2007 001-motivation-to-quantum-computing

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

2cos2

2sin2

A new theoryA new theory

0

1

0

1

2cos2

2sin2

12102

i 12

e02i i

The particle can exist in a linear combination or superposition of the two paths

12)1e(i02

1e ii

Probability Amplitude and Probability Amplitude and MeasurementMeasurement

0

1

0

1

20

If the photon is measured when it is in the state then we get with probability

and |1> with probability of |a1|210 10

21

200

121

20

Quantum OperationsQuantum OperationsThe operations are induced by the apparatus linearly, that is, if

andthen

12

i02112

102i10 1010

12102

i0

12i02

11

12i

2102

12i

1010

Quantum OperationsQuantum Operations

Any linear operation that takes statessatisfying

and maps them to statessatisfying

must be UNITARY

121

20 10 10

10 '1

'0 12'

12'

0

Linear AlgebraLinear Algebra

I1001

uuuu

uuuuUU *

**

1110

0100t

11*01

1000

1110

0100uuuuU

is unitary if and only if


10 10

0

01

1

10

1

010 1

001

corresponds to

corresponds to

corresponds to


2i

21

21

2i

corresponds to

corresponds to

ie0

01


0

corresponds to

01

2i

21

21

2i

ie0

01

2i

21

21

2i

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

21

21

21

21

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

1211

210

201

200

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

yx 2xy

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