Quantum Computing

Vorlesung SS ‘08
Quantum Computing
Physik der Quanten-Informationsverarbeitung
C. Meyer, C.M. Schneider
Vorlesung Quantum Computing SS ‘08
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in the media
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 DWave-Systems CEO Herb Martin.
Vorlesung Quantum Computing SS ‘08
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plan…
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
introduction
quantum mechanical background
basic operations/superposition/entanglement
quantum computing with ion traps
Deutsch-Josza algorithm and its implementation
NMR quantum computing
Shor algorithm and its implementation (15 = 5x3)
magnetic resonance QC in solid state
quantum dots for quantum computing
first experiments
superconducting qubits
quantum error correction
13.
invitation to Research Centre Jülich
Vorlesung Quantum Computing SS ‘08
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technology of computation
Erbaut
Built
Technik
Techniques
1941: Kondrad Zuse
Rechenwerk:
600 relays
Relais
“Processor”: 600
Speicher:
1400relays
Relais
Memory: 1400
Taktfrequenz
5.3 Hertz
Frequency
RechenMultiplikation: 3 sec
Multiplication
Speed
geschwindigkeit Addition:
0,7 sec
Datenformat
22bit-Zahlen
22-bit digits im
Format
Gleitkommaformat
floating point
Weight
Gewicht
~1t
Einsatzgebiete techn.
Berechnungen,
Technical
calculations,
Tasks
Schach
chess
2001: Vandersypen et al.
1016 Moleküle mit je 7
NMR aktiven Atomen
1 Hertz
einfaches logisches
Gatter: ~ 20 ms
4bit-Zahlen
~4t
Rebuilt 1960 by K. Zuse
Primfaktorzerlegung:
Deutsches Museum München
15 = 3 · 5
• 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)
Vorlesung Quantum Computing SS ‘08
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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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Moore’s 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
16-bit
32-bit
Intel 4004
1971
Intel 8080
1974
1985
2000
64-bit
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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quantum effects in silicon
technology barrier silicon
size of viruses and DNA
semiconductor industry
exponential extrapolation
proteins, macro-molecules
minimum size of chip components (nm)
breaking the barrier?
year
source:
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computational power
we want to increase our capability of solving problems
increase
{
speed
accuracy
complexity
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
Vorlesung Quantum Computing SS ‘08
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Turing machine
ə
0
0
0
1
1
1
0
0
0
0
0
b,f
head
• Consists of a stripe and a head
• Stripe consists of symbols “0”, “1”, “blank”, “ə”
• 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
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boolean algebra and logic gates
classical (irreversible) computing
in
1-bit logic gates:
out
gate
identity
x
0
1
Id
0
1
NOT
x
0
1
x
Vorlesung Quantum Computing SS ‘08
NOT x
1
0
NOT x
14
boolean algebra and logic gates
2-bit logic gates:
x
x
x OR y
y
x
y
0
0
1
1
0
1
0
1
x OR y
0
1
1
1
Vorlesung Quantum Computing SS ‘08
x AND y
y
x
y
0
0
1
1
0
1
0
1
x AND y
0
0
0
1
15
Turing Machine
ə
0
0
0
1
1
1
0
0
0
0
0
ə
b
0
b
0
b
0
b
1
b
1
f
1
bf
0
1
bf
0
1
fbf
10
ff
b
0
0
ff
head head head head head head head
head head head head head
what happens, depends on the states of the head
table of states
56 + 7 = 63
Vorlesung Quantum Computing SS ‘08
0
1
ə
b
R,b
R,f
P1,L,b
f
E,R,f
R,f
L,b
R,b
16
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
Can a physical implementation be
found that provides a better solution?
Experiment
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physical system designed for problem
soap bubbles can (theoretically) be used to solve some
optimization problems in NP-complete
quantum
soap bubbles
computer
≠ NDTM
≠ 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
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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
<=>
3.2 - 5.5 V
<=> 1
off
<=>
-0.5 - 0.8 V
<=> 0
quantum mechanical bit (qubit)
| 1  <=>
(
<=> (
<=>
Vorlesung Quantum Computing SS ‘08
1
0
0
1
(
(
| 0  <=>
superposition:
a1| 0  + a2| 1  =
a1
a2
( )
21
quantum parallelism
input
output
{} { }{}
a1 |00>
+
a2 |01>
+
a3 |10>
+
a4 |11>
Vorlesung Quantum Computing SS ‘08
F
a1 F |00>
+
a2 F |01>
+
a3 F |10>
+
a4 F |11>
=
b1 |00>
+
b2 |01>
+
b3 |10>
+
b4 |11>
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quantum computing
quantum-bit (qubit)
0  
a
a1 0 + a2 1 = a1
2
1  
classical bit
1  ON  3.2 – 5.5 V
0  OFF  -0.5 – 0.8 V
calculation
decoherence
preparation
Y0

H

U
read-out

H-1
time

Y|A|Y
time
exponentially faster for Fourier transformation (Shor algorithm)
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important algorithms
algorithm
task
classical
computer
quantum
computer
database search
N data sets
e.g. find no. in phonebook
(60 million data sets)
30 million
steps
7746
steps
1019 steps
89 steps
prime factor decomposition
e.g. 128 bit decoding
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trapped ions
R. Blatt group (Innsbruck) '97 - '00
C. Monroe, D.Wineland, et al. Nature 2000
Vorlesung Quantum Computing SS ‘08
C. Monroe group, Michigan ‘06
25
spin resonance
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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
Vorlesung Quantum Computing SS ‘08
I. Chiorescu et al., Science 2003
28
implementations
atoms or ions in traps
electronic states of atoms/ions
vibrational modes
F
spin resonance (NMR, ESR)
spins in molecules or solid state matrix
hyperfine, exchange, or magnetic dipolar interaction
F
C
C
F
(CH)5 Fe (CO)2
F
C
C
F
spintronic
electron spins in quantum dots
exchange interaction
superconductor electronics
Cooper pairs or flux
Josephson coupling
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from classic to quantum
we live in Hilbert Space H
the state of our world is |y
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