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Quantum Computing Part 1 Quantum Mechanics

Overview

Technological limits

- Gordon Moores law
- The observation made in 1965 by Gordon Moore,

co-founder of Intel, that the number of

transistors per square inch on integrated

circuits had doubled every year since the

integrated circuit was invented. - Moore predicted that this trend would continue

for the foreseeable future. - In subsequent years, the pace slowed down a bit,

but data density has doubled approximately every

18 months, and this is the current definition of

Moore's Law, which Moore himself has blessed. - Most experts, including Moore himself, expect

Moore's Law to hold for at least another two

decades.

Technological limits

- But all good things may come to an end
- We are limited in our ability to increase
- the density and
- the speed of a computing engine.
- Reliability will also be affected
- to increase the speed we need increasingly

smaller circuits (light needs 1 ns to travel 30

cm in vacuum) - smaller circuits ? systems consisting only of a

few particles subject to Heissenberg uncertainty

Energy/operation

- If there is a minimum amount of energy dissipated

to perform an elementary operation, then to

increase the speed, thus the number of operations

performed each second, we require a liner

increase of the amount of energy dissipated by

the device. - The computer technology vintage year 2000

requires some 3 x 10-18 Joules per elementary

operation. - Even if this limit is reduced say 100-fold we

shall see a 10 (ten) times increase in the amount

of power needed by devices operating at a speed

103 times larger than the sped of today's devices.

Power dissipation, circuit density, and speed

- In 1992 Ralph Merkle from Xerox PARC calculated

that a - 1 GHz computer operating at room temperature,

with 1018 gates packed in a volume of about 1 cm3

would dissipate 3 MW of power. - A small city with 1,000 homes each using 3 KW

would require the same amount of power - A 500 MW nuclear reactor could only power some

166 such circuits.

Talking about the heat

- The heat produced by a super dense computing

engine is proportional with the number of

elementary computing circuits, thus, with the

volume of the engine. - The heat dissipated grows as the cube of the

radius of the device. - To prevent the destruction of the engine we have

to remove the heat through a surface surrounding

the device. Henceforth, - our ability to remove heat increases as the

square of the radius - while the amount of heat increases with the cube

of the size of the computing engine.

Quantum Quantum mechanics

- Quantum is a Latin word meaning some quantity.
- In physics it is used with the same meaning as

the word discrete in mathematics, i.e., some

quantity or variable that can take only sharply

defined values as opposed to a continuously

varying quantity. - Quantum mechanics is a mathematical model of the

physical world

I'll bet any quantum mechanic in the service

would give the rest of his life to fool around

with this gadget. - Chief Engineer Quinn,

Forbidden Planet , 1956

Heissenberg uncertainty principle

- Heisenberg uncertainty principle says we cannot

determine both the position and the momentum of a

quantum particle with arbitrary precision. - In his Nobel prize lecture on December 11, 1954

Max Born says about this fundamental principle of

Quantum Mechanics - ... It shows that not only the determinism of

classical physics must be abandoned, but also the

naive concept of reality which looked upon atomic

particles as if they were very small grains of

sand. At every instant a grain of sand has a

definite position and velocity. This is not the

case with an electron. If the position is

determined with increasing accuracy, the

possibility of ascertaining its velocity becomes

less and vice versa.''

Quantum mechanics displays some weird features

that cannot be duplicated by any classical system.

Quantum Weirdness

- Can Weirdness be useful?
- Quantum Computing
- Quantum Encryption
- Quantum teleportation
- Quantum Global Positioning

Electromagnetic Weirdness

- In 1867 Maxwell proposed a theory unifying

electricity, magnetism, and optics.

The theory was weird! Light was propagated as a

wave in a vacuum - no underlying medium.

Only when Marconi invented the radio transmitter

and receiver did the weirdness become accepted

as normal, or even obvious.

Quantum Weirdness

some remarkable quotes

- The quantum is the greatest mystery we've got.

Never in my life was I more up a tree than today.

-John Wheeler - If someone says that he can think about quantum

physics without becoming dizzy, that shows only

that he has not understood anything whatever

about it. - Niels Bohr - A picture may seem extraordinarily strange to you

and after some time not only does it not seem

strange but it is impossible to find what there

was in it that was strange. - GERTRUDE STEIN

concerning modern art.

John Wheeler and friends

Niels Bohr and big money.

