Vorlesung Quantum Computing SS ‘08 1

Quantum ComputingPhysik der Quanten-Informationsverarbeitung

C. Meyer, C.M. Schneider

Vorlesung SS ‘08

Vorlesung Quantum Computing SS ‘08 2

Heise-online: 14.02.2007 12:49 << Vorige | Nächste >>

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.

in the media

Vorlesung Quantum Computing SS ‘08 3

plan…

1. introduction

2. quantum mechanical background

3. basic operations/superposition/entanglement

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. superconducting qubits

12. quantum error correction

13. invitation to Research Centre Jülich

Vorlesung Quantum Computing SS ‘08 4

technology of computation

• 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)

Rebuilt 1960 by K. ZuseDeutsches Museum München

Built

Techniques

Frequency

Speed

Format

Weight

Tasks

“Processor”: 600 relaysMemory: 1400 relays

Multiplication

22-bit digits floating point

Technical calculations,chess

Vorlesung Quantum Computing SS ‘08 5

ENIAC

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

1946

Vorlesung Quantum Computing SS ‘08 6

Moore’s law

Vorlesung Quantum Computing SS ‘08 7

birth of microelectronics

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

Vorlesung Quantum Computing SS ‘08 8

microprocessors

4-bit 8-bit 32-bit 64-bit

Intel 40041971

Intel 80801974 2000

20051985

16-bit

increase power of microprocessors bybus bit width and clock frequency

Vorlesung Quantum Computing SS ‘08 9

microprocessor designhttp://www.offis.de/

Embedded Hardware- / Software-SystemsDr. Jens Appell

Vorlesung Quantum Computing SS ‘08 10

breaking the barrier?

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(n

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source:

quantum effects in silicon

technology barrier silicon

year

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Asemiconductor industry

exponential extrapolation

Vorlesung Quantum Computing SS ‘08 11

computational power

we want to increase our capability of solving problems

speed accuracy complexity

increase 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

Vorlesung Quantum Computing SS ‘08 12

fundamental approach

Question: Is there a general method or process by which one could decide whether a mathematical proposition could be proved?

Answer: No!

On computable numbers, with an application to the Entscheidungsproblem Proceedings of the London Mathematical Society, Series 2, Vol.42 (1936 - 37) pages 230 to 265online available: http://web.comlab.ox.ac.uk/oucl/research/areas/ieg/e-library/sources/tp2-ie.pdf

“Turing Machine”

what is a computer and what kind of problems can it solve?

Vorlesung Quantum Computing SS ‘08 13

Turing machine

head

• Consists of a stripe and a head

ə 0 0 0 1 1 1 0 0 0 0 0

• Stripe consists of symbols “0”, “1”, “blank”, “ə”

b,f

• Head can be in states, e.g. „b“ and „f“• Symbols determine the action of the head:

- Writing/Erasing of symbol- Direction of reading- change of state

Vorlesung Quantum Computing SS ‘08 14

boolean algebra and logic gates

classical (irreversible) computing

gateinout

1-bit logic gates: identity

x NOT x

0 11 0

x Id

0 01 1

NOT

x NOT x

Vorlesung Quantum Computing SS ‘08 15

boolean algebra and logic gates

2-bit logic gates:

y x OR y

0

1

0

1

0

0

1

1

x

0

1

1

1

y x AND y

0

1

0

1

0

0

1

1

x

0

0

0

1

x

yx OR y

x

yx AND y

Vorlesung Quantum Computing SS ‘08 16

Turing Machine

0 1 ə

b R,b R,f P1,L,b R,b

f E,R,f R,f L,b

ə 0 0 0 1 1 1 0 0 0 0 0

head

b

head

b

head

b

head

f

head

f

head

b

head

f

head

f

head

f

head

b

1

head

f

head

b

1

head

b

1

head

b

head

b

head

f

table of states

what happens, depends on the states of the head

ə 0 0 0 1 1 1 0 0 0 0 0

head

f

head

f

head

f

head

b

head

f

56 + 7 = 63

Vorlesung Quantum Computing SS ‘08 17

complexity classes

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

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

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

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

NP: problems that can be solved with a NDTM in polynomial time.

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

Vorlesung Quantum Computing SS ‘08 18

traveling salesman problem

the traveling salesman problem is NP-complete

What is the shortest route betweena given number of cities?

scales exponentially with number of cities for a DTM

Can a physical implementation be found that provides a better solution?

Experiment

Vorlesung Quantum Computing SS ‘08 19

physical system designed for problem

soap bubbles can (theoretically) be used to solve some optimization problems in NP-complete

soap bubbles ≠ NDTMquantum computer ≠ NDTM

[Feynman1982] “ ....certain quantum mechanical effects cannot be simulated efficiently on a classical computer. This observation led to speculation that perhaps computation in general could be done more efficiently if it made use of these quantum effects.”

R. P. Feynman, Int. J. Theor. Phys. 21, 467(1982); Found. Phys. 16, 507(1986)

Vorlesung Quantum Computing SS ‘08 20

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

Vorlesung Quantum Computing SS ‘08 21

quantum bits

conventional bit

on <=> 3.2 - 5.5 V <=> 1

off <=> -0.5 - 0.8 V <=> 0

quantum mechanical bit (qubit)

| 0 <=> <=>

| 1 <=> <=>

10(

(

01(

(

a1| 0 + a2| 1 = a1

a2( )

superposition:

Vorlesung Quantum Computing SS ‘08 22

quantum parallelism

a1 F |00>+

a2 F |01>+

a3 F |10>+

a4 F |11>

}{a1 |00>

+a2 |01>

+a3 |10>

+a4 |11>

}{input

b1 |00>+

b2 |01>+

b3 |10>+

b4 |11>

}{=

output

F

Vorlesung Quantum Computing SS ‘08 23

quantum computing

calculationpreparation read-outtime

classical bit

1 ON 3.2 – 5.5 V

0 OFF -0.5 – 0.8 V

exponentially faster for Fourier transformation (Shor algorithm)

quantum-bit (qubit)

0 1

a10 + a21 =a1a2

H H-1U |A|

time

decoherence

Vorlesung Quantum Computing SS ‘08 24

important algorithms

database search N data sets

e.g. find no. in phonebook (60 million data sets)

30 millionsteps

7746steps

89 steps1019 steps

taskalgorithm classicalcomputer

quantumcomputer

prime factor decomposition

e.g. 128 bit decoding

Vorlesung Quantum Computing SS ‘08 25

trapped ions

C. Monroe, D.Wineland, et al. Nature 2000

R. Blatt group (Innsbruck) '97 - '00

C. Monroe group, Michigan ‘06

Vorlesung Quantum Computing SS ‘08 26

spin resonance

7 mm

Vorlesung Quantum Computing SS ‘08 27

quantum dots

J. R. Petta et al., Science 2005

F. Koppens et al., Nature 2006

Vorlesung Quantum Computing SS ‘08 28

superconductor electronics

Y. Nakamura et al., Nature 1999

I. Chiorescu et al., Science 2003

Vorlesung Quantum Computing SS ‘08 29

implementations

atoms or ions in trapselectronic states of atoms/ionsvibrational modes

spintronicelectron spins in quantum dotsexchange interaction

superconductor electronicsCooper pairs or fluxJosephson coupling

spin resonance (NMR, ESR)spins in molecules or solid state matrixhyperfine, exchange, or magnetic dipolar interaction

(CH)5 Fe (CO)2

F

C C

C C

F

F

F

F

Vorlesung Quantum Computing SS ‘08 30

from classic to quantum

we live in Hilbert Space Hthe state of our world is |