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

http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Fibonacci.html

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

“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

−−=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


Vertex

OperatorPlaquette

Operator

v

p

Qv = 0,1 Bp = 0,1


Trivalent Lattice

−−=p

p

v

v BQH


Vertex

OperatorPlaquette

Operator

v

p

Qv = 0,1 Bp = 0,1


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


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


Vertex Operator: Qv

i

j

k

vvQ ijk= i

j

k

v0001010100 ===

All other 1=ijk

−−=p

p

v

v BQH


−−=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


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

“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.”


6-sided Plaquette: Bp is a 12-qubit Operator

Physical

qubits are

fixed in

space


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

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


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


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

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


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



43 CNOT Gates

24 Single Qubit Gates

6

1

2

4

3

12

9

10

11

7

8

5

p


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=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

0=

1=

Vertex Errors = String Ends

0=vQ

0=

1=

String Ends = Fibonacci Anyons

Fibonacci anyons

1 F-moves

4 F-moves

8 F-moves

32 F-moves

48 F-moves

51 F-moves

56 F-moves

BraidDehn Twist


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.

simulating anyons on a quantum computer

Documents