2 3 4
5 6
7 1 2 3 4
5
7
1 2 3 4
1 2 3 4
5 6
7
5 6
1 2 3 4
5
6
7
1 2 3 4
7
5 6
7 1 2 3 4
5 6
7
1 2 3 4
1 2 3 4
5 6
7
6
1 2 3 4
7
1 2 3 4
5
7
1 2 3 4
5 6
7
5 6
7 1 2 3 4
5 6
7
1 2 3 4
1 2 3 4
5 6
7
6 5
6 6
1 2 3 4
6 5
7
1 2 3 4
5
7
1 2 3 4
5
7
5 6
7 1 2 3 4
5 6
7
1 2 3 4
1 2 3 4
5 6
7
6 6
1 2 3 4
6 5
7 SWAP
1 2 3 4
5
7
1 2 3 4
5
7
5 6
7 1 2 3 4
5 6
7
1 2 3 4
1 2 3 4
5 6
7
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
=
SWA
P
NEB, D.P. DiVincenzo, PRB 2012
Pentagon Equation as a Quantum Circuit
1
1
1
1
1
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
=
SWA
P
NEB, D.P. DiVincenzo, PRB 2012
Pentagon Equation as a Quantum Circuit
Pentagon Equation as a Quantum Circuit
−=
−−
−−
12/1
2/11
ϕϕϕϕ
F2
15 +=ϕ
Simplified Pentagon Circuit
=
SWA
P
F F
F F
F
NEB, D.P. DiVincenzo, PRB 2012
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).
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