Qit Full Report

Chapter-1

ABSTRACT

A new quantum information technology (QIT) could emerge in the future based on current

research in the fields of quantum information processing and communication1-3 (QIPC) In

contrast to conventional IT where quantum mechanics plays a support role in improving the

building blocks fundamental quantum phenomena play a central role for QIPC ndash information is

stored processed and communicated according to the laws of quantum physics This additional

freedom could enable future QIT to perform tasks we will never achieve with ordinary IT This

article provides an introduction to QIPC some indication of the state of play today and some

comments on the future Quantum information technology promises powerful information

processing based on the control of the dynamics of individual particles Research on single

electron dynamics aiming at the coherent electrical manipulation of single electron charge and

spin in semiconductor quantum dots is expected to provide practical quantum information

technology A single electron charge quantum bit (qubit) which has recently been realized in a

double quantum dot is advantageous for flexible control of the quantum state by a high-speed

electrical signal while a single electron spin qubit is expected to have a sufficiently long

decoherence time This paper reviews our research on single electron dynamics for quantum

information processing

Moores Law has set great expectations that the performanceprice ratio of commercially

available semiconductor devices will continue to improve exponentially at least until the end of

the next decade Although the physics of nanoscale silicon transistors alone would allow these

expectations to be met the physics of the metal wires that connect these transistors will soon

place stringent limits on the performance of integrated circuits We will describe a Si-compatible

global interconnect architecture - based on chip-scale optical wavelength division multiplexing -

that could precipitate an optical Moores Law and allow exponential performance gains until

the transistors themselves become the bottleneck Based on similar fabrication techniques and

technologies

1

CHAPTER-2

INTRODUCTION

Today many people are familiar with at least the consequences of Moorersquos Law ndash the fastest

computer in the shops doubles in speed about every 18 months to two years This is because

electronic component devices are shrinking The smaller they get the faster they work and the

closer they can be packed on a silicon chip This exponential progress first noted4by Gordon

Moore a co-founder and former CEO of Intel in 1965 has continued ever since But it cannot

go on forever Hurdles exist for example silicon will hit problems with oxide thinness track

width or whatever 5 new materials or even new paradigms such as self-assembled nano-

devices or molecular electronics will be needed lots of dollars will be needed as Moorersquos

second law tells us that fabrication costs are also growing exponentially However even if all the

hurdles can be overcome we will eventually run into Nature

Very small things do not behave the same way as big ones ndash they begin to reveal their true

quantum nature Following Moorersquos Law an extrapolation of the exponentially decaying number

of electrons per elementary device on a chip gets to one electron per device around 2020 This is

clearly too naiumlve but it gives us a hint Eventually we will get to scales where quantum

phenomena rule whether we like it or not If we are unable to control these effects then data bits

in memory or processors will suffer errors from quantum fluctuations and devices will fail

Clearly this alone makes a strong case for investment in research into quantum devices and

quantum control The results should enable us to push Moorersquos Law to the limit evolving

conventional information technology (IT) as far as it can go However such quantum research

has already shown that the potential exists to do much more ndashrevolution Instead of playing

support act to make better conventional devices let quantum mechanics take centre stage in new

technology that stores processes and communicates information according to the laws of

quantum mechanics

2

CHAPTER-3

LITERATURE REVIEW

Tomorrows computer might well resemble a jug of water

This for sure is no joke Quantum computing is here What was science fiction two decades back

is a reality today and is the future of computing The history of computer technology has

involved a sequence of changes from one type of physical realization to another --- from gears to

relays to valves to transistors to integrated circuits and so on Quantum computing is the next

logical advancement

Todays advanced lithographic techniques can squeeze fraction of micron wide logic gates and

wires onto the surface of silicon chips Soon they will yield even smaller parts and inevitably

reach a point where logic gates are so small that they are made out of only a handful of atoms

On the atomic scale matter obeys the rules of quantum mechanics which are quite different from

the classical rules that determine the properties of conventional logic gates So if computers are

to become smaller in the future new quantum technology must replace or supplement what we

have now

GENERAL CONCEPT OF INFORMATION

In quantum information technology a quantum state is a carrier of information It is transformed

from an initial input state to a final output state by applying an external field eg an

electromagnetic field for acertain period The fundamental transformation of the quantum state

is called a quantum logic gate and it should be a unitary transformation to keep coherency The

quantum state is changed by applying a series of quantum logic gates for the start of quantum

computation when classical digital information is input to the quantum state until the end of the

computation when the output states are finally measured as classical information One may not

measure the intermediate state because that would cause the quantum state to collapse Quantum

computing requires that a high degree of quantum coherence be maintained for along enough

time to complete the computation One can design an algorithm for quantum computation in such

a way that a series of data processing is performed simultaneously in a parallel fashion (quantum

parallelism) Quantum computation is therefore expected to provide extremely efficient

3

calculations for specific problems that cannot be solved efficiently with conventional classical

computers To construct a quantum computer a bunch of single quantum states must be

(i) prepared physically

(ii) initialized

(iii) manipulated coherently

(iv) preserved for a long enough time and

(v) measured individually

Each of these requirements needs further development Below we briefly summarize our

strategies for realizing quantum computers using single-electron dynamics

The building blocks ndash quantum bits

Most information manipulation these days is done digitally so data is processed stored and

communicated as bits The two states of a conventional data bit (but written in suggestive

quantum notation as) |0gt and |1gt take many forms ndash two different voltages across a transistor on

a chip two different orientations of a magnetic domain on a disc or tape two different voltages

propagating down a wire two different light pulses travelling down an optical fibre and so on ndash

dependent upon what is being done with the data At any time a bit is always in state |0gt or state

|1gt hence the name although bits get flipped as data is processed or memory is rewritten

However the quantum analogue of a conventional bit a qubit has rather more freedom It can

sit anywhere in a two-dimensional Hilbert space ndash picture it as the surface of a sphere ndash with a

general state of the form (1) parametrized by two angles A conventional bit only has the choice

of the poles but a qubit can live anywhere on the surface of the sphere States such as (1) are

superposition states they have amplitudes for and thus carry information about the states |0gt and

|1gt at the same time Similarly a collection or register of N qubits can have exponentially

many (2N ) amplitudes whereas the analogous conventional data register can only hold one of

these states at any given time Clearly if it is possible to operate or compute simultaneously

with all the amplitudes of a quantum register there is the possibility of massively parallel

computation based on quantum superpositions12We can read ordinary information without

noticeably changing it ndash you can read a book without harming it and your telephone calls can be

tapped without you knowing The same is simply not so for quantum information If a qubit in

4

state (1) is measured to determine its bit value it will always give the answer 0 or 1 This is a

truly random and irreversible process with respective probabilities of cos2a and sin2a and

afterwards the qubit is left in the corresponding bit state |0gt or |1gt (if it isnrsquot destroyed) It is

thus impossible to read or similarly copy or clone67unknown quantum information without

generally leaving evidence of the intrusion This unavoidable disturbance through quantum

measurement can be used to detect eavesdropping on quantum communications3 and provides

the basis for guaranteed security89

Many types of usable qubit exist or in some cases reasonable approximations where two

orthogonal quantum states (used to represent |0gt and |1gt) are or can be separated from the rest

of the space Examples include two adjacent energy eigenstates of atoms10or ions11 (separated

by a microwave or an optical transition) the vacuum or single photon state of a mode in a small

optical or superconducting microwave cavity12 two orthogonal linear or circular polarizations

of a travelling photon or weak light pulse3 the ldquowhich pathrdquo label of a photon3or atom in an

interferometer the energy eigenstates (up or down) of a spin-12 in a magnetic field13

two adjacent energy eigenstates of an electron or exciton in a quantum dot14 two charge states

of a tiny superconducting island 15 or flux states of a superconducting ring1617 and so on

This list is not at all exhaustive and many more candidate qubits have been proposed and are

under investigation As with realisations of conventional data bits the most appropriate choice is

defined by the application

CONCEPT OF INFORMATION IN QUANTUM COMPUTERS - THE QUBIT

In quantum computers also the basic unit of information is a bit The c-ocept of quantum

computing first arose when the use of an atom as a bit was suggested If we choose an atom as a

physical bit then quantum mechanics tells at apart from the two distinct electronic states (the

excited state and the vid state) the atom can be also prepared in what is known as a coherent gt-

Derposition of the two states This means that the atom can be both in state 0 and state 1

simultaneously It is at this point that the concept of a quantum bit or a 3ubit arises This concept

is the backbone of the idea of quantum computing for the same reason lets see in detail what

actually coherent superposition is

5

COHERENT SUPERPOSITION

In any quantum mechanical system a particular state of the system is represented by a

mathematical function called as the wave function of that state A wave function is a complex

exponential which includes all possible phases of existence of that particular state

Considering any quantum mechanical system let gti and y2 be two wave functions that

represent any two independent states of the system Then quantum mechanics tells us that there

exists a state of the same system that can be represented by the wave function Ciji +

c2v|2This state is called as a superposition of the two states represented by yi and vj2 This

would mean that the system would be in both the states of y-i and i|2 simultaneously All super

positions of two quantum states of a system need not be stable If the superposition is to be

stable then there should be some sort of coherence between the two states that are being super

positioned Such a superposition is called as a coherent superposition

There can be more than one coherent superposition for a particular pair of states of a quantum

mechanical system So in our talk the term coherent superposition would refer to that

superposition which is the most stable one

ADVANTAGE OF USING COHERENT-SUPERPOSITIONED MEMORY

The importance of coherent-superpositioned storage can be understood from the following

example Consider a register composed of three physical bits Any classical register of that type

can store in a given moment of time only one out of eight different numbers ie the register can

be in only one out of eight possible configurations such as 000 001 010 111 Consider the

case of a quantum register at that place Since a qubit can store both the values of 0 amp 1

simultaneously a quantum register composed of three qubits can store in a given moment of

time all the eight numbers in a quantum superposition The catch is that the memory size grows

exponentially when additional bits are added compared to the linear growth in classical

computers Also once the register is prepared in such a superposition operations can be

performed on all the numbers simultaneously

6

ROLE OF COHERENT SUPERPOSITION IN COMPUTING OPERATIONS

We have seen that once a register is prepared in a superposition of different numbers we can

perform operations on all of them For example if qubits are atoms then suitably tuned laser

pulses affect atomic electronic states and evolve initial super positions of encoded numbers into

different super positions During such evolution each number in the superposition is affected and

as the result we generate a massive parallel computation albeit in one piece of quantum

hardware This means that a quantum computer can in only one computational step perform the

same mathematical operation on 2L different input numbers encoded in coherent super positions

of L qubits In order to accomplish the same task any classical computer has to repeat the same

computation 2L times or one has to use 2L different processors working in parallel In other

words a quantum computer offers an enormous gain in the use of computational resources such

as time and memory

But this after all sounds as yet another purely technological progress It looks like classical

computers can do the same computations as quantum computers but simply need more time or

more memory The catch is that classical computers need exponentially more time or memory to

match the power of quantum computers and this is really asking for too much because an

exponential increase is really fast and we run out of available time or memory very quickly