Gertrude Stein by Picasso

Classical versus Quantum Experiments

- Classical Experiments
- Experiment with bullets
- Experiment with waves
- Quantum Experiments
- Two slits Experiment with electrons
- Stern-Gerlach Experiment

Experiment with bullets

(b)

Figure 1 Experiment with bullets

Experiment with bullets

H1 is open H2 is closed

(b)

Figure 1 Experiment with bullets

Experiment with bullets

H1 is closed H2 is open

(b)

Figure 1 Experiment with bullets

Experiment with bullets

H1 is open H2 is open

(b)

Figure 1 Experiment with bullets

Experiment with Waves

H1 is closed H2 is closed

wave source

H1

H2

wall

(a)

Figure 2 Experiments with waves

Experiment with Waves

H1 is open H2 is closed

wave source

H1

H2

wall

(b)

(a)

Figure 2 Experiments with waves

Experiment with Waves

H1 is closed H2 is open

wave source

H1

H2

I2(x)

wall

(b)

(a)

Figure 2 Experiments with waves

Experiment with Waves

H1 is open H2 is open

wave source

H1

H2

wall

This is a result of interference

(a)

Figure 2 Experiments with waves

Two Slit Experiment

Are electrons particles or waves?

Figure 3 Two slit experiment

Two Slit Experiment

Figure 3 Two slit experiment

Two Slit Experiment With Observation

Now we add light source

H1

H2

? Decoherence

wall

(a)

Figure 4 Two slit experiment with observation

Stern-Gerlach Experiment

1922 - a beam of silver atoms directed through an

inhomogeneous magnetic field would be forced into

two beams. Consistent with the possession of an

intrinsic angular momentum and a magnetic moment

by individual electrons.

Stern-Gerlach experiment with spin-1/2 particles

Conclusions From the Experiments

- Limitations of classical mechanics
- Particles demonstrate wavelike behavior
- Effect of observations cannot be ignored
- Evolution and measurement must be distinguished

Can we use these phenomena practically? Quantum

computing and information

Quantum Weirdnesses

- 1. Complementarity Every particle is a wave, and

every wave is a particle. It just depends on how

you measure it.

Leonard Mandel

classical particle

classical wave

quantum photon

Quantum Weirdnesses

- 2. Superposition If Agt represents one state of

the system, and Bgt represents another, then - aAgt bBgt is also a state of the system.
- Examples
- heregt theregt
- live catgt dead catgt
- .

Schrödinger Cat State

Quantum Weirdnesses

- 3. Entanglement The quantum state of a two

particle system in which the states of the

individual particles are inseparable even though

the two particles are separated.

A revolutionary approach to computing and

communication

- We need to consider a revolutionary rather than

an evolutionary approach to computing. - Quantum theory does not play only a supporting

role by prescribing the limitations of physical

systems used for computing and communication. - Quantum properties such as
- uncertainty,
- interference, and
- entanglement
- form the foundation of a new brand of theory,

the quantum information theory where

computational and communication processes rest

upon fundamental physics.

Milestones in quantum physics

- 1900 - Max Plank presents the black body

radiation theory the quantum theory is born. - 1905 - Albert Einstein develops the theory of the

photoelectric effect. - 1911 - Ernest Rutherford develops the planetary

model of the atom. - 1913 - Niels Bohr develops the quantum model of

the hydrogen atom. - 1923 - Louis de Broglie relates the momentum of a

particle with the wavelength - 1925 - Werner Heisenberg formulates the matrix

quantum mechanics. - 1926 - Erwin Schrodinger proposes the equation

for the dynamics of the wave function. - 1926 - Erwin Schrodinger and Paul Dirac show the

equivalence of Heisenberg's matrix formulation

and Dirac's algebraic one with Schrodinger's wave

function. - 1926 - Paul Dirac and, independently, Max Born,

Werner Heisenberg, and Pasqual Jordan obtain a

complete formulation of quantum dynamics. - 1926 - John von Newmann introduces Hilbert spaces

to quantum mechanics. - 1927 - Werner Heisenberg formulates the

uncertainty principle.

