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Quantum Computers

Aditya Kumakale

Akshay Diwakar 

TYEJ II, SVCP,

 Pune.

Abstract

The subject of quantum computing bringstogether ideas from classical information theory,computer science, and quantum physics. Thisreview aims to summarize not just quantumcomputing, but the whole subject of quantuminformation theory. Information can be identified as the most general thing which must propagatefrom a cause to an effect. It therefore has afundamentally important role in the science of 

 physics. However, the mathematical treatment of information, especially information

 processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in

  physics is only now being discovered. This isespecially true in quantum mechanics. Thetheory of quantum information and computing 

 puts this significance on a firm footing, and hasled to some profound and exciting new insightsinto the natural world. Among these are the useof quantum states to permit the securetransmission of classical information (quantumcryptography), the use of quantumentanglement to permit reliable transmission of quantum states (teleportation), the possibility of 

 preserving quantum coherence in the presenceof irreversible noise processes (quantum error 

correction), and the use of controlled quantumevolution for efficient computation (quantumcomputation). The common theme of all theseinsights is the use of quantum entanglement asa computational resource.

It turns out that information theory and quantummechanics fit together very well. In order toexplain their relationship, this review begins with

an introduction to classical information theory and computer science. Basic quantuminformation ideas are next outlined, including qubits, the `no cloning' property and teleportation.

The review concludes with an outline of themain features of quantum information physicsand avenues for future research.

I. Introduction

The acceleration of computing power never seemsslow. With each blink of an eye, there’s a new, faster   processor or a better data-storage-intensive harddrive. But for all their computational might,computers as we know them will eventually bump upagainst the laws of physics. Technology marches onresolutely, shrinking electronic components andcramming more circuitry onto smaller and smaller wafers of silicon. If the current rate of miniaturizationcontinues, computer experts predict that within adecade or two, transistors will dwindle to the size of an atom. But at those dimensions, well-behaved, predictable classical behavior goes out the window,

and the slippery, untenable nature of quantummechanics takes over. In the quantum world, rather than being entities with sharply defined positions andmotions, particles are described by spread-out wavefunctions, seemingly existing in many places at once.So it might seem that the power of computers isdestined to reach a limit. But scientists usually don'ttake such pronouncements at face value--in this case,they have long been aware of 

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a way around this apparent constraint. For within theshadowy quantum world there is more potentialcomputing power than the speediest processor couldever dream of. That power stems from quantum  particles' capability for existing in more than onestate, as well as their ability to become inextricablylinked to each other by a phenomenon known asentanglement. Traditional computers performcalculations, however quickly, in a basicallysequential manner. Their limitations surface in thesimple, yet striking, example of factoring a largenumber. The time a computer spends searching for anumber's factors increases astronomically with the sizeof the number. To factor a 400-digit number, for example, would take a modern computer billions of years. On the other hand, a computer made of quantum particles has a built-in parallelism becausequantum calculations can be performed on the  particles' coexisting states simultaneously. Aquantum computer, then, might factor that 400-digit

number in minutes. Such a completely differentapproach to computing, it seems, truly earns the

designation "paradigm shift.”

II. Concepts

A. Moore’s Law And The Future Of 

Computers.

In 1965 Intel co-founder Gordan Moore noted that processing power (number of transistors and speed)of computer chips was doubling each 18 months or so. This trend has continued for nearly 4 decades.The basic processing unit in a computer chip is thetransistor which acts like a small switch. The binarydigits 0 and 1 are represented by the transistor beingturned off or on.Currently thousands of electrons are used to driveeach transistor. As the processing power increases,the size of each transistor reduces. If Moore's lawcontinues unabated, then each transistor is predictedto be as small as a hydrogen atom by about 2030. Atthe size the quantum nature of electrons in the atoms  becomes significant. It generates errors in thecomputation.

However, rather than be a hindrance, it is possible toexploit the quantum physicsas a new way to docomputation. And this new way opens up fantasticnew computational power  based on the wave nature of quantum particles.

B. Particle-Wave Duality.

We normally think of electrons, atoms and moleculesas particles. But each of these objects can also behaveas waves. This dual particle-wave behavior wasfirst suggested in the 1920's by Louis de Broglie.This concept emerged as follows. Thomas Young'sexperiment with double slits in the early 1800’s showthat light behaves as if it is a wave. But, strikingly,Einstein's explanation of the photoelectric effect in1905 shows that light consists of particles. In 1923 de

Broglie suggested this dual particle-wave propertymight apply to all particles including electrons. Thenin 1926 Davisson and Germer found that electronsscattered off a crystal of nickel behaved as if theywere waves. Since then neutrons, atoms and evenmolecules have been shown to behave as waves. Thewaves tell us where the particle is likely to be found.This dual particle-wave property is exploited inquantum computing in the following way. A wave isspread out in space. In particular, a wave can spreadout over two different places at once. This means thata particle can also exist at two places at once. Thisconcept is called the superposition  principle - the particle can be in a superposition of two places.