SINGLE-ELECTRON DYNEMICS

A quantum dot is a small conductive island (less than 100 nm in diameter) that contains a

tunable number of electrons occupying discrete orbitals Figure 1shows a scanning electron

micrograph of representative samples containing from one to four quantum dots (schematically

illustrated by circles) Since the electron filling in a quantum dot resembles that in normal atoms

quantum dots are often referred to as artificial atoms For example as is true for some atoms a

quantum dot containing a few electrons exhibits magnetic properties depending on the electron

filling Actually quantum dots with a few electrons show spin-polarized states even at zero

magnetic field The great advantage of artificial atoms (quantum dots) is that the element of the

artificial atom (HHe Li and so on) can be changed just by changing external voltagesNTT

7

Basic Research Laboratories have been study VL VR Vsd IsdS D300 nm Drain Quantum dot

Gate Heterostructure Source Vl Vr

(a) (b)

( c ) (d)

Fig 1 Scanning electron micrographs of quantum dot devices (d) Double quantum dot

schematically shown by two circles fabricated by dry etching and fine metal gate patterning (b)

Two sets of double quantum dots (prototype device) (c) A vertical single quantum dot in which

the source and drain electrodes are located above and below

The dynamical response of a single electron in a quantum dot which is called single-electron

dynamics For instance a single electron can be injected into or extracted from a quantum dot by

applying a high speed electrical signal with time accuracy better than 1 ns The injection and

extraction can be detected as a change in the charge on the dot with time accuracy better than 1

micros Moreover an electron in a quantum dot can be excited by applying microwaves which

8

corresponds to optical excitation in atoms This research promises to provide novel semi-

classical electronic devices that can precisely control and measure an electronic current The

more challenging and effective application of this research though is in quantum information

technology in which single electron charge and spin is controlled quantum mechanically

Single electron dynamics in quantum dots is expected to satisfy the requirements for quantum

information technology and lead to useful quantum computers

CHARGE QUANTUM BIT IN A BOUBLE QUANTUM DOT

Consider a double quantum dot in which two quantum dots are separated by a tunneling barrier

The double quantum dot can be regarded as diatomic molecule the two dots are coupled

electrostatically (corresponding to an ionic bond) and quantum mechanically (corresponding to a

covalent bond) The coupling strengths can also be controlled by external voltages Figure 1(a)

shows a typical double quantum dot device fabricated in an AlGaAsGaAs modulation doped

heterostructure The upper and lower dark regions are etched and the adjacent region is depleted

of conductive electrons The bright vertical lines are metal gate electrodes used to deplete the

underlying region The resultant conductive islands (double quantum dot) are schematically

shown by circles The double quantum dot is attached to the source and drain electrodes and can

be investigated by measuring the tunneling currentNow consider the situation where an excess

electron is injected into the double quantum dot (See Fig2(a)) In the classical picture the

electron occupies either the left or right dot at any given time This is analogous to the logical 0

and 1 of a classical bit In the quantum mechanical picture however an electron behaves as if it

occupies the left and right dots simultaneously (superposition of 0 and 1) but the probability of

finding the electron in one of the dots is determined quantum mechanically In quantum

information technology this probability is the quantum information itself and should be

manipulated or preserved during the calculationWe have succeeded in controlling the quantum

state of a double quantum dot by applying a high-speed electrical pulse (from 100 ps to 2 ns) to

one of the electrodes [4] The electron initially injected into the left dot moves back and forth

between the two dots during the pulse which corresponds to a sinusoidal oscillation in

probability between 0 and 100 The probability can be controlled by tailoring the pulse

waveform and is obtained from the tunneling current under the repetition of many pulses A

9

specific pulse waveform that changes the probability from 0 to 100 or from 100 to 0

can be used as a NOT gate which

( a )

( b )

Fig 2 (a) Quantum state of a single electron in a double quantum dot When an excess electron

is injected it can occupy the left dot (0) right dot (1) or both dots simultaneously (0+1) (b) The

probability of finding the electron in the right dotis the most fundamental quantum logic gate

The next experiment would be two-qubit operation on two sets of double quantum dots (See Fig

1(b) and Fig 3(a)) The two double dots are coupled electrostatically to correlate the electron

occupation in one double quantum dot with that in the other For instance the NOT gate

operation at one double quantum dot can be performed only when the electron is located in the

10

left dot of the other double quantum dot This conditional NOT operation (controlled NOT gate)

is another fundamental gate operation in quantum information technology If it is performed

coherently the controlled- NOT gate should work for any superposition state as well One can

design any quantum algorithm by combining a few types of fundamental quantum logic gates (a

universal set of logic gates) Another important requirement for quantum information

technology is the single-shot measurement in which the electron occupation in the double

quantum dot is determined by a single measurement In this case the measurement outcome is

either 0 or 1 whose probability can be controlled by quantum logic gates The electron

occupation in a double quantum dot can be measured with a single electron transistor (SET)

which can be fabricated near the double quantum dot by the same technique We are developing

a high-speed version of the SET operated with a radio frequency signal (RF-SET) and expect it

to work as a single-shot measurement device [5] So far basic technologies (NOT gate

controlled- NOT gate and single shot measurement) for a charge qubit have been summarized

(Fig 3(a)) However a serious problem for the success of quantum information technology is

decoherence the loss of the quantum information the whole computation must to be finished

within this decoherence time For a charge qubit in a double quantum dot the decoherence time

is not very long about 1 ns at present Solving this problem requires further investigation and

refinement

Fig3 Summary of quantum information technologies one- and two-qubit operations and

single-shot measurement for (a) a charge qubit in a double quantum dot

11

( a )

( b )

Fig 4 Summary of quantum information technologies one- and two-qubit operations and

single-shot measurement for (b) a spin qubit in a single quantum dot

SPIN QUBIT INTO A SINGLE QUANTUM DOT

The spin degree of freedom which originates from the spinning of a charged particle is an

alternative way to construct a qubit While the charge qubit is an artificial qubit the electron

spin is a natural one The spin decoherence time of conductive electrons in bulk GaAs crystal

can be longer than 100 ns and electron spin bound to a donor in silicon shows a decoherence

12

time of about 300 μs Electron-spin based quantum computation is motivated by this long

decoherence time Coherent manipulation of electron spins has been studied in many systems

The easiest way is to use electron spin resonance which involves applying a microwave

magnetic field under a static magnetic fieldHowever in contrast to the countless studies on the

ensemble of spins in many materials little work has been done on the manipulation of single-

electron spin A quantum dot containing a single electron spin provides flexible quantum

information storage (See Fig 1(c) and Fig 3(b)) We have developed a novel electrical pump

and probe experiment for a quantum dot and investigated inelastic spin relaxation time in a

quantum dot containing a few electrons (Fig 1(c)) [6] We found that the energy relaxation time

which is the time necessarily to complete the quantum computation including the last

measurement can be longer than 200 μs for a realistic quantum dot structure This result is very

encouraging The NOT gate operation which reverses the spin direction from up (0) to down (1)

or vice versa can be performed using microwave irradiation (electron spin resonance) To

address each electron spin (qubit) in many quantum dots single-spin manipulation and

measurement techniques are essential The effectiveg-factor of each electron spin can be made

different for different quantum dots through g-factor engineeringor a moderate magnetic field

gradient is applied to the quantum dots so that each qubit is addressed by a corresponding

microwave frequency However a typical one-qubit operation using electron spin resonance

requires a relatively long period of time 100 ns for instance because of the weak magnetic

dipole interaction Alternative approaches ie using the optical Stark effect in a specific band

structure or exchange coupling among three electron spins constituting one qubit are suitable

for much faster operations Two-qubit operation which is required to construct a universal set of

logic gates is expected to be achieved by connecting two quantum dots with a tunable tunneling

barrier The exchange coupling between the two spins can be used to swap the two spins (SWAP

gate) The controlled-NOT gate can be constructed by combining a NOT gate and a SWAP gate

Single-shot spin measurement is a challenging technique for quantum information technology

One proposal is based on the spin-dependent tunneling between two quantum dots combined

with an RFSETWhen each of the dots possesses one electron spin before the measurement

tunneling from one dot to the other is allowed if the two electron spins can make a spin pair

(spin singlet state) This spin-dependent tunneling could be measured with an RF-SET in a short

time

13

QUANTUM COMPUTING

The seeds of quantum computing began with Richard Feynman 2 and others in the early 1980s

and it was David Deutsch 28 who first considered in detail the implications of quantum physics

for the theory of computation In 1992 Deutsch and Richard Jozsa came up with an algorithm 29

that showed a clear quantum advantage and the subject really took off in the mid-1990s with the

factoring algorithm of Peter Shor 30 and the searching algorithm of Lov Grover31 which give

significant advantage over their classical counterparts Much of the ldquosecurerdquo communications in

the world today use public key cryptography which is based on factoring a very large number

into its two component primes (or related problems) being practically unbreakable The

construction of a many-thousand qubit quantum computer would thus trash the worldrsquos

communications infrastructure ndash certainly dramatic whether you approve or not Quantum

computers could clearly also do a much better job of simulating quantum systems32 than

conventional IT and so would open up new research capabilities in many fields The search is

still on for more quantum algorithms ndash open problems exist because not everything is amenable

to a naiumlve quantum speed-up The quantum computational advantage arises because (in principle

exponentially) many calculations can run in parallel during the evolution stage However

quantum measurements have to be made to get answers so simple number crunching doesnrsquot get

exponential advantage Rather it is problems that utilise the parallelism through interference that

can gain The factoring algorithm uses the exponential resources and a Fourier transform to find

the (very large) periods of oscillatory functions and the search algorithm offers a square root

reduction in time by effectively searching ldquoamplitudisticallyrdquo rather than

probabilisticallyImplementation research today has progressed to the few qubit and simple

algorithm level The first two-qubit gate was done with an ion trap33 and work has now

progressed to four-ion entanglement 34 and realisation of the Deutsch-Jozsa algorithm35 Atom-

cavity interactions in the optical 3637 and microwave38 domains have got three-qubit

entanglement39 Use of nuclear spin qubits in a molecule in an ensemble nuclear magnetic

resonance approach 4041 has demonstrated a number of simple algorithms42-46 most recently

the factoring of fifteen47 Single superconducting qubits based on charge or phase have been

constructed 48-51 (see figure 1)Many other approaches to quantum computing hardware have

been proposed5253 Examples include photons54 charge14 or spin13 in quantum dots dopant

