Quantum Gates and Quantum Networks

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Quantum Gates and Quantum Networks

• L.C. Kwek

NIE, NanyangTechnological University Local
Driver, QIT Lab (NUS, Singapore) Fujitsu Visiting
Professor, University of Cambridge Talk presented
at the National Taiwan University, Taipei, 14
December 2004
2
QIT Group (Singapore)
Artur Ekert and C.H. Oh

• Janet Anders
• Chia Teck Chee
• Chen Jingling
• Chen Lai Keat
• Choo Keng Wah
• Du Jiangfeng
• Berge Englert
• Feng Xinli
• Ajay Gopinathan
• Darwin Gosal
• Hor Wei Hann
• D. Kaszlikowski
• Christian Kurtsiefer
• L.C. Kwek
• C.H. Lai
• Wayne Lawton
• Lim Jenn Yang
• Antia Lamas Linares

• Alex Ling
• Looi Shiang Yong
• Ivan Marcikic
• Neelima Raitha
• Kuldip Singh
• Tey Meng Khoon
• Tong Dianmin
• Wang Zisheng
• Wu Chunfeng
• 5-10 undergraduate students
• www.quantumlah.org

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3rd Asia Pacific Workshop On Quantum
Information Science and 1st Joint Institute of
Mathematical Sciences -NUS conference on Quantum
Information (3rd Jan 05 to 15th Jan 05)

• http//www.quantumlah.org/workshop/

4

• http//www.lasphys.com/workshops/lasphys05/lphys05
  .htm

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Coverage

• Introduction quantum logic and classical logic
• Single qubit gate
• Two-qubit gate
• Quantum circuits for quantum algorithm
• A simple quantum network
• Experiments

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Computation
OUTPUT
INPUT
1
0
1
1
0
0
0
1
1
0
1
0
Physics Inside (and outside)
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Classical deterministic computation
Physically allowed operations, computational steps
Intermediate configurations
Configuration complete specification of the
state of the computer and data
Final configuration (output)
Initial configuration (input)
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Classical deterministic computation
110
001
101
000
Computational steps moves from one
configuration to another are performed by
elementary operations on bits
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Boolean Networks
0
OR
1
AND
0
OR
1
0
0
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Basic operations logic gates
Logical AND
Wire, identity
1
AND
0
0
1
1
1
1
Output 0 apart from the (1,1) input
0
1
NOT
1
0
Logical OR
0
OR
0
0
X
X
Output 1 apart from the (0,0) input
X
Fan out
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Classical probabilistic computation
Possible outputs
Input
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Quantum computation
Constructive or destructive interference enhance
correct outputs suppress wrong outputs GOOD SIDE
extra computational power BAD SIDE sensitive to
decoherence
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Quantum computation
Initial configuration of the three qubits
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Bits and Qubits
QUBIT
BIT
15
Quantum Boolean Networks
H
H
H
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Quantum operations
H
H
H
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Single qubit gates
Hadamard
H
Continuous set of phase gates
Discrete set of phase gates
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Single qubit interference
H
H
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Any single qubit interference
H
H
INPUT
OUTPUT
in the matrix form
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Any unitary operation on a qubit
H
H
INPUT
OUTPUT
in the matrix form the most general SU(2)
operation on a single qubit
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Two and more qubits
Notation
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Operations on two qubits
Controlled-NOT
Controlled-U
U
U
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Quantum interferometry revisited
H
H
H
H
U
REMEMBER THIS TRICK !
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Phases in a new way
H
H
U
25
Entangled states
H
entangled
separable
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Bell GHZ states
H
H
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Useful decomposition of any U in SU(2)
For any U in SU(2)
Rotation by twice the angle between axis a and
b around the axis perpendicular to a and b
Recall that ?x represents rotation by ? around
axis x
Rotation by ? around some axis a
Rotation by ? around some axis b
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Building controlled-U operations

B
B-1
A-1
U
A
A, A-1, B and B-1 are single qubit operations and
can be constructed from the Hadamard and phase
gates. Controlled-U can be constructed from
single qubit operations and the controlled-NOT
gates. Hence any controlled-U gate can be
constructed from the Hadamard, the
controlled-NOT and phase gates.
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Toffoli Gate

H
H
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Controlled-controlled NOT
Computes logical AND
Quantum adder
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Quantum Networks
H
H
Quantum adder
Quantum Hadamard transform
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Quantum Hadamard Transform
H
H
H
H
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Quantum Hadamard Transform
H
H
H
H
H
H
H
H
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Quantum Network
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(No Transcript)
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f
37
f
38
f
39
f
40
f
41
f
42
f
43
f
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Application 2 Direct measure of purity if r1
r2 r Visibility Tr (r2)
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f
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visibility
f
f
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Possible technology?
IONS TRAP Innsbruck Oxford Munich Boulders
Cavity QED ENS Paris
Linear Optics Singapore?
NMR Oxford, MIT
Josephson Junctions Delft, Karlsruhe,Catania
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Experimental

• Bruker Avance DMR 400 spectrometer
• 9.4 T
• 5mm probe
• C13 sample of alanine
• Resonant frequency of alanine 100 MHz

J.F. Du, P. Zou, M. Shi, L.C. Kwek, J.W. Pan,
C.H. Oh, A. Ekert, D.K.L. Oi and M. Ericsson, An
experimental observation of geometric phases for
mixed states using interferometry, Phys. Rev.
Lett. 91, 100403 (2003) , eprint
arXivquant-ph/0305054