C. Superpositioning

Superpositioning is a big word for an old concept:that two things can overlap each other withoutinterfering with each other. In classical computers,electrons cannot occupy the same space at the sametime, but as waves, they can.

1. Superposition in waves

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Figure 1. is an illustration of two super-imposedwaves A and B To add these waves together numerically, S = A + B, the result would be a wavethat looks like neither of its components. However,one could retrieve either wave be subtracting out theother, as shown. (The wave form S, is shown asdotted to indicate that it is only the apparent waveform; the actual wave form is the combination of twodifferent waves. In the quantum world, the combinedwave form is a set of amplitude probabilities.)

Figure 1. Superpositioning of waves.

D. Bits And Qubits 

The basic data unit in a conventional (or classical)computer is the bit, or binary digit. A bit stores anumerical value of either 0 or 1. An example of how

 bits are stored is given by a CD ROM: ``pits'' and``lands'' (absence of a pit) are used to store the binarydata. In the solid-state world of classical computers,the bit's value corresponds to the presence or absenceof current. Bits can be manipulated by what areknown as logic gates, which transform bit values indesigned ways. For example, a NOT gate changes a bit from 0 to 1, or vice versa. In fact, everything acomputer does, from word processing to modelingthe structure of the universe, can be boiled down tovarious combinations of these simple logic gatesoperating on bits. We could also represent a bit usingtwo different electron orbits in a single atom. In most

atoms there are many electrons in many orbits. Butwe need only consider the orbits available to a singleoutermost electron in each atom .The figure on theright shows two atoms representing the binarynumber 10.The inner orbits represent the number 0and the outer orbits represent the binary number 1.

The position of the electron gives number.

Figure 2. Binary Numbers.

The Figure 2 shows two atoms representing the  binary number 10. The inner orbits represent thenumber 0 and the outer orbits represent the binarynumber 1. The position of the electron gives thenumber stored by the atom. However; a completelynew possibility opens up for atoms. Electrons have awave property which allows a single electron to be intwo orbits simultaneously. In other words, theelectron can be in a superposition of both orbits. Thefigure on the left shows two atoms each with a singleelectron in a superposition of two orbits. Each atomrepresents the binary numbers 0 and 1simultaneously. The two atoms together represent the4 binary numbers 00, 01, 10 and 11 simultaneously norbits To distinguish this new kind data storage froma conventional bit, it is called a quantum bit which isshortened to qubit. Each atom in the figure above is aqubit. The key point is that a qubit can be in a

superposition of the two numbers 0 and 1.Superposition states allow many computations to be performed simultaneously, and give rise to what isknown as quantum parallelism.

Figure 3. Quantum Numbers.

1. Quantum dotsUsing advanced lithographic techniques it is possibleto etch small structures called quantum dots insemiconductor materials. Each such dot, which can

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 be as small as 30nm across (about 30 times the sizeof an atom) can confine a single electron in discreteenergy levels.Thus the quantum dot behaves like alarge artificial atom and can be used as a qubit. Auser can access individual quantum dots usingfocused laser beams which can flip the electron between two discrete energy levels or place it into asuperposition of the two levels. The requiredinteraction between qubits occurs through externallyapplied electric and optical fields.

E. Quantum Parallelism

Quantum Computation is a new field which promisesexponential parallelism. A one bit memory can storeone of the numbers 0 and 1. Likewise a two bitmemory can store one of the binary numbers 00, 01,10 and 11 (i.e. 0, 1, 2 and 3 in base ten) But thesememories can only store a single number (e.g. the

 binary number 10) at a time.As described above, a quantum superposition stateallows a qubit to store 0 and 1 simultaneously Twoqubits can store all the 4 binary numbers 00, 01, 10and 11 simultaneously. Three qubits stores the 8 binary numbers 000, 001, 010, 011, 100, 101, 110and 111 simultaneously. 300 qubits can store morethan 1090 numbers simultaneously. That's more thanthe number of atoms in the visible universe! Thisshows the power of quantum computers: just 300 photons (or 300 ions etc.) can store more numbersthan there are atoms in the universe, and calculationscan be performed simultaneously on each of thesenumbers! A system of 500 qubits, which isimpossible to simulate classically, represents aquantum superposition of as many as 2500 states.Each state would be classically equivalent to a singlelist of 500 1's and 0's. Any quantum operation on thatsystem --a particular pulse of radio waves, for instance, whose action might be to execute acontrolled-NOT operation on the 100th and 101stqubits-- would simultaneously operate on all 2500states. Hence with one fell swoop, one tick of thecomputer clock, a quantum operation could computenot just on one machine state, as serial computers do,  but on 2500 machine states at once! Eventually,however, observing the system would cause it to

collapse into a single quantum state corresponding toa single answer, a single list of 500 1's and 0's, asdictated by the measurement axiom of quantummechanics.

