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The  Quantum  Computer  Roadmap,  and  “going  off  road”  

David  DiVincenzo    27.06.2012  

Varenna  Course  CLXXXIII  

An architecture of the large-scale quantum computer is taking shape

- “roadmap” as given by N. C. Jones et al. How can we do better than Jones (“going off road”)

- Majorana wires? - Direct multiqubit parity measurements - Codes that directly embody universal quantum

computation

Outline

Physical Review X, In press

An architecture of the large-scale quantum computer is taking shape

- “roadmap” as given by N. C. Jones et al. How can we do better than Jones (“going off road”)

- Majorana wires? - Direct multiqubit parity measurements - Codes that directly embody universal quantum

computation

Outline

Proposal  for  realiza>on  B.  M.  Terhal,  F.  Hassler,  and  D.  P.  DiVincenzo,  "From  Majorana  Fermions  to  

Topological  Order,"  arXiv:1201.3757,  Physical  Review  LeRers,  in  press.    

Light-grey areas are superconducting (SC) islands, each with two InAs nanowires on top at the end of which are Majorana bound states: 4 Majorana fermions (in yellow), representing a single qubit. Tunneling coupling of strength !  for Majorana fermions between islands.

arXiv:1205.1910

s1, s2, s3 are the states of the three qubits (0,1) χi is dispersive shift parameter Dispersive coupling is the same for each qubit and the same on both resonators (a and b)

χ=g2/Δ

Wave impedance “looking into” port A (transmission line theory)

Reflection coefficient of full structure NB

(Z0=50Ω)

arg(r(ω)) for different qubit states

θ is the same for all even states (mod 2π) θ is the same for all odd states (mod 2π)

θeven≠θodd

Alternative solution of Mabuchi and coworkers

An architecture of the large-scale quantum computer is taking shape

- “roadmap” as given by N. C. Jones et al. How can we do better than Jones (“going off road”)

- Majorana wires? - Direct multiqubit parity measurements - Codes that directly embody universal quantum

computation

Outline

Quantum  Circuits  for  Measuring  Levin-­‐Wen  Operators  

With  N.E.  Bonesteel          

Florida  State  University  

a

c e b

d

a

e b

d

c

c

e

a

b a

b

a

e b

d

c

a

e b

d

c

a

c e b

d

a

e b

d

c a

b

a

e b

d

c

a

e b

d

c

X 0

c

e

a

b

6

1

2

4

3

12

9

10

11

7

8

5

p

arXiv:1206.6048:

Levin-­‐Wen  Models  

∑∑ −−=p

pv

v BQH

Trivalent lattice: Qubits live on edges

Levin-­‐Wen  Models  

∑∑ −−=p

pv

v BQH

i j

k v vQ ijkδ= i

j

k v

Trivalent lattice: Qubits live on edges

0001010100 === δδδAll other 1=ijkδ

“Doubled Fibonacci” Model Vertex Operator

Levin-­‐Wen  Models  

∑∑ −−=p

pv

v BQH

( ) fmnmns

elmlms

dklkls

cjkjks

bijijs

aninmlkjisijklmn FFFFFFabcdefB ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ = nis,,p

i j

k v vQ ijkδ= i

j

k v

sBp f

a b

d

c

m

j k

l n

i

e

p ( )∑ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′=nmlkji

nmlkjisijklmn abcdefB ,,p f

a b

d

c

m’

j’ k’

l’ n’

i’

e

p

2

10

1 ϕ

ϕ

+

+= pp

p

BBB

Vertex Operator

Plaquette Operator

Horrible 12 spin interaction!

Trivalent lattice: Qubits live on edges

0001010100 === δδδAll other 1=ijkδ

“Doubled Fibonacci” Model

Levin-­‐Wen  Models  

∑∑ −−=p

pv

v BQH

( ) fmnmns

elmlms

dklkls

cjkjks

bijijs

aninmlkjisijklmn FFFFFFabcdefB ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′ = nis,,p

i j

k v vQ ijkδ= i

j

k v

sBp f

a b

d

c

m

j k

l n

i

e

p ( )∑ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′

ʹ′ʹ′ʹ′ʹ′ʹ′ʹ′=nmlkji

nmlkjisijklmn abcdefB ,,p f

a b

d

c

m’

j’ k’

l’ n’

i’

e

p

2

10

1 ϕ

ϕ

+

+= pp

p

BBB

Vertex Operator

Plaquette Operator

Horrible 12 spin interaction!

Trivalent lattice: Qubits live on edges

0001010100 === δδδAll other 1=ijkδ

“Doubled Fibonacci” Model

“Fibonacci”  Levin-­‐Wen  Model  

•  Excitations are Fibonacci anyons: Universal quantum computation can be carried out purely by braiding.

•  Active approach: Ground states of Fibonacci Levin-Wen model can be used as a quantum code (the Fibonacci code). Qv and Bp are stabilizers which are measured to diagnose errors.

