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Quantum Computing Quantum Computing
Marek Perkowski
Part of Computational Part of Computational Intelligence Course 2007 Intelligence Course 2007
Introduction to Quantum Introduction to Quantum Logic and Logic and
Reversible/Quantum Circuits Reversible/Quantum Circuits
Marek Perkowski
Historical Background and LinksHistorical Background and LinksQuantum
Computation&
QuantumInformation
ComputerScience
InformationTheory
CryptographyQuantum
Mechanics
Study of information processing tasks that can be accomplished using quantum mechanical systems
Digital Design
What is quantum What is quantum computation?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 Why bother with quantum computation?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 The power of quantum computationcomputation
• 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 Nobody understands quantum mechanicsmechanics
“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 Absurd but taken seriously (not just quantum mechanics but also quantum mechanics but also
quantum computation)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 splitterA beam splitter
Half of the photons leaving the light source arrive at detector A; the other half arrive at detector B.
A beam-splitterA beam-splitter
0
1
0
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 interferometerAn 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 countPossibilities 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 interferenceCalculating 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 interferenceDouble slit interference
Quantum Interference : Amplitudes Quantum Interference : Amplitudes are added and not intensities !are added and not intensities !
Interference in the interferometer Interference in the interferometer
Arrows flip 180o at mirrors, rotate 90o counter-clockwise when reflected from beam splitters
Quantum InterferenceQuantum Interference
0
1
0
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?
Two beam-splitters
More experimental dataMore experimental data
0
1
0
1
2
cos2
2
sin2
A new theoryA new theory
0
1
0
1
2
cos2
2
sin2
12
10
2
i 1
2
e0
2
i i
The particle can exist in a linear combination or superposition of the two paths
12
)1e(i0
21e ii
Probability Amplitude and Probability Amplitude and MeasurementMeasurement
0
1
0
1
2
0
If the photon is measured when it is in the state then we get with probability
and |1> with probability of |a1|210 10
2
1
2
00
12
1
2
0
Quantum OperationsQuantum OperationsThe operations are induced by the apparatus linearly, that is, if
andthen
1
2
i0
2
11
2
10
2
i10 1010
12
10
2
i0
12
i0
2
11
12
i
2
10
2
1
2
i1010
Quantum OperationsQuantum Operations
Any linear operation that takes statessatisfying
and maps them to statessatisfying
must be UNITARY
12
1
2
0 10 10
10 '1
'0 1
2'1
2'0
Linear AlgebraLinear Algebra
I10
01uu
uu
uu
uuUU *
**
1110
0100t
11*01
1000
1110
0100
uu
uuU
is unitary if and only if
Linear AlgebraLinear Algebra
10 10
0
0
1
1
1
0
1
010 1
0
0
1
corresponds to
corresponds to
corresponds to
Linear AlgebraLinear Algebra
2
i
2
12
1
2
i
corresponds to
corresponds to
ie0
01
Linear AlgebraLinear Algebra
0
corresponds to
0
1
2
i
2
12
1
2
i
ie0
01
2
i
2
12
1
2
i
AbstractionAbstractionThe 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 and and unitary operations like
and
0 1
H
where corresponds toH
2
1
2
12
1
2
1
and corresponds to
ie0
01
An arrangement like
0
is represented with a network like
H H0
More than one qubitMore than one qubit
10 10
If we concatenate two qubits
11100100 11011000
10 10
we have a 2-qubit system with 4 basis states0000 0110 1001 1111
and we can also describe the state as
or by the vector
1
0
1
0
11
01
0
00
1
More than one qubitMore than one qubitIn general we can have arbitrary superpositions
11011000 11100100
12
11
2
10
2
01
2
00
where there is no factorization into the tensor product of two independent qubits.These states are called entangled.
EntanglementEntanglement
• 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 systemsMeasuring multi-qubit systems
If we measure both bits of
we get with probability
11011000 11100100
yx2
xy
MeasurementMeasurement ||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).
SourcesSources
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
Origin of slides: John Hayes, Peter Shor, Martin Lukac, Mikhail Pivtoraiko, Alan Mishchenko, Pawel Kerntopf, Mosca, Ekert