III. Quantum Computers

A quantum computer is a device for  computation thatmakes direct use of  quantum mechanical  phenomena, 

such as superposition and entanglement, to performoperations on data. Quantum computers are differentfrom traditional computers based on transistors. The basic principle behind quantum computation is thatquantum properties can be used to represent data and  perform operations on these data.  A classicalcomputer has a memory made up of  bits, where each  bit represents either a one or a zero. A quantumcomputer maintains a sequence of  qubits. A singlequbit can represent a one, a zero, or, crucially,any quantum superposition of these; moreover, a pair of qubits can be in any quantum superposition of 4states, and three qubits in any superposition of 8. Ingeneral a quantum computer with n qubits can be inan arbitrary superposition of up to 2n different statessimultaneously (this compares to a normal computer that can only be in one of these 2n states at any onetime). A quantum computer operates by manipulatingthose qubits with a fixed sequence of quantum logicgates. The sequence of gates to be applied is called

a quantum algorithm. As our technology rushesforward, several factors work together to push ustoward the quantum computing world These factorsare: scaling in size, energy consumption, economicsof building leading edge computers, and newapplications that are available with quantumcomputers that cannot be executed with classicalcomputers. At the current rate of chipminiaturization, energy efficiency, and economics,the classical computer of the year 2020 , wouldcontain a CPU running at 40 GHz with 160 Gb RAM,and run on 40 watts of power.

A.  Energy

The speed of chips is also rising exponentially.Faster, more densely-packed, andcloser transistors cause thermodynamic problems.Advances in energy efficiency are requiredto keep the chips from melting during use.Fortunately, energy efficiencies are increasing, andthe thermodynamic problems are being resolved.These energy advances are also pushing the physicsof chips into the quantum realm. Quantum computersare reversible, therefore there is theoretically no netenergy consumption. Quantum reversibility means

that quantum computers drive themselves forward ininfinitesimal (reversible) steps, much the same waythat molecules of perfume would diffuse from a perfume bottle. Quantum computer programs are not"run", but are said to "evolve," as they process the program’s inputs to outputs. Incidentally,reversibility also means that the inputs of a quantumcomputer can be implied from the outputs; the program can be run backwards .The argument for 

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energy inevitability is a "carrot-and-stick" argument:the energy inefficiencies drive us away from classicalcomputers, and the appeal for energy-free (or at least,much reduced energy consumption) computing drivesus toward quantum computers.

B. Economics

Besides the energy factors at the micro level of computing, there are large-scale economic factors pushing us to a more energy-efficient means of computing: 5% of the entire national power  production is consumed by computing machinery.With "fossil fuels continuing to dwindle, fission power in disfavor with the public, and fusion power still many decades away, the drain computers imposeon our power supply could become significant."The cost to build a semiconductor plant doublesevery three years. By extrapolating that trend to the

year 2020, a semiconductor plant will cost $1 trillionto build, which is 5% of the U.S. GNP. Based onMotorola’s sales figures, a similar company wouldneed $10 trillion in annual sales to support that kindof construction.Japan, in its bid for software and hardware globaldominance, has allocated huge funds for quantumcomputer research., VP of Hewlett-Packard, reportedthat 70% of all quantum computer research is beingdone by the Japanese. They have included quantumcomputers as an integrated step of their globalacquisition strategy.

IV. Applications

A. Quantum Entanglement

Quantum computers also utilize another aspect of quantum mechanics known as entanglement. One problem with the idea of quantum computers is that if you try to look at the subatomic particles, you could bump them, and thereby change their value. But inquantum physics, if you apply an outside force to twoatoms, it can cause them to become entangled, andthe second atom can take on the properties of the firstatom. So if left alone, an atom will spin in all

directions; but the instant it is disturbed it choosesone spin, or one value; and at the same time, thesecond entangled atom will choose an opposite spin,or value. This allows scientists to know the value of the qubits without actually looking at them, whichwould collapse them back into 1's or 0's.Albert Einstein, Boris Podolski, and Nathan Rosenknew that the state vectors of certain quantumsystems were correlated, or "entangled" with eachother. If one changes the state vector of one system,

the corresponding state vector of the other system isalso changed, instantaneously, and independently of the medium through which some communicatingsignal must travel.Throughout all of history previously, all physical phenomenon was dependent on some force, and some particle to carry that force, and therefore the speed of light restriction applied. For example, electrostaticforces are carried by the electron, gravitational forcesare carried by the graviton, etc. However, withentanglement, quantum systems are correlated insome way that does not involve a force, and the speedof light restriction does not apply. The actualmechanism of how one system affects the other isstill unknown.