Question: How hard is it to measure Qv and Bp?

Koenig, Kuperberg, Reichardt, Ann. of Phys. (2010).

1

2

3

v

X X X X 0

1

2

3

v1 Q−

Quantum  Circuit  for  Measuring  Qv  

That was easy! What about Bp?

NB: 4-qubit Toffoli gate

A  Useful  Resource:    The  F  Move  a b

c d

e Fdb !eace

!e" e’

b

d

a

c

Koenig, Kuperberg, Reichardt, Ann. of Phys. (2010).

Levin-Wen Ground State

A  Useful  Resource:    The  F  Move  a b

c d

e e’

b

d

a

c

Levin-Wen Ground State

Koenig, Kuperberg, Reichardt, Ann. of Phys. (2010).

Fdb !eace

!e"

A  Useful  Resource:    The  F  Move  a b

c d

e e’

b

d

a

c

Levin-Wen Ground State Levin-Wen Ground State on New Lattice

Koenig, Kuperberg, Reichardt, Ann. of Phys. (2010).

Fdb !eace

!e"

F  Quantum  Circuit  

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

−−

−−

12/1

2/11

ϕϕ

ϕϕF

a b

c d

e e’

b

d

a

c

215 +

=ϕ

a

c

e

b

dX

X F

X

X

X =

Fdb !eace

!e"

First gate: Locally equivalent to 5-qubit Toffoli gate

F is a pi-rotation around a funny axis

Pentagon  Equa>on  

1 2 3 4

5 6

7

Pentagon  Equa>on  

1 2 3 4

5 6

7

1 2 3 4

5 6

7

Pentagon  Equa>on  1 2 3 4

1 2 3 4

5 6

7

5 6

7

1 2 3 4

5 6

7

Pentagon  Equa>on  

1 2 3 4

5 6

7

1 2 3 4

5

6

7

1 2 3 4

5 6

7

1 2 3 4

5 6

7

Pentagon  Equa>on  

1 2 3 4

5

6

7

1 2 3 4

5

6

7

1 2 3 4

5 6

7

1 2 3 4

5 6

7

1 2 3 4

5 6

7

Pentagon  Equa>on  

1 2 3 4

5

6

7

1 2 3 4

6 5

7

1 2 3 4

5 6

7

1 2 3 4

5

6

7

1 2 3 4

5 6

7

1 2 3 4

5 6

7

Pentagon  Equa>on  1 2 3 4

1 2 3 4

5 6

7

5 6

7

1 2 3 4

5

6

7

1 2 3 4

5

6

7

1 2 3 4

6 5

7

1 2 3 4

5 6

7

SWAP

3

4

5

6

1

2

7

c

e

a

b

d

b

e

d

c

a

d

e

c

a

b

a

e

b

d

c

d

e

c

a

b

=

SWA

P

Pentagon  Equa>on  as  a  Quantum  Circuit  

Equality holds if Qv = +1 on each vertex

3

4

5

6

1

2

7

c

e

a

b

d

b

e

d

c

a

d

e

c

a

b

a

e

b

d

c

d

e

c

a

b

=

SWA

P

Pentagon  Equa>on  as  a  Quantum  Circuit  

1111

1

Pentagon  Equa>on  as  a  Quantum  Circuit  

= F

F

F

F

F SWA

P

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

−−

−−

12/1

2/11

ϕϕ

ϕϕF

215 +

=ϕ

Simplified Pentagon Circuit

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

Reducing  a  PlaqueRe  to  a  Tadpole  

Koenig, Reichardt, Vidal, Phys. Rev. B (2009).

S

X X =

a

b

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

11

11

2 ϕ

ϕ

ϕS

a

b

X 0

p1 B−

a

b

a

b

Measuring  Bp  for  a  Tadpole  is  Easy!  

S Quantum Circuit

Restoring  the  PlaqueRe  

Restoring  the  PlaqueRe  

Restoring  the  PlaqueRe  

Restoring  the  PlaqueRe  

Restoring  the  PlaqueRe  

Restoring  the  PlaqueRe  

3 4 5 6

1 2

9 10 11 12

7 8

X 0

a

ceb

d

a

eb

dc

ce

a

bab

a

eb

d

c

a

eb

d

c

a

ceb

d

a

eb

dc

ce

a

bab

a

eb

d

c

a

eb

d

c

p1 B−

Quantum  Circuit  for  Measuring  Bp  

6

1

2

4

3

12

9

10

11

7

8

5

p

82 Toffoli gates 43 CNOT gates 26 Single Qubit gates

or 20 n-qubit Toffoli gates 10 CNOT gates 24 Single Qubit gates

An architecture of the large-scale quantum computer is taking shape

- “roadmap” as given by Cody Jones et al. How can we do better than Cody Jones (“going off road”)

- Majorana wires? - Direct multiqubit parity measurements - Codes that directly embody universal quantum

computation

Outline