14

nuclear (or electronic) spins in the solid state5556 spins in fullerene cages57 trapped

electrons58 (see figure 2) quantum Hall systems59magnetic molecules or nano-crystals60 (see

figure 3) and electrons on liquid helium61 From the perspective of scalability in qubit number

solid state approaches which build on the wealth of existing fabrication techniques have much

appeal It is certainly also the case that most qubit successes to date do not seem to be easily

scalable The flip side is that solid state systems generally suffer more decoherence so it will be

a very big challenge to reduce this to the level required for error correction and fault tolerant

operation

ALGORITHMS FOR QUANTUM COMPUTERS

In order to solve a particular problem computers follow a precise set of instructions that can be

mechanically applied to yield the solution to any given instance of the problem A specification

of this set of instructions is called an algorithm Examples of algorithms are the procedures

taught in elementary schools for adding and multiplying whole numbers when these procedures

are mechanically applied they always yield the correct result for any pair of whole numbers

Some algorithms are fast (eg multiplication) other are very slow (eg factorizations) Consider

for example the following factorization problem X = 29083 How long would it take you using

paper and pencil to find the two whole numbers which should be written into the two boxes (the

solution is unique) Probably about one hour At the same time solving the reverse

problem127x129 = again using paper and pencil technique takes less than a minute All because

we know fast algorithms for multiplication but we do not know equally fast ones for

factorization

What really counts for a fast or a usable algorithm according to the standard definition is

not the actual time taken to multiply a particular pairs of number but the fact that the time does

not increase too sharply when we apply the same method to ever larger numbers The same

standard text-book method of multiplication requires little extra work when we switch from two

3 - digit numbers to two 30 - digit numbers By contrast factoring a thirty digit number using the

simplest trial division method is about 1013 times more time or memory consuming than

factoring a three digit number The use of computational resources is enormous when we keep

increasing the number of digits The largest number that has been factorized as a mathematical

15

challenge ie a number whose factors were secretly chosen by mathematicians in order to

present a challenge to other mathematicians had 129 digits No one can even conceive of how

one might factorize say thousand-digit numbers the computation would take much more that the

estimated age of the universe

Apart form the standard definitions of a fast or a usable algorithm computer scientists have a

rigorous way of defining what makes an algorithm fast (and usable) or slow (and unusable) For

an algorithm to be fast the time it takes to execute the algorithm must increase no faster than a

polynomial function of the size of the input Informally think about the input size as the total

number of bits needed to specify the input to the problem for example the number of bits

needed to encode the number we want to factorize If the best algorithm we know for a particular

problem has the execution time (viewed as a function of the size of the input) bounded by a

polynomial then we say that the problem belongs to class P Problems outside class P are known

as hard problems Thus we say for example that multiplication is in P whereas factorization is

not in P and that is why it is a hard problem Hard does not mean impossible to solve or non-

computable - factorization is perfectly computable using a classical computer however the

physical resources needed to factor a large number are such that for all practical purposes it can

be regarded as intractable Purely technological progress can only increase the computational

speed by a fixed multiplicative factor which does not help to change the exponential dependence

between the size of the input and the execution time Such change requires inventing new better

algorithms Although quantum computation requires new quantum technology its real power lies

in new quantum algorithms which allow exploiting quantum superposition that can contain an

exponential number of different terms Quantum computers can be programmed in a

qualitatively new way For example a quantum program can incorporate instructions such as

and now take a superposition of all numbers from the previous operations this instruction is

meaningless for any classical data processing device but makes lots of sense to a quantum

computer As the result we can construct new algorithms for solving problems some of which

can turn difficult mathematical problems such as factorization into easy ones

16

DEMONSTRATING QUANTUM COMPUTING

Due to technical obstacles till date a quantum computer has not yet been realized But the

concepts and ideas of quantum computing has been demonstrated using various methods Here

we discuss four of the most important technologies that are used to demonstrate quantum

computing They are

I Nuclear Magnetic Resonance

ii Ion Trap

iii Quantum Dot

iv Optical Methods

While reading the following top four technologies two things should be kept in mind The first

is that the list will change over time Some of the approaches valuable for exploring quantum

computing in the laboratory are fundamentally un-scalable and so will drop out of contention

over the next few years The second thing to keep in mind is that although there are a

bewildering number of proposed methods for demonstrating quantum computing (a careful

search will yield many more options that what is listed here) all of them are variations on three

central themes

(a) manipulating the spin of a nucleus or subatomic particle

(b) manipulating electrical charge

( c ) manipulating the polarization of a photon

In variation a a qubit is derived from superposition of up and down spins In variation b a

qubit is derived from superposition of two or more discrete locations of the charge In the last

variation a qubit is derived from the superposition of polarization angles Of the three the

manipulation of the spin is generally viewed as the most promising for practical large-scale

application Lets now see each of these techniques in detail

17

Nuclear Magnetic Resonance

Using nuclear magnetic resonance (NMR) techniques invented in the 1940s and widely used in

chemistry and medicine today these spins can be manipulated initialized and measured Most

NMR applications treat spins as little bar magnets whereas in reality the naturally well-

isolated nuclei are non-classical objects

A Nuclear Magnetic Resonance (NMR) quantum computer is based on control of nuclear spin

In demonstrations to date this has been achieved by manipulating the nuclear spins of several

atoms making up a molecule - the most recent effort using a molecule of five fluorine atoms and

two carbon atoms The spin manipulation is accomplished by application of magnetic pulses

within a magnetic field produced by the NMR chamber

The quantum behavior of the spins can be exploited to perform quantum computation Magnetic

field produced by the NMR chamber can be used to manipulate the spin state of the nucleus The

manipulation of the spin is accomplished by application of magnetic pulses in the chamber The

entanglement of spins required to establish a qubit is created by the chemical bonds between

neighboring atoms within a molecule - within 1018 molecules to be more precise since a

measurable signal (relative to the background noise from kinetic energy) is achieved by using a

test tube of processing liquid rather than a single molecule

For example consider the carbon and hydrogen nuclei in a chloroform molecule to be

representing two qubits Applying a radio-frequency pulse to the hydrogen nucleus addresses

that qubit and causes it to rotate from a |0gt state to a superposition state Interactions through

chemical bonds allow multiple-qubit logic to be performed In this manner applying newly

developed techniques to allow bulk samples with many molecules to be used small-scale

quantum algorithms have been experimentally demonstrated with molecules such as Alanine an

amino acid This includes the quantum search algorithm and a predecessor to the quantum

factoring algorithm

The major drawback of this method is scalability the signal strength of the answer decreases

exponentially with the number of qubits

18

ii Ion Trap

An Ion Trap quantum computer is also based on control of nuclear spin (although using vibration

modes or phonons has also been considered) In this approach the individual ions are as the

name implies trapped or isolated by means of an electromagnetic field which is produced by

means of an electromagnetic chamber

Ordinarily the energy difference between different spin states in nuclei is so small relative to the

kinetic energy of the ions that they are not measurable In the prior technique (NMR) this

problem was overcome by operating simultaneously on a large number of atoms But in this

case the solution is a bit different The trapped ions are cooled to the point where motion is

essentially eliminated They are then manipulated by laser pulses and a qubit arises from the

superposition of lower and higher energy spin states

This technique is potentially scalable but a great disadvantage is that it requires a cryogenic

environment - not to mention that to date no more than single qubit systems have been

demonstrated

iii Quantum Dot

A quantum dot is a particle of matter so small that the addition or removal of an electron changes

its properties in some useful way All atoms are of course quantum dots but multi-molecular

combinations can have this characteristic In biochemistry quantum dots are called redox

groups Quantum dots typically have dimensions measured in nanometers where one nanometer

is 109 meter or a millionth of a millimeter

A Quantum Dot quantum computer can involve manipulation of electrical charge spin or

energy state - the Australians have a patent on a spin based version The idea is that a small

number of electrons or possibly an individual electron is confined with a quantum dot the

quantum dot typically being a small hill of molecules grown on a silicon substrate A computer

would be made up of a regular array of such dots As with the prior two methods the most

popular approach is to have spin up counted as zero spin down counted as one and use a

superposition of spin states to create the qubit Techniques for self assembly of large arrays of

quantum dots have already been demonstrated and can be done using the industry standard

19

silicon substrate Thus of all the approaches listed here this one seems to have the highest

potential for commercial scalability

iv Optical

As the name indicates an optical quantum computer uses the two different polarizations of a

light beam to represent two logical states As an example we can consider the polarization of a

light beam in the vertical plane to represent a logical 1 and the polarization of the beam in the

horizontal plane to represent a logical 0 An Optical quantum computer would be based on

manipulating the polarization of individual photons Entanglement is achieved by coincident

creation of identical photons Identical photons in this context would mean photons having the

same energy as well as same polarization The superposition of polarization or phase state is

manipulated using polarizing lenses phase shifters and beam splitters

It was originally believed that non-linear optics would be required in order to create a working

quantum computer and this was considered to be the major technical obstacle to a photon-based

approach However recent theoretical advances indicate that linear optics is sufficient Several

laboratories are working on a practical demonstration The new stumbling block has been

creating beam splitters of sufficient accuracy The entire process can be carried out at room

temperature and there is no reason in principle that it is not scalable to large numbers of

qubytes

ADVANTAGES OF QUANTUM COMPUTING

Quantum computing principles use the principle of coherent superposition storage As stated in

the above example it is quite remarkable that all eight numbers are physically present in the

register but it should be no more surprising than a qubit being both in state 0 and 1 at the same-

time If we keep adding qubits to the register we increase its storage capacity exponentially ie

three qubits can store 8 different numbers at once four qubits can store 16 different numbers at

once and so on in general L qubits can store 2L numbers at once Once the register is prepared

in a superposition of different numbers we can perform operations on all of them For example if

qubits are atoms then suitably tuned laser pulses affect atomic electronic states and evolve initial

super positions of encoded numbers into different super positions During such evolution each

number in the superposition is affected and as the result we generate a massive parallel

