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Quantum Computing Abstract Imagine a computer whose memory is exponentially larger than its apparent physical size; a computer that can manipulate an exponential set of inputs simultaneously; a computer that computes in the twilight zone of Hilbert space. You would be thinking of a quantum computer. Relatively few and simple concepts from quantum mechanics are needed to make quantum computers a possibility. The subtlety has been in learning to manipulate these concepts. Is such a computer an inevitability or will it is too difficult to build? The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, the review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the EPR experiment described. The EPR-Bell correlations and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory, and, arguably, quantum from classical physics. Basic quantum information ideas are described, including key distribution, teleportation, data compression, quantum error correction, the universal quantum computer and quantum algorithms. The common theme of all these ideas is the use of quantum entanglement as a computational resource. Experimental methods for small quantum processors are briefly sketched, concentrating on ion traps, high Q cavities, and NMR. The review concludes with an outline of the main features of quantum information physics, and avenues for future research. Introduction What is quantum computing? It's something that could have been thought up a long time ago - an idea whose time has come. For any physical theory one can ask: what sort of machines will do useful computation? or, what sort of processes will count as useful computational acts? Alan Turing thought about this in 1936 with regard (implicitly) to classical mechanics, and gave the world the paradigm classical computer: the Turing machine. But even in 1936 classical mechanics was known to be false. Work is now under way mostly theoretical, but tentatively, hesitantly groping towards the practical - in seeing what quantum mechanics means for computers and computing. The basic idea behind quantum computing in the merging of computer science, information theory and quantum mechanics from classical physics for better efficiency of operation of systems. Information is the most important element of any system.

Information can be represented in many ways. It can be analyzed provided we know how it was encoded. For example, the two statements ``the quantum computer is very interesting'' and ``l'ordinateur quantique est tres interessant'' have something in common, although they share no words. The thing they have in common is their information content. Essentially the same information could be expressed in many other ways, for example by substituting numbers for letters in a scheme such as a -> 97, b -> 98, c -> 99 and so on, in which case the English version of the above statement becomes 116 104 101 32 113 117 97 110 116 117 109... . It is very significant that information can be expressed in different ways without losing its essential nature, since this leads to the possibility of the automatic manipulation of information: a machine need only be able to manipulate quite simple things like integers in order to do surprisingly powerful information processing, from document preparation to differential calculus, even to translating between human languages. We are familiar with this now, because of the ubiquitous computer, but even fifty years ago such a widespread significance of automated information processing was not foreseen. However, there is one thing that all ways of expressing information must have in common: they all use real physical things to do the job. Spoken words are conveyed by air pressure fluctuations, written ones by arrangements of ink molecules on paper, even thoughts depend on neurons (Landauer 1991). The rallying cry of the information physicist is ``no information without physical representation!'' Conversely, the fact that information is insensitive to exactly how it is expressed, and can be freely translated from one form to another, makes it an obvious candidate for a fundamentally important role in physics, like energy and momentum and other such abstractions. However, until the second half of this century, the precise mathematical treatment of information, especially information processing, was undiscovered, so the significance of information in physics was only hinted at in concepts such as entropy in thermodynamics. It now appears that information may have a much deeper significance. Historically, much of fundamental physics has been concerned with discovering the fundamental particles of nature and the equations which describe their motions and interactions. It now appears that a different programme may be equally important: to discover the ways that nature allows, and prevents, information to be expressed and manipulated, rather than particles to move. For example, the best way to state exactly what can and cannot travel faster than light is to identify information as the speed-limited entity. In quantum mechanics, it is highly significant that the state vector must not contain, whether explicitly or implicitly, more information than can meaningfully be associated with a given system. Among other things this produces the wave function symmetry requirements which lead to Bose Einstein and Fermi Dirac statistics, the periodic structure of atoms, and so on.