Milestones in computing and information theory

- 1936 - Alan Turing dreams up the Universal

Turing Machine, UTM. - 1936 - Alonzo Church publishes a paper asserting

that every function which can be regarded as

computable can be computed by an universal

computing machine''. Church Thesis. - 1946 - A report co-authored by John von Neumann

outlines the von Neumann architecture. - 1948 - Claude Shannon publishes A Mathematical

Theory of Communication. - 1961 - Rolf Landauer decrees that computation is

physical and studies heat generation. - 1973 - Charles Bennet studies the logical

reversibility of computations. - 1981 - Richard Feynman suggests that physical

systems including quantum systems can be

simulated exactly with quantum computers. - 1982 - Peter Beniof develops quantum mechanical

models of Turing machines. - 1984 - Charles Bennet and Gilles Brassard

introduce quantum cryptography. - 1985 - David Deutsch reinterprets the

Church-Turing conjecture. - 1993 - Bennet, Brassard, Crepeau, Josza, Peres,

Wooters discover quantum teleportation. - 1994 - Peter Shor develops a clever algorithm for

factoring large numbers.

Deterministic versus probabilistic photon behavior

The puzzling nature of light

- If we start decreasing the intensity of the

incident light we observe the granular nature of

light. - Imagine that we send a single photon.
- Then either detector D1 or detector D2 will

record the arrival of a photon. - If we repeat the experiment involving a single

photon over and over again we observe that each

one of the two detectors records a number of

events. - Could there be hidden information, which controls

the behavior of a photon? - Does a photon carry a gene and one with a

transmit'' gene continues and reaches detector

D2 and another with a reflect'' gene ends up at

D1?

Each detector detects single photons. Why? What

is a hidden information that controls this

In an attempt to solve this puzzle we design this

setup

The puzzling nature of light (contd)

- Consider now a cascade of beam splitters.
- As before, we send a single photon and repeat the

experiment many times and count the number of

events registered by each detector. - According to our theory we expect the first beam

splitter to decide the fate of an incoming

photon - the photon is either reflected by the first beam

splitter or transmitted by all of them. - Thus, only the first and last detectors in the

chain are expected to register an equal number of

events. - Amazingly enough, the experiment shows that all

the detectors have a chance to register an event.

Why all these detectors detect light?

Polarization experiment

- Polarization of light electric field vector is

confined to a single direction. This is a

quantum-mechanical property of photons. - Light source Assume each photon has a random

polarization - Filter A Horizontally polarized

Polarization experiment

- Polarization of light electric field vector is

confined to a single direction. This is a

quantum-mechanical property of photons. - Light source Assume each photon has a random

polarization - Filter A Horizontally polarized
- Filter B Vertically polarized

Polarization experiment

- Polarization of light electric field vector is

confined to a single direction. This is a

quantum-mechanical property of photons. - Light source Assume each photon has a random

polarization - Filter A Horizontally polarized
- Filter B Vertically polarized
- Filter C polarized at 45 degrees

Polarization experiment explanation

- Photons polarization state can be modeled by a

unit vector pointing in the appropriate direction - Let the basis vectors be ? and ?
- Any polarization can be expressed as
- where a and b are complex numbers such that

- Measurement postulate of quantum mechanics
- Any device measuring a quantum system has an

associated orthonormal basis with respect to

which the measurement takes place. - Measurement of a state transforms the state into

one of these basis vectors. - The probability that the state is measured as

basis vector u is the norm of the amplitude of

the component of the original state in the

direction of u.

- Example
- Let be the polarization

state of a photon. - Then this state will be measured as ? with

probability a2 - and as ? with probability b2 .
- These are indeed probabilities, since

. - The measurement will also change the state to the

result - of the measurement.
- Its original value cannot be determined.

- Back to the polarization experiment
- Original light source produces photons with

random polarizations - Thus 50 are measured as horizontal, and let

through. (Why?) - Those photons are now in the horizontal state,

- None are let through by the vertical filter.

- Now let the 45o filter be put behind the

horizontal filter. The 45o filter measures

photon polarization with respect to a new basis - Photons in state ? will be measured as

with probability 1/2. So half will get through.

- Likewise, filter B will let half of these

through. - Total photons getting through all three filters

(1/2)3 1/8.

Quantum Mechanics

Quantum mechanics vs. Classical

Mechanics

Formulated to explain the behavior of microscopic

systems.

Formulated to explain the behavior of macroscopic

objects.

Newtons second law Integrate twice ? x(t).

Two constants of integration ? two additional

pieces of information required to uniquely define

the state of the system (e.g. xo and vo). The

state of the system is defined by FORCES,

POSITIONS, VELOCITIES Knowledge of the initial

state of the system can predict future states

precisely.