20

computation albeit in one piece of quantum hardware This means that a quantum computer can

in only one computational step perform the same mathematical operation on 2L different input

numbers encoded in coherent super positions of L qubits In order to accomplish the same task

any classical computer has to repeat the same computation 2L times or one has to use 2L

different processors working in parallel In other words a quantum computer offers an enormous

gain in the use of computational resources such as time and memory It looks like classical





WHAT WILL QUANTUM COMPUTERS BE GOOD AT

These are the most important applications currently known

Acirccent Cryptography Perfectly secure communication

Acirccent Searching especially algorithmic searching (Gravers algorithm)

Acirccent Factorizing large numbers very rapidly (Shors algorithm)

Acirccent Simulating quantum-mechanical systems efficiently

OBSTACLES AND RESEARCH

The field of quantum information processing has made numerous promising advancements since

its conception including the building of two- and three-qubit quantum computers capable of

some simple arithmetic and data sorting However a few potentially large obstacles stijl remain

that prevent us from just building one or more precisely building a quantum computer that

can rival todays modern digital computer Among these difficulties error correction

decoherence and hardware architecture are probably the most formidable

Decoherence

We have seen that if a superposition of any two states of a quantum -mechanical system is to be

stable over a period of time there should be some sort of coherence between the states that are

21

being superpositioned But still no superposition of any pair of given states are perfectly stable

to any extent This is because of the existence of a property by name decoherence which forces a

quantum - mechanically superpositioned system to decay from a given quantum coherent -

superpositioned state into an incoherent state as it entangles or interacts with the surroundings

The final effect is that the coherence between the two states that are superpositioned would be

gradually lost For the same reason no quantum memory can at present be used to hold data

that is to be used for operations that take a long time The time taken by the system to lose the

coherence between the two states is known as decoherence time

ii Error Correction

Error correction is rather self explanatory but what errors need correction The answer is

primarily those errors that arise as a direct result of decoherence or the tendency of a quantum

computer to decay from a given quantum state into an incoherent state as it interacts or

entangles with the state of the environment These interactions between the environment and

qubits are unavoidable and induce the breakdown of information stored in the quantum

computer and thus errors in computation Before any quantum computer will be capable of

solving hard problems research must devise a way to maintain decoherence and other potential

sources of error at an acceptable level

Thanks to the theory (and now reality) of quantum error correction first proposed in 1995 and

continually developed since small scale quantum computers have been built and the prospects of

large quantum computers are looking up Probably the most important idea in this field is the

application of error correction in phase coherence as a means to extract information and reduce

error in a quantum system without actually measuring that system In 1998 researches at Los

Alamos National Laboratory and MIT managed to spread a single bit of quantum information

(qubit) across three nuclear spins in each molecule of a liquid solution of alanine or

trichloroethylene molecules They accomplished this using the techniques of nuclear magnetic

resonance (NMR) This experiment is significant because spreading out the information actually

made it harder to corrupt Quantum mechanics tells us that directly measuring the state of a qubit

invariably destroys the superposition of states in which it exists forcing it to become either a 0 or

1

22

The technique of spreading out the information allows researchers to utilize the property of

entanglement to study the interactions between states as an indirect method for analyzing the

quantum information Rather than a direct measurement the group compared the spins to see if

any new differences arose between them without learning the information itself This technique

gave them the ability to detect and fix errors in a qubits phase coherence and thus maintain a

higher level of coherence in the quantum system This milestone has provided argument against

skeptics and hope for believers At this point only a few of the benefits of quantum computation

and quantum computers are readily obvious but before more possibilities are uncovered theory

must be put to the test In order to do this devices capable of quantum computation must be

constructed Quantum computing hardware is however still in its infancy

As a result of several significant experiments nuclear magnetic resonance (NMR) has become

the most popular component in quantum hardware architecture Only within the past year a

group from LOS Alamos National Laboratory and MIT constructed the first experimental

demonstrations of a quantum computer using nuclear magnetic resonance (NMR) technology

Currently research is underway to discover methods for battling the destructive effects of

decoherence to develop optimal hardware architecture for designing and building a quantum

computer and to further uncover quantum algorithms to utilize the immense computing power

available in these devices Naturally this pursuit is intimately related to quantum error correction

codes and quantum algorithms so a number of groups are doing simultaneous research in a

number of these fields

To date designs have involved ion traps cavity quantum electrodynamics (QED) and NMR

Though these devices have had mild success in performing interesting experiments the

technologies each have serious limitations Ion trap computers are limited in speed by the

vibration frequency of the modes in the trap NMR devices have an exponential attenuation of

signal to noise as the number of qubits in a system increases Cavity QED is slightly more

promising however it still has only been demonstrated with a few qubits The future of quantum

computer hardware architecture is likely to be very different from what we know today

however the current research has helped to provide insight as to what obstacles the future will

hold for these devices

23

iii Lack of Reliable Reading Mechanism

The techniques that exist till date have a big problem that trying to read from a super positioned

qubit would invariably make it to lose its superpositioned state and make it to behave just as a

classical bit - ie it would store only one among the values of 0 and 1

Also if we are given a quantum register comprising of n bits of which m bits are superpositioned

ones none among the reading mechanisms available today is able to determine which value from

the superposition is to be read out ie if we are given a 3 - bit register that contains say 4 values

in a particular superposition (let them be 4567) the reading mechanisms available today are

unable to determine which how to access a specific value from the superposition

QUANTUM CRYTOGRAPHY

3Around 1970 Stephen Wiesner 62 realised quantum mechanics could be useful for

cryptography and in 1984 Charles Bennett and Gilles Brassard proposed the well known

BB84scheme 63 for quantum key distribution Many developments and new protocols have

followed64The basic idea is for Alice and 5Bob to share a secret key and to use this as a one-

time-pad to communicate securely ndash quantum mechanics guarantees the security of the key In

BB84 Alice sends to Bob photons chosen randomly from the four states of two overlapping qubit

bases (eg two orthogonal linear polarisations and right and left circular polarizations) and Bob

measures in one of the two bases chosen at randomAfter accumulating data using public

communication 65 and sacrificing some of the bits they can then identify what to keep (the raw

key ndash when Bob used the correct basis) locate and correct errors and scramble and reduce their

correct bits (privacy amplification) to distil a shared secret key Like Bob any eavesdropper

(Eve) has to measure66 the qubits ndash she has to play ldquoguess the basisrdquo and so cannot avoid

introducing errors into the raw key If Eve reads the lot Alice and Bob know this and bin the raw

key if Eve reads only a fraction they can use the rest to distil some guaranteed secure bitsThe

first prototype system 67 ran in 1989 Since then many developments have taken quantum

cryptography out of the laboratory and towards actual technology using qubits embodied in

weak laser pulses or photons sent from Alice to Bob through standard telecommunications

optical fibres or even free space Fibre examples work over useful distances68-72 can operate

alongside conventional communications through multiplexing73 can use multiple Bobs74 have

24

used entangled photons75-79 (see figure 4) and have shared a secret distributed between Bob and

Charlie80 The working distance is now up to 67 km with a ldquoplug amp playrdquo system81 (see figure

5) Free space systems have also been developed82-87 with the aim of secure communications

to and via satellites The distance is currently up to 234 km at altitude in the Alps88 which

makes quantum communication to near-Earth orbit satellites look feasible Research continues to

improve sources and detectors which would enhance all forms of quantum cryptosystem ndash for

example systems are now operating with ldquoon-demandrdquo single photons89 On the theory and

protocols side research continues to see just what can and canrsquot be done securely

by quantum means Clearly key distribution can for example it is known that bit commitment

canrsquot9091 and open problems in between remain

QUANTUM TELEPORTATION

The theory for quantum teleportation was laid out 92 in 1993 by Charles Bennett Gilles

Brassard Claude Crepeau Richard Jozsa Asher Peres and Bill Wootters The basic idea is that

if Alice and Bob share a pair of entangled qubits (as in (2)) they can use this as a resource to

offer a teleportation service Alice takes an unknown qubit from a customer performs a two-

qubit gate on this and qubit A and then measures both She transmits the results (two bits) to

Bob by a conventional communication channel The results uniquely identify one of four single-

qubit operations to Bob one of which is ldquodo nothingrdquo Once he has performed the identified

operation on qubit B it is left in the state of the qubit supplied by the customer There is no

instantaneous signalling as the two bits have to be sent to Bob so relativity is happy and no

quantum copy has been made as all record of the state is destroyed at Alicersquos end Amusingly

and of course very much in principle Alice doesnrsquot have to know where Bob is provided that she

broadcasts her bits

25

Also if the customer supplies half of an entangled pair the outcome is entanglement between

two qubits who have never met From 1997 a number of experiments demonstrating the

principles of teleportation have been performed93-98 The details differ but their basis is to

distribute entanglement using photon qubits (or light pulses) and to use this for the teleportation

of a quantum state from A to B Currently it is not possible to teleport the unknown state of a

customer qubit (for example another photon) with complete success because of the difficulty in

realising the required two-qubit gate at A on demand Research continues towards this goal

There is certainly an incentive because teleportation underpins the concept of a quantum

repeater99 which could be used to extend the working distance of quantum cryptosystems A

recent step towards this has been the demonstration of teleportation through 2 km 6of optical

fibre100 (see figure 6) It isnrsquot ldquoon demandrdquo but it does show that teleportation is progressing

beyond the confines of a single laboratory

26

Prospects for QIT

Quantum cryptography works around Ipswich (UK) under Lake Geneva and between

mountains in the Alps (Europe) in the Los Alamos desert (USA) and in numerous other places

worldwide You can buy a working fibre system101 and secure satellite communications may

well emerge over the next few years Few-qubit demonstration quantum computers exist