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. Today's 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. The point is, however, that quantum technology can offer much more than cramming more and more bits to silicon and multiplying the clock-speed of microprocessors. It can support entirely new kind of computation with qualitatively new algorithms based on quantum principles! A bit is a fundamental unit of information, classically represented as a 0 or 1 in your digital computer. Each classical bit is physically realized through a macroscopic physical system, such as the magnetization on a hard disk or the charge on a capacitor. A document, for example, comprised of n-characters stored on the hard drive of a typical computer is accordingly described by a string of 8n zeros and ones. Herein lies a key difference between your classical computer and a quantum computer. Where a classical computer obeys the well understood laws of classical physics, a quantum computer is a device that harnesses physical phenomenon unique to quantum mechanics (especially quantum interference) to realize a fundamentally new mode of information processing. In a quantum computer, the fundamental unit of information (called a quantum bit or qubit), is not binary but rather more quaternary in nature. This qubit property arises as a direct consequence of its adherence to the laws of quantum mechanics which differ radically from the laws of classical physics. A qubit can exist not only in a state corresponding to the logical state 0 or 1 as in a classical bit, but also in states corresponding to a blend or superposition of these classical states. In other words, a qubit can exist as a zero, a one, or simultaneously as both 0 and 1, with a numerical coefficient representing the probability for each state. This may seem counterintuitive because everyday phenomenon is governed by classical physics, not quantum mechanics -- which takes over at the atomic level. To explain what makes quantum computers so different from their classical counterparts we begin by having a closer look at a basic chunk of information namely one bit. From a physical point of view a bit is a physical system which can be prepared in one of the two different states representing two logical values --- no or yes, false or true, or simply 0 or 1 For example, in digital computers, the voltage between the plates in a capacitor represents a bit of information: a charged capacitor denotes bit value 1 and an uncharged capacitor bit value 0. One bit of information can be also encoded using two different polarizations of light or two different electronic states of an atom. However, if we choose an atom as a physical bit then quantum mechanics tells us that apart from the two distinct electronic states the atom can be also prepared in a coherent superposition of the two states. This means that the atom is both in state 0 and state 1. Experimental validation

In an experiment like that in figure a, where a photon is fired at a half-silvered mirror, it can be shown that the photon does not actually split by verifying that if one detector registers a signal, then no other detector does. With this piece of information, one might think that any given photon travels vertically or horizontally, randomly choosing between the two paths. However, quantum mechanics predicts that the photon actually travels both paths simultaneously, collapsing down to one path only upon measurement. This effect, known as single-particle interference, can be better illustrated in a slightly more elaborate experiment, outlined in figure b. Figure b depicts an interesting experiment that demonstrates the phenomenon of singleparticle interference. In this case, experiment shows that the photon always reaches detector A, never detector B! If a single photon travels vertically and strikes the mirror, then, by comparison to the experiment in figure a, there should be an equal probability that the photon will strike either detector A or detector B. The same goes for a photon traveling down the horizontal path. However, the actual result is drastically different. The only conceivable conclusion is therefore that the photon somehow traveled both paths simultaneously; creating interference at the point of intersection that destroyed the possibility of the signal reaching B. This is known as quantum interference and results from the superposition of the possible photon states, or potential paths. So although only a single photon is emitted, it appears as though an identical photon exists and travels the 'path not taken,' only detectable by the interference it causes with the original photon when their paths come together again. If, for example, either of the paths are blocked with an absorbing screen, then detector B begins registering hits again just as in the first experiment! This unique characteristic, among others, makes the current research in quantum computing not merely a continuation of today's idea of a computer, but rather an entirely new branch of thought. And it is because quantum computers harness these special characteristics that give them the potential to be incredibly powerful computational devices.

