Implementing quantum algorithms using quantum logic gates made from optical components'

1
Implementing quantum algorithms using quantum
logic gates made from optical components.

• Mark Tame

2
Promising Technologies for Quantum Computing

• Ion Traps
• Quantum Dots
• Josephson Junctions
• Nuclear Spins in Silicon / Molecules
• Linear Optics (KLM proposal)
• E. Knill, R. Laflamme and G. J. Milburn, "A
  scheme for efficient quantum computation with
  linear optics",Nature 409, 46 (2001).

3
Three Key Principles in KLM Proposal

• Conditional Non-linear Sign (NS) gates for two
  photon states (today)
• Teleportation to achieve efficiency
• Error Correction to achieve scalability

4
Conditional Non-linear Sign (NS) gates for two
photon states

• Part I Optical Components
• Part II Quantum Logic Gates
• Part III Solving Quantum Algorithms

5
Part I - Optical Components
6
Dual Rail Basis
b
a
b
a
Qubit
As opposed to
7
Beamsplitter
8
Phase Shifter
b
b
p
a
a
9
Hadamard gate
b
b
a
a
H
gt
gt
yin
yout
10
Kerr gate
Classical
Quantum
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Part II Quantum Logic Gates
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CNOT Version I (non-linear components)
d
d
0
0
c
c
1 L
gt
1 L
gt
K
1
1
B
B
b
b
1
0
a
a
0 L
gt
p
1 L
gt
p
0
1
H
H
UCN
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CNOT Version I (non-linear components)
A B A B
0 0 1 1
0 0 1 1
0 1 0 1
0 1 1 0
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Non-linear Sign change gate ( NS gate )
yc
gt
c2 2
gt
c0 0
gt
c1 1
gt


Btcb
c
c
Bba
Bba
b
b
1
gt
q2
a
a
0
gt
q1
q1
y 1
y 2
y 3
gt
y 4
gt
gt
gt
c
c
NS
b
b
1
gt
1
gt
Detection Condition (D.C)
a
a
0
gt
0
gt
15
Non-linear Sign change gate ( NS gate )
16
Non-linear Sign change gate ( NS gate )
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CNOT Version II (linear components based on
KLM NS gate proposal)
b
b
0
0
gt
1
1
gt
NS
D. C
gt
0
0
gt
1 L
gt
1 L
gt
a
a
Bad
Bad
1
1
Bba
Bba
NS
d
d
1
gt
1
gt
0
1
D. C
0
gt
0
gt
c
c
0 L
gt
1 L
gt
0
1
y x
y y
gt
gt
18
CNOT Version II (linear components)
A B A B
0 0 1 1
0 0 1 1
0 1 0 1
0 1 1 0
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Theoretical
Experimental
J. L. OBrien, G. J. Pryde, A. G. White, T. C.
Ralph and D. Branning, Demonstration of an
all-optical quantum computer controlled-NOT
gate",Nature 426, 264 (2003).
The next step will be to incorporate the gate
reported here in simple optical circuits to
demonstrate simple algorithms and error
correcting schemes.
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Part III Solving Quantum Algorithms
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Deutschs Problem
Classical
Requires at least 2 attempts to find out if f(x)
is balanced or constant
Quantum
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Quantum Optical Circuit for Deutschs Problem
B
B
d
d
1
0 L
gt
x
gt
x
gt
c
c
p
p
0
H
Uf
H
B
b
b
0
1 L
gt
y
gt
y f(x)
gt
a
a
p
1
H
Depends on f(x)
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Quantum Optical Circuit for generalised
Deutsch/Jozsa Algorithm
Classical
Requires around this many attempts to find out
if f(x) is balanced or constant
Quantum
Efficiency problems from Detection Conditions for
NS gates in Quantum CNOT gates
/- 1 if f(x) is const 0 if f(x) is
balanced
Amplitude for all zero states
is
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Next Time

• Teleportation to achieve efficiency
• Error Correction to achieve scalability
• Decoherence-free subspaces

References E. Knill, R. Laflamme and G. J.
Milburn, "A scheme for efficient quantum
computation with linear optics",Nature 409, 46
(2001). J. L. OBrien, G. J. Pryde, A. G. White,
T. C. Ralph and D. Branning, Demonstration of
an all-optical quantum computer controlled-NOT
gate",Nature 426, 264 (2003). D. Deutsch and R.
Jozsa, "Rapid solution of problems by quantum
computation", Proceedings of the Royal Society
of London A 439 (1992) 553-558. M. A. Nielsen
and I. L. Chuang, Quantum Computation and
Quantum Information, Cambridge University Press,
Cambridge (2000).