Download motivation-to-quantum

Part of Computational
Intelligence Course 2011
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).
What to remember?
1.
2.
3.
4.
5.
The concept of qubit.
Tensor (Kronecker) Products.
Matrix notation for gates.
Analysis of a circuit from quantum gates.
Measurements of single qubit and multi-qubit
circuits.
6. Hadamard and Phase gates.
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