Quantum Computing Physik der QuantenInformationsverarbeitung

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Quantum ComputingPhysik der Quanten-Informationsv
erarbeitung
Vorlesung SS 08

• C. Meyer, C.M. Schneider

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in the media
14.02.2007 1249
Heise-online 14.02.2007 1249 ltlt Vorige
Nächste gtgt Erster Quantenprozessor der Welt
vorgestellt Das kanadische Start-up D-Wave
Systems hat in Kalifornien einen Quantenprozessor
mit 16 Qubits vorgestellt. Die Qubits werden von
je einer kreisförmigen supraleitenden
Stromschleife aus dem Metall Niob dargestellt.
Die Betriebstemperatur des Prozessors beträgt 5
Millikelvin, 0,005 Grad über dem absoluten
Nullpunkt. "Unser Durchbruch in der
Quantentechnologie ist ein wichtiger Fortschritt
bei der Lösung wirtschaftlicher und
wissenschaftlicher Probleme, die bislang nur
schwer in den Griff zu bekommen waren", erklärte
D-Wave-Systems CEO Herb Martin.
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plan
1. introduction 2. quantum mechanical
background 3. basic operations/superposition/entan
glement 4. quantum computing with ion
traps 5. Deutsch-Josza algorithm and its
implementation 6. NMR quantum computing 7. Shor
algorithm and its implementation (15
5x3) 8. magnetic resonance QC in solid
state 9. quantum dots for quantum
computing 10. first experiments 11. superconductin
g qubits 12. quantum error correction
13. invitation to Research Centre Jülich
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technology of computation
Rebuilt 1960 by K. Zuse Deutsches Museum München

• 2-state (binary) logic 0 and 1
• state is defined by a switch open closed
• logic operations array of switches (gates)
• mechanical switches (Zuse Z1)
• electromechanical relays (Zuse Z3)

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ENIAC
1946

• Electronic Numerical Integrator And Computer
• 17468 vacuum tubes
• weight 20 t, power consumption 150 kW

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Moores law
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birth of microelectronics

• 1947 invention of transistor
• 1958 invention of integrated circuit (TI)
• 1971 first microprocessor (4004)

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microprocessors
4-bit
8-bit
32-bit
64-bit
16-bit
Intel 4004 1971
Intel 8080 1974
2000
1985
2005
increase power of microprocessors by bus bit
width and clock frequency
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microprocessor design
http//www.offis.de/
Dr. Jens Appell
Embedded Hardware- / Software-Systems
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breaking the barrier?
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computational power

• we want to increase our capability of solving
  problems

what is a complex problem?
Is there a subset of -2,-3,15,14,7,-10 which
adds up to 0?
Easy verify that sum-2,-3,15,-10
0 Difficult identify this subset
Similar problem find prime factors of 1601
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fundamental approach
Question Is there a general method or process by
which one could decide whether a mathematical
proposition could be proved? Answer
No!
what is a computer and what kind of problems can
it solve?
Turing Machine
On computable numbers, with an application to the
Entscheidungsproblem Proceedings of the London
Mathematical Society, Series 2, Vol.42 (1936 -
37) pages 230 to 265 online available
http//web.comlab.ox.ac.uk/oucl/research/areas/ieg
/e-library/sources/tp2-ie.pdf
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Turing machine

• Consists of a stripe and a head

• Symbols determine the action of the head
• Writing/Erasing of symbol
• Direction of reading
• change of state

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boolean algebra and logic gates
classical (irreversible) computing
1-bit logic gates
identity
NOT
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boolean algebra and logic gates
2-bit logic gates
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Turing Machine
1
1
1
what happens, depends on the states of the head
table of states
56 7 63
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complexity classes

• Deterministic Turing Machine (DTM) models all
  classical computers
• therefore called universal

P problems that can be solved with a DTM in
polynomial time

• Probabilistic Turing Machine (PTM)
• actions are carried out with certain probability

ZPP problems that can be solved with a PTM with
zero probability of error in polynomial time.

• Non-Deterministic Turing Machine (NDTM)
• multiple computation paths (computation tree)

NP problems that can be solved with a NDTM in
polynomial time.
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traveling salesman problem

• the traveling salesman problem is NP-complete

What is the shortest route between a given number
of cities?
? scales exponentially with number of cities for
a DTM
Experiment
Can a physical implementation be found that
provides a better solution?
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physical system designed for problem
soap bubbles can (theoretically) be used to solve
some optimization problems in NP-complete
soap bubbles ? NDTM
quantum computer ? NDTM
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Quantum Turing Machine

• Read, write, and shift operations are
  accomplished by quantum interactions
• Tape and head exist each in a quantum state
• symbols 0 or 1 are replaced by qubits, which
  can hold a quantum superposition of 0? and 1?

The quantum Turing machine can encode many inputs
to a problem simultaneously, and then it can
perform calculations on all the inputs at the
same time. This is called quantum parallelism.
David Deutsch, Proceedings of the Royal Society
of London A 400 (1985), 97
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quantum bits
conventional bit on ltgt 3.2 - 5.5 V ltgt 1 off
ltgt -0.5 - 0.8 V ltgt 0
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quantum parallelism
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quantum computing
classical bit 1 ? ON ? 3.2 5.5 V 0 ?
OFF ? -0.5 0.8 V
quantum-bit (qubit) ?0? ? ? ?1? ? ?
calculation
preparation
read-out
time
exponentially faster for Fourier transformation
(Shor algorithm)
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important algorithms
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trapped ions
R. Blatt group (Innsbruck) '97 - '00
C. Monroe, D.Wineland, et al. Nature 2000
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spin resonance
7 mm
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quantum dots
F. Koppens et al., Nature 2006
J. R. Petta et al., Science 2005
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superconductor electronics
Y. Nakamura et al., Nature 1999
I. Chiorescu et al., Science 2003
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implementations
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from classic to quantum
we live in Hilbert Space H the state of our world
is y?