1. Collapse Of State Vector

When two quantum systems are created while

conserving some property, their state vectors arecorrelated, or entangled. For example, when two photons are created, and their spin conserved, as itmust, one photon has a spin of 1 and a spin of -1. Bymeasuring one of the state vectors of the photon, thestate vector collapses into a knowable state.Instantaneously and automatically, the state vector of the other photon collapses into the other knowablestate.When one photon’s spin is measured and found to be1, the other photon’s spin of -1 immediately becomesknown too. There are no forces involved and noexplanation of the mechanism.

B. Quantum Teleportation

The principle of entanglement enables a phenomenoncalled quantum teleportation. This kind of teleportation does not involve moving an entity fromone physical location to another, as may be found inmany popular science fiction stories, but adestruction of the original and recreation of anidentical duplicate at another location.

C. Encryption Technology

The speed of quantum computers also jeopardizes theencryption schemes that rely on impracticably-longtimes to decrypt by brute force methods. Encryptionschemes that may take millions of years to guess andcheck are now vulnerable to quantum computers thatmay reach a solution within one year. Manygovernments, included ours, use such encryptionschemes for national security. They are veryinterested in any technology that

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 puts that at risk. As a result, the Office of NavalResearch, the CIA, and DARPA, are sinkinghuge funds into quantum computer research. DARPAis funding $5 million for a QuantumInformation and Computing Institute, and the CIA isfunding an unknown amount for factoringof large integers, a fundamental part of encryptiontechnology.

D. Ultra-secure and Super-dense

Communications

It is possible to transmit information without a signal path by using newly-discovered quantum principles,quantum teleportation. There is no way to interceptthe path and extract information. Ultra-securecommunication is also possible by super-denseinformation coding, which is a new technology in itsown right. Quantum bits can be used to allow more

information to be communicated per bit than thesame number of classical bits.

E. Improved Error Correction and

Error Detection

Through similar processes that support ultra-secureand super-dense communications, the existing bitstreams can be made more robust and secure byimprovements in error correction and detection.Recovering informational from a noisy transmission path will also be a lucrative and useful practice.

F. True Randomness

Classical computers do not have the ability togenerate true random numbers. The random number generators on today’s computers are pseudo-randomgenerators—there is always a cycle or a trend,however subtle. Quantum computers can generatetrue randomness, thus give more veracity to programsthat need true randomness in their processing.Randomness plays a significant part of applicationswith a heavy reliance on statistical approaches, for simulations, for code making, randomized algorithmsfor problems solving, and for stock market

 predictions, to name a few.With the global forces of computer competition,encryption technology for national security, newapplications, and the thermodynamics of computer systems changing as they are, there is a rush towardthe new quantum technology to produce the firstviable quantum computer.

V. Conclusion

With classical computers gradually approaching their limit, the quantum computer promises to deliver anew level of computational power. With them comesa whole new theory of computation that incorporates

the strange effects of quantum mechanics andconsiders every physical object to be some kind of quantum computer. A quantum computer thus has thetheoretical capability of simulating any finite physical system and may even hold the key tocreating an artificially intelligent computer.The quantum computers power to performcalculations across a multitude of parallel universesgives it the ability to quickly perform tasks thatclassical computers will never be able to practicallyachieve. This power can only be unleashed with thecorrect type of algorithm, a type of algorithm that isextremely difficult to formulate. Some algorithms

have already been invented; they are proving to havehuge implications on the world of cryptography. Thisis because they enable the most commonly usedcryptography techniques to be broken in a matter of seconds. Ironically, a spin off of quantum computing,quantum communication allows information to besent without eavesdroppers listening undetected.For now at least, the world of cryptography is safe because the quantum computer is very difficult toimplement. The most successful experiments only being able to add one and one together.Nobody cantell if the problems being experienced by researcherscan be overcome.The very thing that makes them

 powerful, their reliance on quantum mechanics, alsomakes them extremely fragile.

VI. References

• "Quantum Computing with Molecules" article

in Scientific American  by Neil Gershenfeld andIsaac L. Chuang.

• Gregg Jaeger (2006). Quantum Information: An

Overview.

• David P. DiVincenzo (2000). "The Physical

Implementation of QuantumComputation". Experimental Proposals for 

Quantum Computation.