Download Motivation to Quantum Computing.

Part of Computational
Intelligence Course 2007
Quantum Computing
Marek Perkowski
Introduction to Quantum
Logic and
Reversible/Quantum Circuits
Marek Perkowski
Historical Background and Links
Quantum
Computation
&
Quantum
Information
Study of information
processing tasks that can
be accomplished using
quantum mechanical
systems
Cryptography
Quantum
Mechanics
Computer
Science
Information
Theory
Digital
Design
What is quantum
computation?
•
•
Computation with coherent atomic-scale
dynamics.
The behavior of a quantum computer is
governed by the laws of quantum
mechanics.
Why bother with quantum
computation?
• Moore’s Law: We hit the quantum level
2010~2020.
• Quantum computation is more powerful
than classical computation.
• More can be computed in less time—the
complexity classes are different!
The power of quantum
computation
• In quantum systems possibilities count,
even if they never happen!
• Each of exponentially many possibilities
can be used to perform a part of a
computation at the same time.
Nobody understands quantum
mechanics
“No, you’re not going to be able to understand it. . .
. You see, my physics students don’t understand it
either. That is because I don’t understand it.
Nobody does. ... The theory of quantum
electrodynamics describes Nature as absurd from
the point of view of common sense. And it agrees
fully with an experiment. So I hope that you can
accept Nature as She is -- absurd.
Richard Feynman
Absurd but taken seriously (not just
quantum mechanics but also
quantum computation)
• Under active investigation by many of the top
physics labs around the world (including CalTech,
MIT, AT&T, Stanford, Los Alamos, UCLA, Oxford,
l’Université de Montréal, University of Innsbruck,
IBM Research . . .)
• In the mass media (including The New York Times,
The Economist, American Scientist, Scientific
American, . . .)
• Here.
Quantum
Logic
Circuits
A beam splitter
Half of the photons leaving the light source arrive at
detector A;
the other half arrive at detector B.
A beam-splitter
0
0
1
1
50%
50%
The simplest explanation is that the beam-splitter
acts as a classical coin-flip, randomly sending each
photon one way or the other.
An interferometer
• Equal path lengths, rigid mirrors.
• Only one photon in the apparatus at a time.
• All photons leaving the source arrive at B.
• WHY?
Possibilities count
• There is a quantity that we’ll call the “amplitude” for each
possible path that a photon can take.
• The amplitudes can interfere constructively and destructively,
even though each photon takes only one path.
• The amplitudes at detector A interfere destructively; those at
detector B interfere constructively.
Calculating interference
• Arrows for each possibility.
• Arrows rotate; speed depends on frequency.
• Arrows flip 180o at mirrors, rotate 90o counter-clockwise
when reflected from beam splitters.
• Add arrows and square the length of the result to determine
the probability for any possibility.
Double slit interference
Quantum Interference : Amplitudes
are added and not intensities !
Interference in the interferometer
Arrows flip 180o at
mirrors, rotate 90o
counter-clockwise
when reflected
from beam
splitters
Quantum Interference
Two beam-splitters
0
0
1
1
100%
The simplest explanation must be wrong, since it would
predict a 50-50 distribution.
How to create a mathematical model that would explain
the previous slide and also help to predict new
phenomena?
More experimental data
0
1


0 sin  2 
2  
1 cos  
2
2

A new theory
The particle can exist in a linear combination or
superposition of the two paths
0
1
i
1
0 
1
2
2


0 sin  2 
2  
1 cos  
2
2

i
ei
0 
1
2
2
ei  1
i(ei  1)
0 
1
2
2
Probability Amplitude and
Measurement
If the photon is measured when it is in the state
2
0 0  1 1 then we get 0
with probability  0
and |1> with probability of |a1|2
0

1

0
1
0
2
1
2
2
 0  1
2
1
Quantum Operations
The operations are induced by the apparatus linearly,
i
1
that is, if
0 
0 
1
2
and
then
2
1
i
1 
0 
1
2
2
 i
 0 0  1 1   0 
0 
 2
i

  0
 1
2

1


1   1 
2 

1 

 0   0
2

1
i

0 
1
2
2 
1
i 
 1
1
2
2
Quantum Operations
Any linear operation that takes states
2
2
satisfying
0 0  1 1
 0  1  1
and maps them to states
'0 0  '1 1 satisfying
must be UNITARY
'
0

2
'
1

2
1
Linear Algebra
u00 u01 
U

 u10 u11 
is unitary if and only if

u
u
u


00
01
t
00
UU  


u10 u11  u01*
*
u10   1 0
I
*   

u11  0 1 

*
Linear Algebra
0
corresponds to
1
corresponds to
0 0  1 1
corresponds to
1
 
0
0
 
1
1
 0   0 
 0    1     
0
 1   1 
Linear Algebra
corresponds to

corresponds to






i
2
1
2
1 

2
i 

2
1 0 


i 
0 e 
Linear Algebra
0

corresponds to






i
2
1
2
1 

2
i 

2

1
0

 


i  
0 e  


i
2
1
2
1 

2
i 

2
1
 
0
Abstraction
The two position states of a photon in a
Mach-Zehnder apparatus is just one
example of a quantum bit or qubit
Except when addressing a particular physical
implementation, we will simply talk about
“basis” states 0 and 1
and unitary operations like
H
and

where
H
corresponds to
and

corresponds to






1
2
1
2
1 

2
1 

2
1 0 


i 
0 e 
An arrangement like
0

is represented with a network like
0
H

H
More than one qubit
If we concatenate two qubits

0
0  1 1
 
0
0  1 1

we have a 2-qubit system with 4 basis states
0 0  00
0 1  01
1 0  10
1 1  11
and we can also describe the state as
00 00  01 01  10 10  11 11
or by the vector
  0 0 


  01   0   0 
           
 1 0  1  1
 
 1 1
More than one qubit
In general we can have arbitrary superpositions
00 0 0  01 0 1  10 1 0  11 1 1
2
2
2
2
 00   01  10  11  1
where there is no factorization into the tensor
product of two independent qubits.
These states are called entangled.
Entanglement
• Qubits in a multi-qubit system are not
independent—they can become
“entangled.”
• To represent the state of n qubits we
use 2n complex number amplitudes.
Measuring multi-qubit systems
If we measure both bits of
00 0 0  01 0 1  10 1 0  11 1 1
we get
x y
with probability
 xy
2
Measurement
• ||2, for amplitudes of all states matching an output
bit-pattern, gives the probability that it will be read.
• Example:
0.316|00› + 0.447|01› + 0.548|10› + 0.632|11›
–The probability to read the rightmost bit as 0 is |0.316|2 +
|0.548|2 = 0.4
• Measurement during a computation changes the state of
the system but can be used in some cases to increase
efficiency (measure and halt or continue).
Sources
Origin of slides: John Hayes, Peter Shor, Martin
Lukac, Mikhail Pivtoraiko, Alan Mishchenko, Pawel
Kerntopf,
Ekert
Mosca, Mosca,
Hayes,
Ekert,
Lee Spector
in collaboration with
Herbert J. Bernstein, Howard Barnum, Nikhil Swamy
{lspector, hbernstein, hbarnum,
nikhil_swamy}@hampshire.edu}
School of Cognitive Science, School of Natural Science
Institute for Science and Interdisciplinary Studies (ISIS)
Hampshire College