The most exciting really new feature of quantum computing is quantum parallelism. A quantum system is in general not in one "classical state", but in a "quantum state" consisting (crudely speaking) of a superposition of many classical or classical-like states. This superposition is not just a figure of speech, covering up our ignorance of which classical-like state it's "really" in. If that was all the superposition meant, you could drop all but one of the classical-like states (maybe only later, after you deduced retrospectively which one was "the right one") and still get the time evolution right. But actually you

need the whole superposition to get the time evolution right. The system really is in some sense in all the classical-like states at once! If the superposition can be protected from unwanted entanglement with its environment (known as decoherence), a quantum computer can output results depending on details of all its classical-like states. This is quantum parallelism - parallelism on a serial machine. And if that wasn't enough, machines that would already, in architectural terms, qualify as parallel can benefit from quantum parallelism too - at which point the mind begins to seriously boggle! The Potential and Power of Quantum Computing Let us start by describing the problem at hand: factoring a number N into its prime factors (e.g., the number 51688 may be decomposed as ). A convenient way to quantify how quickly a particular algorithm may solve a problem is to ask how the number of steps to complete the algorithm scales with the size of the ``input'' the algorithm is fed. For the factoring problem, this input is just the number N we wish to factor; hence the length of the input is . (The base of the logarithm is determined by our numbering system. Thus a base of 2 gives the length in binary; a base of 10 in decimal.) `Reasonable' algorithms are ones which scale as some small-degree polynomial in the input size (with a degree of perhaps 2 or 3). On conventional computers the best known factoring algorithm runs in steps. This algorithm, therefore, scales exponentially with the input size . For instance, in 1994 a 129 digit number (known as RSA129) was successfully factored using this algorithm on approximately 1600 workstations scattered around the world; the entire factorization took eight months . Using this to estimate the prefactor of the above exponential scaling, we find that it would take roughly 800,000 years to factor a 250 digit number with the same computer power; similarly, a 1000 digit number would require years (significantly lon ger than the age of the universe). The difficulty of factoring large numbers is crucial for publickey cryptosystems, such as ones used by banks. There, such codes rely on the difficulty of factoring numbers with around 250 digits. Recently, an algorithm was developed for factoring numbers on a quantum computer which runs in steps where is small. This is roughly quadratic in the input size, so factoring a 1000 digit number with such an algorithm would require only a few million steps. The implication is that public key cryptosystems based on factoring may be breakable. To give you an idea of how this exponential improvement might be possible, we review an elementary quantum mechanical experiment that demonstrates where such power may lie hidden. The two-slit experiment is prototypic for observing quantum mechanical behavior: A source emits photons, electrons or other particles that arrive at a pair of slits.

These particles undergo unitary evolution and finally measurement. We see an interference pattern, with both slits open, which wholly vanishes if either slit is covered. In some sense, the particles pass through both slits in parallel. If such unitary evolution were to represent a calculation (or an operation within a calculation) then the quantum system would be performing computations in parallel. Quantum parallelism comes for free. The output of this system would be given by the constructive interference among the parallel computations. In a traditional computer, information is encoded in a series of bits, and these bits are manipulated via Boolean logic gates arranged in succession to produce an end result. Similarly, a quantum computer manipulates qubits by executing a series of quantum gates, each a unitary transformation acting on a single qubit or pair of qubits. In applying these gates in succession, a quantum computer can perform a complicated unitary transformation to a set of qubits in some initial state. The qubits can then be measured, with this measurement serving as the final computational result. This similarity in calculation between a classical and quantum computer affords that in theory, a classical computer can accurately simulate a quantum computer. In other words, a classical computer would be able to do anything a quantum computer can. So why bother with quantum computers? Although a classical computer can theoretically simulate a quantum computer, it is incredibly inefficient, so much so that a classical computer is effectively incapable of performing many tasks that a quantum computer could perform with ease. The simulation of a quantum computer on a classical one is a computationally hard problem because the correlations among quantum bits are qualitatively different from correlations among classical bits, as first explained by John Bell. Take for example a system of only a few hundred qubits, this exists in a Hilbert space of dimension ~1090 that in simulation would require a classical computer to work with exponentially large matrices (to perform calculations on each individual state, which is also represented as a matrix), meaning it would take an exponentially longer time than even a primitive quantum computer. Richard Feynman was among the first to recognize the potential in quantum superposition for solving such problems much faster. For example, a system of 500 qubits, which is impossible to simulate classically, represents a quantum superposition of as many as 2500 states. Each state would be classically equivalent to a single list of 500 1's and 0's. Any quantum operation on that system --a particular pulse of radio waves, for instance, whose action might be to execute a controlled - NOT operation on the 100th and 101st qubits-- would simultaneously operate on all 2500 states. Hence with one fell swoop, one tick of the computer clock, a quantum operation could compute not 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 to a single answer, a single list of 500 1's and 0's, as dictated by the measurement axiom of quantum mechanics. The reason this is an exciting result is because this answer, derived from the massive quantum parallelism achieved through superposition, is the equivalent of performing the same operation on a classical super computer with ~10150 separate processors (which is of course impossible)!! Early investigators in this field were naturally excited by the potential of such immense computing power, and soon after realizing its potential, the hunt was on to find something