However useful many-qubit machines are still a long way off and we donrsquot yet know what form

they might take If QIT develops it is unlikely to displace conventional IT and more likely to

work with it addressing specific tasks In the future you are more likely to buy a PC with some

quantum chips in it rather than chucking your existing machine in the bin (or recycler) in favour

of a new wholly quantum one Teleportation of a single qubit works down about 2 km of optical

fibre However I very much doubt whether any of us will ever walk into a teleport and utter the

immortal words ldquoBeam me up Scottyrdquo

That said simpler teleportation could play a very important future role in distributing

quantum information between processors or effectively stringing out entanglement for long-

distance quantum communications Present day IT companies measure their annual revenue in

billions of dollars If such mass-market scale or consumer QIT is to emerge in the future new

quantum applications software and protocols will be needed Hardware development is certainly

necessary but certainly also not sufficient

The development of large-scale quantum processors will likely be very expensive so this

investment will need the promise of a market This means the quantum algorithms theory and

protocols folk cannot now put their feet up and simply leave things to the hardware scientists

and engineers Much further research is needed in all aspects of the field if QIT is to become a

reality

QUANTUMALGORITHMS

The main quantum algorithms are Quantum circuit based algorithmThe Deutsch Oracle

The DeutschJozsa OracleThe Simon OracleShorrsquos AlgorithmGroverrsquos Algorithm

Adiabatic algorithm

Measurement based algorithm

Topological quantum field theory(TQFT) algorithm

27

13 ENEMIES OF QUANTUM COMPUTING

There are two known enemies of quantum computing

a) Decoherence

If we keep on putting quantum gates together into circuits we will quickly run into some serious

practical problems The more interacting qubits are involved the harder it tends to be to engineer

the interaction that would display the quantum interference Apart from the technical difficulties

of working at single-atom and single-photon scales one of the most important problems is that of

preventing the surrounding environment from being affected by the interactions that generate

quantum superposition The more components the more likely it is that quantum computation

will spread outside the computational unit and will irreversibly dissipate useful information to

the environment This process is called ldquoDecoherencerdquo Even though we try to isolate the

quantum system from the environment much as we can we cannot supply total isolation

Therefore the interaction of the quantum system and the environment result in ldquoDecoherencerdquo of

the quantum state which is equivalent to a partial measurement of the state by the environment

b) Gate Inaccuracies

Decoherence is not the only problem with quantum computing Gates whether they are classical

or quantum are not perfect The gates are usually combined together So small errors in gates

can combine together during computation and eventually causing failure and it is not clear how

to correct these small errors

The simplest example of error correcting code is a repetition code replacing the bit we want to

protect by 3 copies of the bit

0 rarr (000)

1rarr (111)

Now an error may occur that causes one of the three bits to flip If itrsquos the first bit say

(000) rarr (100)

(111) rarr (011)

Now in spite of the error the bit can be encoded correctly by majority voting

28

CHAPTER- 4

TITLE OF WORK

Quantum information science combines two of the great scientific and technological revolutions

of the 20th century quantum mechanics and information theory According to the National

Science and Technology Councilrsquos 2008 report ldquoA Federal Vision for Quantum Information

Sciencerdquo quantum information science will enable a range of exciting new possibilities

including greatly improved sensors with potential impact for mineral exploration improved

medical imaging and a revolutionary new computational paradigm that will likely lead to the

creation of computation device capable of efficiently solving problems that cannot be solved on a

classical computer

One of the fundamentally important research areas involved in quantum information science is

quantum communications which deals with the exchange of information encoded in quantum

states of matter or quantum bits (known as qubits) between both nearby and distant quantum

systems Our Quantum Communication project performs core research on the creation

transmission processing and measurement of optical qubits ndash the quantum states of photons

with particular attention to application to future information technologies

Single photons at telecommunication wavelengths can be detected with higher efficiency with

our frequency up-conversion detector

29

In the past few years we have undertaken an intensive study of quantum key distribution (QKD)

systems for secure communications Specifically we demonstrated high-speed QKD systems

that generate secure keys for encryption and decryption of information using a one-time pad

cipher and extended them into a 3-node quantum communications network We have

demonstrated the strengths and observed the limitations of QKD systems and networks One

such limitation is the effective communication distance of a point-to-point QKD system which is

about 100 km Quantum repeaters represent a promising solution to this distance limitation It

enables quantum information exchange between two distant quantum systems including quantum

computers Though quantum repeaters are conceptually feasible there are tremendous challenges

to their development Our goal in this area is to identify the problems find potential solutions

and evaluate their capabilities and limitations for future quantum communication applications

In summary we perform research and development (RampD) in quantum communication and

related measurement areas with an emphasis on applications in information technology Our

RampD is aimed to promote US innovation industrial competitiveness and enhance the nationrsquos

security This website shows the footprint of our RampD efforts in the past few years

For more information concerning this program please contact project leader Dr Xiao Tang

(xiaotangnistgov)

Keywords quantum communication quantum measurement science entangled photons

quantum teleportation and repeaters free space optics quantum cryptography photon

sourcedetectors

The History of Quantum Money

But can one actually exploit the No-Cloning Theorem to achieve classically-impossible

cryptographic tasks This question was first asked by Wiesner [39] in a remarkable paper

written around 1970 (but only published in 1983) that arguably founded quantum information

science In that paper Wiesner proposed a scheme for quantum money that would be physically

impossible to clone In Wiesnerrsquos scheme each ldquobanknoterdquo would consist of a classical serial

30

number s together with a quantum state |ψsi consisting of n unentangled qubits each one |0i |1i

|0iradic+2|1i or |0iradicminus2|1i

with equal probability The issuing bank would maintain a giant database which stored a

classical description of |ψsi for each serial number s Whenever someone wanted to verify a

banknote he or she would take it back to the bankmdashwhereupon the bank would use its

knowledge of how |ψsi was prepared to measure each qubit in the appropriate basis and check

that it got the correct outcomes On the other hand it can be proved [31] that someone who did

not know the appropriate bases could copy the banknote with success probability at most (34)n

Though historically revolutionary Wiesnerrsquos money scheme suffered at least three drawbacks

( 1 ) The ldquoVerifiability Problemrdquo The only entity that can verify a banknote is the bank that

printed it

(2) The ldquoOnline Attack Problemrdquo A counterfeiter able to submit banknotes for verification

and get them back afterward can easily break Wiesnerrsquos scheme

(3) The ldquoGiant Database Problemrdquo The bank needs to maintain a database with an entry

for every banknote in circulation

In followup work in 1982 Bennett Brassard Breidbart and Wiesner [14] (henceforth BBBW)

at least showed how to eliminate the giant database problem namely by generating the state

|ψsi =_ψfk(s)

using a pseudorandom function fk with key k known only by the bank Unlike Wiesnerrsquos

original scheme the BBBW scheme is no longer information-theoretically secure a counterfeiter

can recover k given exponential computation time On the other hand a counterfeiter cannot

break

31

CHAPTER- 5

MATHEMATICAL ANALYSIS

Quantum information science is an interdisciplinary research endeavour that brings together

computer scientists mathematicians physicists chemists and engineers to develop

revolutionary information processing and communication technologies that are infeasible

without exploiting the principles of quantum mechanics The importance of quantum information

was first widely recognized in 1982 when Feynman conjectured that a quantum computer would

efficiently simulate quantum systems and a universal Turing machine (ldquoclassical computerrdquo)

could not

In the mid‐1990s Shor showed that the quantum computer could efficiently determine the

factors of large numbers whereas this problem is believed to be intractable on a classical

computer Even earlier in 1984 Bennett and Brassard proposed an information theoretically

secure key distribution technique through public channels as opposed to standard methods that

are only computationally secure Originally proposed in 1984 quantum cryptography has since

become commercial technology

Quantum information technology is thus ldquodisruptiverdquo both technically and also at a fundamental

level both to physics and to computer science Quantum information leads to a violation of the

strong Church‐Turing thesis and could enable information‐theoretic security over public

channels Moreover quantum computing and quantum cryptography damage and ameliorate

respectively information security

Theoretical research in quantum information relies on sophisticated mathematical methods and

advances rely on concomitant developments of new mathematics hence the need for strong

mathematical research in quantum information

32

Areas of interest for the PIMS CRG on MQI

This CRG is ideally suited to address and make significant progress in the areas such as the three

noted below

Models of quantum computing

Quantum computing is technologically challenging so different models for implementing

quantum algorithms are of great interest The first model treated unitary gates on single qubits

(quantum binary digits) and on pairs of qubits a few gates of these types can be used to construct

any quantum circuit efficiently with bounded error Subsequently remarkably different circuits

were devised that serve the same end such as adiabatic quantum computing topological

quantum computing quantum computing with oscillators (so‐called ldquocontinuous variable

quantum computingrdquo) and one‐way quantum computing

Proving that these models are equivalent especially under the fault‐tolerant conditions of error

correction is generally difficult Not only are models of quantum computing valuable in

exploring different physical realizations but also these models are conceptually quite different

and have inspired new quantum algorithms by forcing new ways to think about circuits In

addition for one‐way quantum computation beautiful and unexpected connections with graph

theory are beginning to emerge and they require further exploration

Error correction

The theory of quantum error correction showed that quantum computing errors could be

efficiently corrected despite Heisenberg limits to quantum measurements Quantum error

correction was then shown to follow easily from ldquoclassicalrdquo error correction theory which made

devising error correction protocols relatively straightforward Quantum error correction was also

shown to underlie the security of quantum key distribution by proving that ldquoclassicalrdquo

cryptographic privacy amplification of quantum keys protected against quantum eavesdroppers

Although quantum error correction is accepted as strategy against errors and faults the overhead

of error correction is typically high and new strategies are sought to reduce the resource

overhead For example quantum error correction can be assisted by providing consumable

entanglement resources thereby reducing the number of extra qubits and gates Also more