Quantum Mechanics

- Designed to describe the behavior observed for

microscopic particles and systems. - Behavior
- discrete rather than continuous energy levels
- photons, electrons and nuclei exhibit both wave

and particle nature. - History
- 1925 Werner Heisenberg, Max Born and Pascual

Jordon introduced matrix-based mathematical

formalism to describe observed quantum mechanical

phenomena - 1926 Erwin Schrödinger introduced a

differential equation and its solution that

equivalently describes equation observed quantum

mechanical phenomena

The postulates of quantum mechanics (QM)

- Postulate I
- For any possible state of a system, there is a

function y of the coordinates of the parts of the

system and time that completely describes the

system.

Y Is called a wave function. For two particles

system,

The wave function square Y2 is proportional to

probability. Since Y may be complex, we are

interested in YY, where Y is the complex

conjugate (i ? -i) of Y. The quantity YYdt is

proportional to the probability of finding the

particles of the system in the volume element, dt

dxdydz.

that is the probability of finding the particle

in the universe is 1 ? normalization condition.

The postulates of quantum mechanics (QM)

- Orthogonality of two wave functions

Example sinq and cosq are orthogonal functions.

Fourier series expansion sin(nq) and cos(nq)

orthogonal functions

The Wavefunction

Postulate I For any possible state of a system,

there is a function y of the coordinates of the

parts of the system and time that completely

describes the system

- Y(t) f ig
- f and g are real functions of coordinates and

time - An abstract, complex quantity but related to

physically measurable quantities - State is dependent on coordinates (spatial and

spin) and time - The time-dependent Schrödinger equation
- A single integration with respect to time is

required to obtain Y(t), so that only one

constant of integration is required to predict

future states of the system.

The Wavefunction

How can we describe the state of a quantum

mechanical system such as nuclear spins? Complex

wavefunction Y(t) Y(t,t) description of

all knowable information about the state of the

system What if we want to know if the system

is in a given state Y(t) at time t? The

probability that the system is in the state given

by Y(t) at time t is P Y(t)Y(t) Y2 For

example, for a single particle at time t', the

wavefunction is Y(x,y,z,t'), and the probability

that time t', the particle is in a given volume

of space (dxdydz) is given by Y(x,y,z,t')2

dxdydz

Time dependence

Spatial and spin coordinates (independent of time)

probability density

Complex Conjugate

Complex Conjugate Y Y f ig Y f

ig (replace i with i) YY (f ig) (f ig)

f2 ifg ifg (i2)g f2 g2

real, non-negative (as P should be!)

Normalization Condition

Since the system must exist in some state at

time, t, if we integrate over all coordinates of

the system (t represents the generalized

coordinates, which may include spatial

coordinates and spin state), the probability

density is 1. Normalization condition ?

Y(t)Y(t)dt 1 i.e. the probability of finding

the particle somewhere in space is one. The

Schrödinger equation describes the evolution in

time of a given system

The Schrödinger Equation

constant, often omitted

Hamiltonian FORCES

The Hamiltonian, H, represents the forces acting

on the system, which can be time-dependent or

time-independent.

The time-independent Schrödinger equation

If the Hamiltonian is time-independent, then the

Schrödinger equation can be solved by separation

of variables. Y(t) y(t) f(t) The Hamiltonian

acts only on the generalized coordinate part of

Y(t), y(t), since H is independent of time (i.e.

f(t) acts as a constant). Choosing units so

that h 1 Multiplying both sides by ? Y(t)

dt

time indep.

time- dep.

1

The time-independent Schrödinger equation

Noting that the wave function is normalized and

multiplying by i To simplify things, lets

define So that Now the goal is to solve

this differential equation.

The time-independent Schrödinger equation

The solution is This can be seen by

differentiating with respect to t Now we can

write

The time-independent Schrödinger equation

Differentiating with respect to

time Multiplying both sides by i Comparing

to the original form of the Schrödinger equation,

we get Which is the time-independent

Schrödinger equation. The time dependence can be

thought of as a phase factor that cancels when

the probability distribution is calculated

The time-independent Schrödinger equation

We have just shown that H is an operator E is

an energy ? is the wave function

Probabilistic nature of QM

- If Y is a solution to the Schrödinger equation,

then so is cY (c arbitrary constant), and Sci?i

is also a solution, where each ?i is a possible

state of the system. - Y isnt really a physical wave. It is an

abstract mathematical entity that yields

information about the state of the system. - Y gives information on the probabilities for

possible outcomes of measurements of the systems

physical properties.

Quantum mechanics says a lot, but does not

really bring us any closer to the secrets of the

Old One. I, at any rate, am convinced that He

does not throw dice. Albert Einstein

Operators

- Postulate II
- With every physical observable q there is

associated an operator Q, which when operating

upon the wavefunction associated with a definite

value of that observable will yield that value

times the wavefunction F, i.e. QF qF.

