Download - Simulating Anyons on a Quantum Computer
Simulating Anyons on a Quantum Computer
Nick Bonesteel
Florida State University
CINT Annual Meeting, 9/24/18
Fractionalized quasiparticles = Non-Abelian anyons
A degenerate Hilbert space whose dimensionality is
exponentially large in the number of quasiparticles.
States in this space can only be distinguished by global
measurements provided quasiparticles are far apart.
Essential features:
A perfect place to hide quantum information!
“Non-Abelian” FQH States (Moore & Read ‘91)
Topological Quantum Computation
MMM
M
aa
aa
1
111
iY=f
Ytime
iY
fY
Matrix depends only on the topology of the braid swept out by
anyon world lines!
Robust quantum computation?
Kitaev, ‘97 ; Freedman, Larsen, and Wang, ‘01
Topological Quantum Computation
iY
fY
MMM
M
aa
aa
1
111
iY=f
Y
time
Matrix depends only on the topology of the braid swept out by
anyon world lines!
Robust quantum computation?
Kitaev, ‘97 ; Freedman, Larsen, and Wang, ‘01
Possible Non-Abelian FQH States
= 12/5: Possible Read-Rezayi
“Parafermion” state. Read & Rezayi, ‘99
Charge e/5 quasiparticles
are Fibonacci anyons.Slingerland & Bais ’01
Universal for Quantum Computation!Freedman, Larsen & Wang ’02
J.S. Xia et al., PRL (2004). = 5/2: Probable Moore-Read Pfaffian
state.
Charge e/4 quasiparticles
are Ising AnyonsMoore & Read ’91
Not Universal for Quantum Computation
Single Qubit Operations
time
U
NEB, L. Hormozi, G. Zikos, & S. Simon, PRL 2005
L. Hormozi, NEB, & S. Simon, PRL 2009
X
Two Qubit Gates (CNOT)
NEB, L. Hormozi, G. Zikos, & S. Simon, PRL 2005
L. Hormozi, NEB, & S. Simon, PRL 2009
=
01
10X
X
XX
X
U
U
Quantum Circuit
Braid
“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
W. Feng, NEB, D.P. DiVincenzo, In preparation
2D Array of Qubits
Trivalent Lattice
Trivalent Lattice
Trivalent Lattice
−−=p
p
v
v BQH
“Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005
Trivalent Lattice
−−=p
p
v
v BQH
“Fibonacci” Levin-Wen Model
Vertex
Operator
v
Qv = 0,1
Levin & Wen, PRB 2005
Trivalent Lattice
−−=p
p
v
v BQH
“Fibonacci” Levin-Wen Model
Vertex
OperatorPlaquette
Operator
v
p
Qv = 0,1 Bp = 0,1
Levin & Wen, PRB 2005
Trivalent Lattice
−−=p
p
v
v BQH
“Fibonacci” Levin-Wen Model
Vertex
OperatorPlaquette
Operator
v
p
Qv = 0,1 Bp = 0,1
Levin & Wen, PRB 2005
0],[ =pv BQ
0],[ =vv QQ
0],[ =pp BB
Trivalent Lattice
−−=p
p
v
v BQH
Vertex
OperatorPlaquette
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
p
v
v BQH
Vertex
OperatorPlaquette
Operator
v
p
Ground State
Qv = 1 on each vertex
Bp = 1 on each plaquette
Excited States are Fibonacci
AnyonsQv = 0,1 Bp = 0,1
“Fibonacci” Levin-Wen Model Levin & Wen, PRB 2005
Vertex Operator: Qv
i
j
k
vvQ ijk= i
j
k
v0001010100 ===
All other 1=ijk
−−=p
p
v
v BQH
“Fibonacci” Levin-Wen Model
−−=p
p
v
v BQHVertex Operator: Qv
i
j
k
vvQ ijk= i
j
k
v
0=
1=
1v =Q
on each vertex
“branching”
loop states.
0001010100 ===
All other 1=ijk
“Fibonacci” Levin-Wen Model
Plaquette Operator: Bp
Very Complicated 12-qubit Interaction!
−−=p
p
v
v BQH
1p =B
on each plaquette
superposition
of loop states
( ) fmn
mns
elm
lms
dkl
kls
cjk
jks
bij
ijs
aninmlkjis
ijklmn FFFFFFabcdefB
= nis
,
,p
sBpf
a
b
d
c
m
j
k
l
n
i
e
p ( )
=
nmlkji
nmlkjis
ijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1
+
+=
pp
p
BBB
Plaquette Operator: Bp −−=p
p
v
v BQH
1p =B
on each plaquette
superposition
of loop states
Very Complicated 12-qubit Interaction!