33

sophisticated methods such as belief propagation can be used to correct errors This CRG will

develop significantly better quantum error correction protocols

Quantum algorithms

Quantum computers are of revolutionary importance because they would make some

computational problems efficiently solvable that are regarded as intractable on a classical

computer So far the class of problems that are converted from intractable to tractable by a

quantum computer is small and this CRG will seek to expand the class of such problems by

developing new quantum algorithms The graph isomorphism problem is of special interest to

members of the CRG and its development will link to topological models of quantum

computing

In summary members of the proposed CRG have expertise and active research in many areas of

quantum information including the three important topics listed above models of quantum

computing error correction and algorithms The CRG will pursue these areas by bringing

together their complimentary expertise and holding workshops with the worldrsquos leading

scientists in the field

These early ideas about quantum money inspired the field of quantum cryptography [13]

But strangely the subject of quantum money itself lay dormant for more than two decades

even as interest in quantum computing exploded However the past few years have witnessed

a ldquoquantum money renaissance rdquo Some recent work has offered partial solutions to the

verifiability problem for example Mosca and Stebila [32] suggested that the bank use a blind

quantum computing protocol to offload the verification of banknotes to local merchants

while Gavinsky [23] proposed a variant of Wiesnerrsquos scheme that requires only classical

communication between the merchant and bank

However most of the focus today is on a more ambitious goal namely creating what

Aaronson [3] called public-key quantum money or quantum money that anyone could

authenticate not just the bank that printed it As with public-key cryptography in the 1970s it is

far from obvious a priori whether public-key quantum money is possible at all Can a bank

publish a description of a quantum circuit that lets people feasibly recognize a state |ψi but does

not let them feasibly prepare or even copy |ψi Aaronson [3] gave the first formal treatment of

34

public-key quantum money as well as related notions such as copy-protected quantum software

He proved that there exists a quantum oracle relative to which secure public-key quantum money

is possible Unfortunately that result though already involved did not lead in any obvious way

to an explicit (or ldquoreal-worldrdquo) quantum money scheme4 He raised as an open problem whether

secure public-key quantum money is possible relative to a classical oracle In the same paper

Aaronson also proposed an explicit scheme based on random stabilizer states but could not

offer any evidence for its security

And indeed the scheme was broken about a year afterward by Lutomirski et al [30] using an

algorithm for finding planted cliques in random graphs due to Alon Krivelevich and Sudakov

[7] Recently Farhi et al [22] took a completely different approach to public-key quantum

money They proposed a quantum money scheme based on knot theory where each banknote is a

superposition over exponentially-many oriented link diagrams Within a given banknote all the

link diagrams L have the same Alexander polynomial p (L) (a certain knot invariant)5 This p

(L) together with a digital signature of p (L) serves as the banknotersquos ldquoclassical serial numberrdquo

Besides the unusual mathematics employed the work of Farhi et al [22] (building on [30]) also

developed an idea that will play a major role in our work That idea is to construct public-key

quantum money schemes by composing two ldquosimplerrdquo ingredients first objects that we call

mini-schemes and second classical digital signature schemes

The main disadvantage of the knot-based scheme which it shares with every previous scheme is

that no one can say much about its securitymdashother than that it has not yet been broken and that

various known counterfeiting strategies fail Indeed even characterizing which quantum states

Farhi et alrsquos verification procedure accepts remains a difficult open problem on which progress

seems likely to require major advances in knot theory In other words there might be states that

look completely different from ldquolegitimate banknotesrdquo but are still accepted with high

probability

In followup work Lutomirski [29] proposed an ldquoabstractrdquo version of the knot scheme which

gets rid of the link diagrams and Alexander polynomials and simply uses a classical oracle to

achieve the same purposes

35

In talks beginning in 20021 the author often raised the following question

Suppose a function f [n] rarr [n] is a permutation rather than far from a permutation

Is there a small ( polylog (n)-qubit) quantum proof |f i of that fact which can be verified

using polylog (n) quantum queries to f

In this paper we will answer the above question in the negative As a consequence we will

obtain an oracle A such that SZKA 6 sub QMAA This implies for example that any QMA

protocol for graph non-isomorphism would need to exploit something about the problem

structure beyond its reducibility to the collision problem Given that the relativized SZK versus

QMA problem remained open for eight years our solution is surprisingly simple We first use

the in-place amplification procedure of Marriott and Watrous [12] to ldquoeliminate the witnessrdquo and

reduce the question to one about quantum algorithms with extremely small acceptance

probabilities We then use a relatively-minor adaptation of the polynomial degree argument that

was used to prove the original collision lower bound Our proof actually yields an oracle A such

that SZKA 6 sub A0PPA where A0PP is a class defined by Vyalyi [15] that sits between QMA

and PP

Despite the simplicity of our result to our knowledge it constitutes the first nontrivial lower

bound on QMA query complexity where ldquonontrivialrdquo means that it doesnrsquot follow immediately

from earlier results unrelated to QMA2 We hope it will serve as a starting point for stronger

results in the same vein 2 Preliminaries We assume familiarity with quantum query complexity

as well as with complexity classes such as QMA (Quantum Merlin-Arthur) QCMA (Quantum

Merlin-Arthur with classical witnesses) and SZK (Statistical Zero-Knowledge) See Buhrman

and de Wolf [9] for a good introduction to quantum query complexity and the Complexity Zoo3

for definitions of complexity classes We now define the main problem we will study

Problem 1 (Permutation Testing Problem or PTP) Given black-box access to a function f

[n] rarr [n]

and promised that either

(i) f is a permutation (ie is one-to-one) or

(ii) f differs from every permutation on at least n8 coordinates

36

The problem is to accept if

(i) holds and reject if

(ii) holds

1See for example Quantum Lower Bounds wwwscottaaronsoncomtalkslowerppt The Future

(and Past) of Quantum Lower Bounds by Polynomials wwwscottaaronsoncomtalksfutureppt

The Polynomial Method in Quantum and Classical Computing 2From the BBBV lower bound

for quantum search [6] one immediately obtains an oracle A such that coNPA 6sub

QMAA for if there exists a witness state |i that causes a QMA verifier to accept the all-0 oracle

string then thatsame |i must also cause the verifier to accept some string of Hamming weight 1

Also since QMA sube PP relative to all oracles the result of Vereshchagin [14] that there exists an

oracle A such that AMA 6 sub PPA implies an A such that AMA 6 sub QMAA as well

In the above definition the choice of n8 is arbitrary it could be replaced by cn for any

0 lt c lt 1

As mentioned earlier Aaronson [1] defined the collision problem as that of deciding whether f

is one-to-one or two-to-one promised that one of these is the case In this paper we are able to

prove a QMA lower bound for PTP but not for the original collision problem

Fortunately however most of the desirable properties of the collision problem carry over to

PTP As an example we now observe a simple SZK protocol for PTP

Proposition 2 PTP has an (honest-verifier) Statistical Zero-Knowledge proof protocol requiring

O (log n) time and O (1) queries to f

Proof The protocol is the following to check that f [n] rarr [n] is one-to-one the verifier picks

an input x isin [n] uniformly at random sends f (x) to the prover and accepts if and only if the

prover returns x Since the verifier already knows x it is clear that this protocol has the zero-

knowledge property If f is a permutation then the prover can always compute fminus1 (f (x)) so the

protocol has perfect completeness If f is n8-far from a permutation then with at least 18

probability the verifier picks an x such that f (x) has no unique preimage in which case the

prover can find x with probability at most 12 So the protocol has constant soundness

37

21 Upper Bounds

To build intuition we now give a simple QMA upper bound for the collision problem Indeed

this will actually be a QCMA upper bound meaning that the witness is classical and only the

verification procedure is quantum

Theorem 3 For all w isin [0 n] there exists a QCMA protocol for the collision problemmdashie for

verifying that f [n] rarr [n] is one-to-one rather than two-to-onemdashthat uses a w log n-bit classical

witness and makes O _min npnw n13o_ quantum queries to f

Proof If w = O 1048576n13_ then the verifier V can just ignore the witness and solve the problem in

O 1048576n13_ queries using the Brassard-Hoslashyer-Tapp algorithm [8] So assume w ge Cn13 for some

suitable constant C

The witness will consist of claimed values fprime (1) fprime (w) for f (1) f (w) respectively

Given this witness V runs the following procedure

(Step 1) Choose a set of indices X sub [w] with |X| = O (1) uniformly at random Query f (x) for

each x isin X and reject if there is an x isin X such that f (x) 6= fprime (x)

(Step 2) Choose a set of indices Y sub w + 1 n with |Y | = nw uniformly at random Use

Groverrsquos algorithm to look for a y isin Y such that f (y) = fprime (x) for some x isin [w] If such a y

is found then reject otherwise accept

Clearly this procedure makes O _pnw_ quantum queries to f For completeness notice

that if f is one-to-one and the witness satisfies fprime (x) = f (x) for all x isin [w] then V accepts

with probability 1 For soundness suppose that Step 1 accepts Then with high probability

we have fprime (x) = f (x) for at least (say) a 23 fraction of x isin [w] However as in the analysis

of Brassard et al [8] this means that if f is two-to-one then with high probability a Grover

search over nw randomly-chosen indices y isin w + 1 n will succeed at finding a y such

that f (y) = fprime (x) = f (x) for some x isin [w] So if Step 2 does not find such a y then V has

verified to within constant soundness that f is one-to-one

For the Permutation Testing Problem we do not know whether there is a QCMA protocol that

satisfies both T = o 1048576n13_ and w = o (n log n) However notice that if w = (n log n) then the

witness can just give claimed values fprime (1) fprime (n) for f (1) f (n) respectively In that

case the verifier simply needs to check that fprime is indeed a permutation and that fprime (x) = f (x) for

O (1) randomly-chosen values x isin [n] So if w = (n log n) then the QMA QCMA and MA

query complexities are all T = O (1) 3 Main Result

38

In this section we prove a lower bound on the QMA query complexity of the Permutation

Testing Problem Given a QMA verifier V for PTP the first step will be to amplify V rsquos success

probability For this we use the by-now standard procedure of Marriott and Watrous [12] which

amplifies without increasing the size of the quantum witness

Lemma 4 (In-Place Amplification Lemma [12]) Let V be a QMA verifier that uses a w-qubit

quantum witness makes T oracle queries and has completeness and soundness errors 13 Then

for all s ge 1 there exists an amplified verifier V primes that uses a w-qubit quantum witness makes

O (Ts) oracle queries and has completeness and soundness errors 12s

Lemma 4 has a simple consequence that will be the starting point for our lower bound