H

http//hyperphysics.phy-astr.gsu.edu/hbase/quantum

/qmoper.html

Eigenvalues and Eigenfunctions

Eigenvalue equations An operator acts on a

function to give another function Âf(x)

g(x) Where Â is an operator. For example, an

operator can be defined where â represents

multiplying by a âf(x) af(x) Depending upon

the operator, the new function can be very

different from the original function. However,

in a special case, the new function is a multiple

of the original function Âf(x) lf(x) In this

case, f(x) is said to be an eigenfunction of Â

with the associated eigenvalue, l. In the case

of the Schrödinger equation with a

time-independent Hamiltonian, H is the operator,

Y is the eigenfunction and E is the eigenvalue H

Y EY

Operators

- (1) The operators are linear, which means that
- O(Y1 Y2) OY1 OY2
- The linear character of the operator is related

to the superposition of states and waves

reinforcing each other in the process - (2) The second property of the operators is that

they are Hermitian (the 19th century French

mathematician Charles Hermite). - Hermitian matrix is defined as the transpose of

the complex conjugate () of a matrix is equal to

itself, i.e. (M)T M

In QM, the operator O is Hermitian if

C. Hermite

http//commons.wikimedia.org/wiki/ImageCharles_He

rmite_circa_1887.jpg

Hermitian Operators

- The adjoint of an operator (A) satisfies the

equation - yA ly
- Hermitian operators are self-adjoint (A A)
- This has several implications
- Eigenvalues for Hermitian operators are real
- Eigenfunctions for Hermitian operators form a

complete orthonormal set - ?yiyjdt ?yjyidt dij Kronecker delta
- Hilbert space a complete set of N orthonormal

functions which constitutes a basis set

Eigenvalues of QM operator

- Postulate III
- The permissible values that a dynamical variable

may have are those given by OF aF, where F is

the eigenfunction of the QM operator (Hermitian)

O that corresponds to the observable whose

permissible real values are a. - The is postulate can be stated in the form of an

equation as

O F a F

operator wave function eigenvalue

wave function

Example Let F e2x and Od/dx ? dF/dx

d(e2x)/dx 2 e2x ? F is an eigenfunction of the

operator d/dx with an eigenvalue of 2.

Quantum Mechanics

Ayi liyi The result of making a measurement of

A is one of the eigenvalues of A. That is, only a

limited set of outcomes are possible (discrete

nature of quantum mechanics). What is the value

we might expect to measure? Expectation

Value The expectation value is the average

magnitude of a property sampled over an ensemble

of identically prepared systems. The expectation

value, ltAgt, is the scalar product of Y and

AY ltAgt ?YAYdt If the wavefunction is an

eigenfunction of the operator (Y yn) ltAgt

?YAYdt ?ynAyndt ln?ynyndt ln

Expectation Value

If a system is in state Y(t,t), the average of

any physical observable C at time t is ltCgt

?yCydt If one makes a large number of

measurements of C with identical initial state

Y(t,0), then one obtains a set of values C1, C2,

, CN. The average of C is given by the rule A

postulate of quantum mechanics is that the

integral and summation above provide the same

value, which is the expectation value.

Quantum Mechanics

In general Y ? yn, but Hence

probability that cj is obtained in a single

measurement

Quantum Mechanics

What does this mean? When A is measured for a

single member of an ensemble, the result is one

of the eigenvalues of A, but which one cannot be

predicted in advance. The result means that the

eigenvalue lj will be obtained in a single

measurement with the probability of cj2. So, for

a single measurement, there are specified values

of A that are possible, but over an ensemble, the

expectation value ltAgt can be a continuous value.

Example Particle in a box

- Infinite potential well
- Particle mass m in a box length L
- U(x)0, if 0ltxltL,
- U(x)8, if xlt0 or- xgtL
- Boundary conditions on y
- Y(0)0Y(L)

Particle in a box

- Second derivative proportional to the function

with - sign - Possible solutions sin(kx) and cos(kx)

k-wave vector

Particle in a box

- Lets satisfy boundary conditions

Wave vector is quantized!

Particle in a box

- Quantum number n

Energy is quantized!

We are not done yet, We dont know A

Particle in a box

- We know for sure that the particles is somewhere

in the box - Probability to find the particle in 0ltxltL is 1
- Unitarity condition

Particle in a box