( ) fmn
mns
elm
lms
dkl
kls
cjk
jks
bij
ijs
aninmlkjis
ijklmn FFFFFFabcdefB
= nis
,
,p
sBpf
a
b
d
c
m
j
k
l
n
i
e
p ( )
=
nmlkji
nmlkjis
ijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1
+
+=
pp
p
BBB
Plaquette Operator: Bp −−=p
p
v
v BQH
1p =B
on each plaquette
superposition
of loop states
Very Complicated 12-qubit Interaction!
( ) fmn
mns
elm
lms
dkl
kls
cjk
jks
bij
ijs
aninmlkjis
ijklmn FFFFFFabcdefB
= nis
,
,p
sBpf
a
b
d
c
m
j
k
l
n
i
e
p ( )
=
nmlkji
nmlkjis
ijklmn abcdefB ,
,p f
a
b
d
c
m’
j’
k’
l’
n’
i’
e
p
2
10
1
+
+=
pp
p
BBB
“Surface Code” Approach
vp
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
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
eba
edcF 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
F
2
15 +=
a
c
e
b
d =F
XX
X
XX
a d
b c
e
e
eba
edcF e’
a d
b c
F Quantum Circuit
NEB, D.P. DiVincenzo, PRB 2012
a
c
e
b
d
1
2
3
4
5a d
b c
e
e
eba
edcF e’
a d
b c
F Quantum Circuit
1 5 4
32
1 5 4
32
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
1
1
2
S
Measuring Bp for a Tadpole is Easy!
S Quantum Circuit
a
b a
b
0
p1 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
3456
12
9101112
78
b
aec
d
b
ec
da
ae
b
cab
b
c
d
a
b
aec
d
b
ec
da
ae
b
cab
b
ec
d
a
b
ec
d
a
p1 B−
a
b
c
d
ee
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
3456
12
9101112
78
b
aec
d
b
ec
da
ae
b
cab
b
c
d
a
b
aec
d
b
ec
da
ae
b
cab
b
ec
d
a
b
ec
d
a
p1 B−
a
b
c
d
ee
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
Bp = ?
5
4
3
2
1
6 12
7
8 9
10
11
Plaquette Swapping
Freshly initialized
“tadpole” in LW
ground state
Bp = ?
5
4
3
2
1
612
7
8 9
10
1113
14
15
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1
Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1Bp = ?
Plaquette Swapping
Bp=1
Measure Bp
Plaquette Swapping
0
p1 B−
3456
12
9101112
78
b
a
dc
e
s
X
b
a
c
de
b
a
c
d
e
b
a
c
d
e
b
a
c
d
e
b
c
d
e
ae
bc
a
a
e
b
c
X
a
b
a
b0
0
0
Plaquette Swapping Circuit
= SWAPW. Feng, NEB, D.P. DiVincenzo, In Preparation
Bp=1
Measure Bp
After Swapping
Bp=1
Bp =1
No Error
After Swapping
Bp=1
Simply Remove Swapped Plaquette
After Swapping
Bp=1
Bp =0
Error
After Swapping
Bp=1
Bp =0
Error
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
Bp=1
Move “Packaged” Error Using F-Moves
After Swapping
0=
1=
Vertex Errors = String Ends
0=vQ
0=
1=
String Ends = Fibonacci Anyons
Fibonacci anyons
1 F-moves
1 F-moves
4 F-moves
4 F-moves
8 F-moves
8 F-moves
8 F-moves
32 F-moves
48 F-moves
48 F-moves
51 F-moves
51 F-moves
56 F-moves
56 F-moves
56 F-moves
56 F-moves
BraidDehn Twist
Konig, Kuperberg, Reichardt, Ann. Phys. 2010
Conclusions
A “conventional” quantum computer can be used to “simulate” a
topological quantum computer based on braiding anyons.
David DiVincenzo Juelich/Aachen
Steve Simon Oxford
Layla Hormozi BNL
Georgios Zikos ASML
Collaborators Former Students
Support: US DOE
Thanks to:
The ability to redraw the abstract lattice of the anyon theory using
the F move is a fundamental resource for this kind of quantum
computation.