Lemma 5 (Guessing Lemma) Suppose a language L has a QMA protocol which makes T queries

and uses a w-qubit quantum witness Then there is also a quantum algorithm for L (with no

witness) that makes O (Tw) queries accepts every x isin L with probability at least 0752w and

accepts every x isin L with probability at most 0252w

Proof Let V prime s be the amplified verifier from Lemma 4 Set s = w + 2 and consider running

V primes with the w-qubit maximally mixed state Iw in place of the QMA witness |xi Then given

any yes-instance x isin L Pr _V prime s (x Iw) accepts_ ge1 2w Pr _V prime s (x |xi) accepts_ ge 1 minus

2minuss2wge0752w while given any no-instance x isin L Pr _V prime s (x Iw) accepts_ le12s le0252w

Now let Q be a quantum algorithm for PTP which makes T queries to f Then just like in

the collision lower bound proofs of Aaronson [1] Aaronson and Shi [4] and Kutin [11] the

crucial

fact we will need is the so-called ldquoSymmetrization Lemmardquo namely Qrsquos acceptance probability

4

can be written as a polynomial of degree at most 2T in a small number of integer parameters

characterizing f

In more detail call an ordered pair of integers (m a) valid if

(i) 0 le m le n

(ii) 1 le a le n minus m and

(iii) a divides n minus m

Then for any valid (m a) let Sma be the set of all functions f [n] rarr [n] that are one-toone

on m coordinates and a-to-one on the remaining n minus m coordinates (with the two ranges not

intersecting so that |Imf| = m + nminusm

39

a ) The following version of the Symmetrization Lemma is

a special case of the version proved by Kutin [11]

Lemma 6 (Symmetrization Lemma [1 4 11]) Let Q be a quantum algorithm that makes T

queries to f [n] rarr [n] Then there exists a real polynomial p (m a) of degree at most 2T such

that p (m a) = E fisinSma hPr hQf acceptsii for all valid (m a)

Finally we will need a standard result from approximation theory due to Paturi [13]

Lemma 7 (Paturi [13]) Let q R rarr R be a univariate polynomial such that 0 le q (j) le _

for all integers j isin [a b] and suppose that |q (lceilxrceil) minus q (x)| = (_) for some x isin [a b] Then

deg (q) = _p(x minus a + 1) (b minus x + 1)_

Intuitively Lemma 7 says that deg (q) = 1048576radicb minus a_ if x is close to one of the endpoints of the

range [a b] and that deg (q) = (b minus a) if x is close to the middle of the range

We can now prove the QMA lower bound for PTP

Theorem 8 (Main Result) Let V be a QMA verifier for the Permutation Testing Problem which

makes T quantum queries to the function f [n] rarr [n] and which takes a w-qubit quantum

witness |f i in support of f being a permutation Then Tw = 1048576n13_

Proof Assume without loss of generality that n is divisible by 4 Let = 0252w Then by

Lemma 5 from the hypothesized QMA verifier V we can obtain a quantum algorithm Q for the

PTP that makes O (Tw) queries to f and that satisfies the following two properties

(i) Pr _Qf accepts_ ge 3 for all permutations f [n] rarr [n]

(ii) Pr _Qf accepts_ le for all f [n] rarr [n] that are at least n8-far from any permutation

Now let p (m a) be the real polynomial of degree O (Tw) from Lemma 6 such that

p (m a) = EfisinSma hPr hQf acceptsii

for all valid (m a) Then p satisfies the following two properties

(irsquo) p (m 1) ge 3 for all m isin [n] (For any f isin Sm1 is one-to-one on its entire domain)

(iirsquo) 0 le p (m a) le for all integers 0 le m le 3n4 and a ge 2 such that a divides n minusm (For in

this case (m a) is valid and every f isin Sma is at least n8-far from a permutation)

So to prove the theorem it suffices to show that any polynomial p satisfying properties (irsquo) and

(iirsquo) above has degree 1048576n13_

Let g (x) = p (n2 2x) and let k be the least positive integer such that |g (k)| gt 2 (such a

k must exist since g is a non-constant polynomial) Notice that g (12) = p (n2 1) ge 3 that

g (1) = p (n2 2) le and that |g (i)| le 2 for all i isin [k minus 1] By Lemma 7 these facts together

40

imply that deg (g) = _radick_ Now let c = 2k and let h (i) = p (n minus ci c) Then for all integers

i isin _ n4c nc _ we have 0 le h (i) le since (n minus ci c) is valid n minus ci le 3n4 and c ge 2 On the

other hand we also have h _ n 2c_ = p _n2

c_ = p _n2 2k_ = g (k) gt 2

By Lemma 7 these facts together imply that deg (h) = (nc) = (nk)

Clearly deg (g) le deg (p) and deg (h) le deg (p) So combining

deg (p) = _maxnradickn ko_ = _n13_

41

CHAPTER-6

FUTURE OF QUANTUM INFORMATION TECHNOLOGY

For some time now we have been immersed in the information age - if youre reading this

online youre proving the point Our immersion in the information age is reflected in almost

every social setting it has become hard not to find at least one face if not several faces aglow

with the cool blue light of a smartphone screen Google Facebook and Twitter have changed the

way we learn about the world and each other

The digitisation of information has brought about Marshall McLuhans global village in which

nearly everyone is - or soon will be - fully connected The explosive growth of information

technology has transformed our world into one that is now draped in a network of fiber-optic

cables dotted with cell-phone towers and encircled by a drone army of communication satellites

all of which enable the flow of digital information around the globe

Privacy security knowledge and power

A major unresolved question is the role or even the possibility of private communication and

information security in this brave new digital world - a topic in which governments corporations

and private citizens all have a vested interest This issue has come sharply into focus as various

governments attempt to monitor the communications of their citizens and even threaten to ban

some services such as Blackberrys BBM because of the incredibly secure encryption such

services provide to end users But its important to emphasise that private and secure

communication is broadly relevant to users of the internet For example if I want to do some

online banking I want to be sure my financial data is protected from online prying eyes

A fundamental scientific question is this how much privacy is even possible over networks

controlled and monitored by others Under what physical conditions can Person A communicate

privately with Person B over a public network Fortunately we currently have efficient

encryption systems to reliably protect online activities such as personal banking But practical

systems can be cracked depending on the resources the would-be eavesdroppers have at their

disposal

42

One of the most widely used encryption schemes on the internet today is the RSA scheme The

basic idea is that one can encode information with a key - which is some very large number - that

is made publicly available With RSA the encoded information can only be decoded by someone

who knows the two prime factors that when multiplied together produce this very large number

While it is easy to multiply two numbers together it turns out to be extremely difficult to find the

two prime factors that are multiplied to create a large product This difficulty is what enables

privacy I make my locking-key publicly available and anyone who wants to send me

information privately encodes that information with this locking-key If I do not disclose the

prime factors to anyone - that is if I keep my unlocking key private - then only I can decipher

the encoded message To anyone else the message looks like random binary gibberish

How secure is this The answer depends on a number of practical considerations but

fundamentally RSA remains only as secure as the difficulty of finding the two prime factors for

the locking-key For large enough locking-key numbers this problem is believed to be unfeasibly

hard to solve - even with vast amounts of conventional computing power - because finding the

prime factors gets exponentially more difficult as you increase the number of digits in the key

43

A multinational research group is trying to transmit quantum bits to an orbiting satellite

[GALLOGETTY]

What can quantum physics do for you or to you

Information is an abstract concept - it can comprise names numbers dates places almost

anything The important point is that any information is ultimately represented as some physical

quantity - it is always encoded in some physical medium whether the physical medium involves

sound waves from one persons mouth to anothers ear blotches of ink on paper pulses of light in

a fiber-optic cable or the magnetised regions of a hard drive When the physical medium is

manipulated according to the laws of classical physics - for example the laws of classical

electromagnetism which completely describe conventional computers - then these laws imply

certain physical limits on how the encoded information can be manipulated and accessed And

44

when the information is encoded in physical media that obey the laws of quantum physics then a

different set of rules describes how the information can be manipulated and accessed

The laws of quantum physics are now well established as the appropriate rules that govern the

way that world works However the special features of the quantum laws that make them

different from the classical laws are typically only manifest when we manipulate objects at the

level of individual atoms and photons So the idea of quantum information technology is based

on the possibility of encoding information at this tiny scale But before we discuss the practical

issues associated with this technological challenge lets first address the following question how

do the unique features of the quantum laws - and any quantum technology based upon them -

affect information privacy

The quantum information age

The quantum information age was born about 20 years ago from a somewhat unexpected union

between quantum physicists and computing scientists One of the major insights that brought

quantum information to the forefront of science was the discovery by Peter Shor that a quantum

computer (a computer built out of components that can be manipulated according to the full

extent allowed by the laws of quantum physics) could easily solve the factoring problem Hence

a quantum computer if one could be built would spell an end to the security of the most

practical encryption method used for private communication today

Shors algorithm and a host of other quantum algorithms that have been discovered subsequently

have stimulated a major global research effort investigating practical ways to build a large-scale

quantum computer However there are major technological obstacles to realising large-scale

quantum computers It turns out that the same special features of quantum mechanics that give

power to quantum computing also make them tricky to build Quantum systems are fragile fickle

and tough to control While small-scale quantum computers consisting of up to a dozen quantum

bits or qubits have been realised in the most advanced research labs currently there is no

known technological pathway to building a large-scale quantum computer with thousands or

even tens of thousands of qubits which would be required to crack present-day encryption

45

The quantum world taketh but also giveth

Quantum technology creates a threat to the possibility of private communication using current

encryption methods - but interestingly it also provides a new and more secure solution to

achieving private communication While a quantum computer would break current practical

encryption schemes quantum technology also enables a new means of establishing

unconditionally secure private communication through a protocol known as quantum key

distribution which was actually discovered a decade before Shors algorithm

Quantum key distribution exploits one of the fundamental features of quantum mechanics known

as the Heisenberg uncertainty principle This principle holds that when dealing with quantum

systems it is impossible to observe one property of that system without disturbing some other

property The significance of this for private communication is this if a Sender A transmits some

(random) data to a Receiver B using quantum bits encoded in the right way then the receiver

can always detect whether an eavesdropper has snooped on the transmission

If no eavesdropper is detected then B is certain that the random data is private and this private

random data can then be used to establish a secure communication channel over a regular

