Quantum Computing - Advanced Theory of Computation - Lecture Slides, Slides of Theory of Computation

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QuantumComputing

Overview

^

Introduction

^

Data Representation

Computational Complexity Implementation Technologies Quantum Computer Languages

Introduction to quantum mechanics ^

Quantum mechanics is a more fundamental theorythan Newtonian mechanics and classicalelectromagnetism

^

It provides accurate and precise descriptions formany phenomena that these "classical" theoriessimply cannot explain on the atomic andsubatomic level

What is a quantum computer?

^

A quantum computer is a machine that performs calculations based on the laws of quantum mechanics,which is the behavior of particles at the sub-atomiclevel.

Computer technology is makingdevices smaller and smaller…

…reaching a point where classicalphysics is no longer a suitablemodel for the laws of physics.

Physics and Computation

•^

Information is stored in a physical medium,and manipulated by physical processes.

-^

The laws of physics dictate the capabilities ofany information processing device.

-^

Designs of “classical” computers are implicitlybased in the

classical

framework for physics

•^

Classical physics is known to be wrong orincomplete… and has been replaced by a morepowerful framework:

quantum mechanics

“No, you’re not going to be able to understand it...

. You see, my physics students don’t understandit either. That is because I don’t understand it.Nobody

does.

The

theory

of

quantum

electrodynamics describes Nature as absurdfrom the point of view of common sense. And itagrees

fully with an experiment. So I hope that

you can accept Nature as She is -- absurd. Richard Feynman

Nobody understands quantummechanics

consider a setup involving a photon source,

a half-silvered mirror (beamsplitter),and a pair of photon detectors. photonsource

beamsplitter

detectors

A simple experiment in optics

… consider a modification of the experiment…

100%

The simplest explanation is wrong!

The simplest explanation forthe modified setup would stillpredict a 50-50 distribution…

full mirror

The “weirdness” of quantum mechanics

Classical probabilities…

Consider a computation tree for a simple two-step (classical) probabilisticalgorithm, which makes a coin-flip at each step, and whose output is 0 or 1:

1 2

1 2 1 2 1 2 1 2

(^0101)

The probability of the computation followinga given path is obtained by multiplying theprobabilities along all branches of thatpath… in the example the probability thecomputation follows the red path is

1 4 1 2 1 2

 

The probability of the computation giving theanswer 0 is obtained by adding theprobabilities of all paths resulting in 0:

(^12) (^14) (^1 )

 

… consider a modification of the experiment…

The simplest explanation forthe modified setup would stillpredict a 50-50 distribution…

full mirror

Explanation of experiment

0

0 1 2

1 1 2

100%

0 1 0 1 2 0 1 2





(^10)

1 1 2 1 1 2





Representation of Data ^

Quantum computers, which have not been built yet, would be based onthe strange principles of quantum mechanics, in which the smallestparticles of light and matter can be in different places at the same time. ^

In a quantum computer, one "qubit" - quantum bit - could be both 0 and1 at the same time. So with three qubits of data, a quantum computercould store all eight combinations of 0 and 1 simultaneously. Thatmeans a three-qubit quantum computer could calculate eight timesfaster than a three-bit digital computer. ^

Typical personal computers today calculate 64 bits of data at a time. Aquantum computer with 64 qubits would be 2 to the 64th power faster,or about 18 billion billion times faster. (Note: billion billion is correct.)

Representation of Data

• Qubits

A physical implementation of a qubit could use the two energylevels of an atom. An excited state representing |1> and aground state representing |0>.

ExcitedState GroundState

Nucleus

Light pulse offrequency

^ for time interval t Electron

State |0>

State |1>

Representation of Data - Superposition

A single qubit can be forced into a

superposition

of the two states

denoted by the addition of the state vectors:

Where



and



are complex numbers and

| 

|

+ |



|^

= 1

1

2

1

2

1

2

2

2

A qubit in superposition is in both of the

states |1> and |0 at the same time