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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
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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).
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2
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v
X X X X 0
1
2
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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
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Pentagon Equa>on
1 2 3 4
5 6
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1 2 3 4
5 6
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Pentagon Equa>on 1 2 3 4
1 2 3 4
5 6
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5 6
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1 2 3 4
5 6
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Pentagon Equa>on
1 2 3 4
5 6
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1 2 3 4
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1 2 3 4
5 6
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1 2 3 4
5 6
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Pentagon Equa>on
1 2 3 4
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1 2 3 4
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1 2 3 4
5 6
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1 2 3 4
5 6
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1 2 3 4
5 6
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Pentagon Equa>on
1 2 3 4
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1 2 3 4
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1 2 3 4
5 6
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1 2 3 4
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1 2 3 4
5 6
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1 2 3 4
5 6
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Pentagon Equa>on 1 2 3 4
1 2 3 4
5 6
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5 6
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1 2 3 4
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1 2 3 4
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1 2 3 4
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1 2 3 4
5 6
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SWAP
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SWA
P
Pentagon Equa>on as a Quantum Circuit
Equality holds if Qv = +1 on each vertex
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Pentagon Equa>on as a Quantum Circuit
1111
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Pentagon Equa>on as a Quantum Circuit
= F
F
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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
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X 0
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p1 B−
Quantum Circuit for Measuring Bp
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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