(classical) network Unlike the RSA scheme currently in use private communication with

quantum key distribution remains secure even if an adversary has access to a quantum computer

The technological threshold for creating and using quantum communication in practice is much

lower than that for creating a practical quantum computer In fact we already have the

technology - researchers have shown that it is possible to transmit quantum bits over hundreds of

kilometers using commercial-grade fiber-optic cables Moreover there are already private

companies offering quantum cryptographic systems For example quantum key distribution was

used to establish secure communication during the federal election in Switzerland in 2007

Moreover a multi-national research group led by one of my colleagues Thomas Jennewein at

the Institute for Quantum Computing is now undertaking a research program to transmit

quantum bits to an orbiting communications satellite which would enable quantum key

distribution on a truly global scale

46

Of course the extra security afforded by quantum key distribution is currently unnecessary for

most applications Current encryption methods and the information it protects can be decrypted

only in the future from an adversary who gains eventual access to a quantum computer

Although for most applications this level of security is not relevant for others the threat of this

future technology can be a serious security concern

We live at a time when rapid developments in conventional information technology have led to

an equally rapidly adapting social and political landscape surrounding private communication

over public networks The advent of quantum information technology will further shape the

future of communication privacy in our expanding global village

47

CHAPTER-7

CONCLUSION

The field of quantum information has typically concerned itself with the manipulation of discrete

systems such as quantum bits or ldquoqubitsrdquo However many quantum variables such as position

momentum or the quadrature amplitudes of electromagnetic fields are continuous leading to the

concept of continuous quantum information Initially quantum information processing with

continuousvariables seemed daunting at best ill-defined atworst Nonetheless the first real

success came with theexperimental realization of quantum teleportation for optical fields This

was soon followed by a flood of activity to understand the strengths and weaknesses of this type

of quantum information and how it may be processed The next major breakthrough was the

successful definition of a notion of universal quantum computation over continuous variables

suggesting that such variables are as powerful as conventional qubits for any class of

computation In some ways continuous-variable computation may not be so different from qubit-

based computation In particular limitations due to finite precision make quantum floating-point

operations like their classical counterparts effectively discrete Thus we might expect a

continuous-variable quantum computer to perform no better than a discrete quantum computer

However for some tasks continuous-variable quantum computers are nonetheless more efficient

Indeed in many protocols especially those relating to communication they only require linear

operations together with classical feedforward and detection This together with the large

bandwidths naturally available to continuous (optical) 61 variables appears to give them the

potential for a significant advantage

However notwithstanding these successes the very practical optical cv

approach when solely based upon Gaussian transformations such as beam-splitter and squeezing

transformations feed-forward and homodyne detections is not sufficient for implementing more

advanced or ldquogenuinerdquo quantum information protocols Any more sophisticated quantum

protocol that is truly superior to its classical counterpart requires a non- Gaussian element This

may be included on the level of the measurements for example via state preparation conditioned

upon the number of photons detected in a subset of the Gaussian modes Alternatively one may

48

directly apply a non-Gaussian operation which involves a highly nonlinear optical interaction

described by a Hamiltonian at least cubic in the mode operators Though being a significant first

step communication protocols in which this non-Gaussian element is missing cannot fully

exploit the advantages offered by quantum mechanics For example the goals in the Gaussian

protocols of cv quantum teleportation and dense coding are reliable transfer of quantum

information and increase of classical capacity respectively However in both cases preshared

entanglement is required Using this resource via teleportation fragile quantum information can

be conveyed through a classical communication channel without being subject to decoherence in

a noisy quantum channel In entanglement-based dense coding using an ideal quantum channel

more classical information can be transmitted than directly through a classical channel For

transferring quantum information over long distances however entanglement must be

distributed through increasingly noisy quantum channels Hence entanglement distillation is

needed and for this Gaussian resources and Gaussian operations alone do not suffice Similarly

ldquotruerdquo quantum coding would require a non-Gaussian decoding step at the receiving end In

general any cv quantum computation that is genuinely quantum and hence not efficiently

simulatible by a classical computer must contain a non-Gaussian element Among the

communication protocols cv quantum key distribution appears in some sense exceptional

because even in a purely Gaussian implementation it may well enhance the security compared to

classical key distribution schemes The experiments accomplished so far in cv quantum

information reflect the observations of the preceding paragraphs Gaussian state preparation

including (multiparty) entangled states and Gaussian state manipulation are techniques well

understood and implemented in many laboratories around the globe However in order to come

closer to real applications both for long-distance quantum communication and for quantum

computationa new generation of experiments is needed crossing the border between the

Gaussian and non-Gaussian worlds Beyond this border techniques from the more ldquotraditionalrdquo

single-photon based discrete-variable domain will have to be incorporated into the cv

approaches In fact a real-world application of optical quantum communication and

computation possibly including atom-light quantum interfaces and atomic quantum memories

will most likely combine the assets of both approaches the continuous-variable one and that

based on discrete variables

49

BIBLIOGRAPHY

[1] S Aaronson Quantum lower bound for the collision problem In Proc ACM STOC pages

635ndash642 2002 quant-ph0111102

[2] S Aaronson Quantum computing postselection and probabilistic polynomial-time Proc

Roy Soc London A461(2063)3473ndash3482 2005 quant-ph0412187

[3] S Aaronson and G Kuperberg Quantum versus classical proofs and advice Theory of

Computing 3(7)129ndash157 2007 Previous version in Proceedings of CCC 2007

quant-ph0604056

50

REFERENCES

The links referred are

[1] wwwieeecom

[2] wwwgooglecom

[3] httpw3antdnistgovqinindexshtml

[4] httpwwwaljazeeracomindepthopinion20120220122237159922635html

[5] M A Nielsen and I L Chuang ldquoQuantum Computation and Quantum Informationrdquo

(Cambridge 2000)

[6] H Takayanagi H Ando and T Fujisawa ldquoFrom Quantum Effects to Quantum Circuits

Toward the Quantum Computerrdquo NTT REVIEW Vol 12 No 1 pp 17-25 2000

[7] T Fujisawa ldquoSingle electron dynamicsrdquo to be published in Encyclopedia of Nanoscience

and Nanotechnology

51

CHAPTER-2

INTRODUCTION

Today many people are familiar with at least the consequences of Moorersquos Law ndash the fastest

computer in the shops doubles in speed about every 18 months to two years This is because

electronic component devices are shrinking The smaller they get the faster they work and the

closer they can be packed on a silicon chip This exponential progress first noted4by Gordon

Moore a co-founder and former CEO of Intel in 1965 has continued ever since But it cannot

go on forever Hurdles exist for example silicon will hit problems with oxide thinness track

width or whatever 5 new materials or even new paradigms such as self-assembled nano-

devices or molecular electronics will be needed lots of dollars will be needed as Moorersquos

second law tells us that fabrication costs are also growing exponentially However even if all the

hurdles can be overcome we will eventually run into Nature

Very small things do not behave the same way as big ones ndash they begin to reveal their true

quantum nature Following Moorersquos Law an extrapolation of the exponentially decaying number

of electrons per elementary device on a chip gets to one electron per device around 2020 This is

clearly too naiumlve but it gives us a hint Eventually we will get to scales where quantum

phenomena rule whether we like it or not If we are unable to control these effects then data bits

in memory or processors will suffer errors from quantum fluctuations and devices will fail

Clearly this alone makes a strong case for investment in research into quantum devices and

quantum control The results should enable us to push Moorersquos Law to the limit evolving

conventional information technology (IT) as far as it can go However such quantum research

has already shown that the potential exists to do much more ndashrevolution Instead of playing

support act to make better conventional devices let quantum mechanics take centre stage in new

technology that stores processes and communicates information according to the laws of

quantum mechanics

2

CHAPTER-3

LITERATURE REVIEW






logical advancement







have now














3




(ii) initialized


























4





























5



























6












as time and memory
















7



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

CHAPTER-3

LITERATURE REVIEW






logical advancement







have now














3




(ii) initialized


























4





























5



























6












as time and memory
















7



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51




(ii) initialized


























4





























5



























6












as time and memory
















7



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51





























5



























6












as time and memory
















7



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



























6












as time and memory
















7



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51












as time and memory
















7



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



(a) (b)

( c ) (d)










8






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51






























9



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



( a )

( b )








10


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51


















refinement



11

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

( a )

( b )








12































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51































time

13

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

QUANTUM COMPUTING






























14








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51








operation













factorization









15




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51




























16





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51





computing They are


ii Ion Trap

iii Quantum Dot

iv Optical Methods







central themes









17

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

























factoring algorithm



18

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

ii Ion Trap













demonstrated

iii Quantum Dot














19



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



iv Optical















qubytes












20
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51
























Decoherence



21









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51









ii Error Correction




















1

22






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51






























23










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51










QUANTUM CRYTOGRAPHY




















24























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51























broadcasts her bits

25













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51













26

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

Prospects for QIT






















reality

QUANTUMALGORITHMS



Adiabatic algorithm



27



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



a) Decoherence



















0 rarr (000)

1rarr (111)


(000) rarr (100)

(111) rarr (011)


28

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

CHAPTER- 4

TITLE OF WORK








classical computer









29

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

















(xiaotangnistgov)



sourcedetectors







30











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51











printed it







|ψsi =_ψfk(s)




break

31

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

CHAPTER- 5








could not















32



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



noted below















Error correction











33



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



Quantum algorithms







computing




















34

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

























probability




35















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51















and PP










[n] rarr [n]




36



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51



(ii) holds










0 lt c lt 1















37

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

21 Upper Bounds









suitable constant C






















38




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51




















crucial


4


characterizing f


(i) 0 le m le n






39
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51
































40




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51




c_ = p _n2 2k_ = g (k) gt 2




41

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

CHAPTER-6


























disposal

42
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51
















43


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51


[GALLOGETTY]










44




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51




























45




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51




























46










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51










47

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

CHAPTER-7

CONCLUSION



























48































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51































49

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

BIBLIOGRAPHY


635ndash642 2002 quant-ph0111102





quant-ph0604056

50

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

REFERENCES


[1] wwwieeecom

[2] wwwgooglecom




(Cambridge 2000)




and Nanotechnology

51

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