interesting for a quantum computer to do. Peter Shor, a research and computer scientist at AT&T's Bell Laboratories in New Jersey, provided such an application by devising the first quantum computer algorithm. Shor's algorithm harnesses the power of quantum superposition to rapidly factor very large numbers (on the order ~10200 digits and greater) in a matter of seconds. The premier application of a quantum computer capable of implementing this algorithm lies in the field of encryption, where one common (and best) encryption code, known as RSA, relies heavily on the difficulty of factoring very large composite numbers into their primes. A computer which can do this easily is naturally of great interest to numerous government agencies that use RSA -- previously considered to be "uncrackable" -- and anyone interested in electronic and financial privacy. Encryption, however, is only one application of a quantum computer. In addition, Shor has put together a toolbox of mathematical operations that can only be performed on a quantum computer, many of which he used in his factorization algorithm. Furthermore, Feynman asserted that a quantum computer could function as a kind of simulator for quantum physics, potentially opening the doors to many discoveries in the field. Currently the power and capability of a quantum computer is primarily theoretical speculation; the advent of the first fully functional quantum computer will undoubtedly bring many new and exciting applications. Now we push the idea of superposition of numbers a bit further. 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 i.e. the register can be in only one out of eight possible configurations such as 000, 001, 010, ... 111. A quantum register composed of three qubits can store in a given moment of time all eight numbers in a quantum superposition (Fig. 2). This 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 i.e. 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 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. Let us have a closer look at this issue. 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 (e.g. multiplication) other are very slow (e.g. factorisation, playing chess). Consider, for example, the following factorisation 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. Solving the reverse problem 127 x 129 =? , 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 factorisation. 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 three digit numbers to two thirty digits numbers. By contrast, factoring a thirty digit number using the simplest trial divison 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 factorised as a mathematical challenge, i.e. 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. Skipping details of the computational complexity we only mention that 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 factorizations is not in P and that is why it is a hard problem. Hard does not mean ``impossible to solve'' or ``noncomputable'' --- factorizations 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. It worth pointing out that computer scientist has carefully constructed the definitions of efficient and inefficient algorithms trying to avoid any reference to a physical hardware. According to the above definition factorizations is a hard problem for any classical computer regardless its make and the clock-speed. Have a look at Fig.3 and compare a modern computer with its ancestor of the nineteenth century, the Babbage differential engine. The technological gap is obvious and yet the Babbage engine can perform the same computations as the modern digital computer. Moreover, factoring is equally difficult both for the Babbage engine and top-of-the-line connection machine; the execution time grows exponentially with the size of the number in both cases. Thus 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 factorizations, into easy ones! Brief history The idea of a computational device based on quantum mechanics was first explored in the 1970's and early 1980's by physicists and computer scientists such as Charles H. Bennett of the IBM Thomas J. Watson Research Center, Paul A. Benioff of Argonne National Laboratory in Illinois, David Deutsch of the University of Oxford, and the late Richard P. Feynman of the California Institute of Technology (Caltech). The idea emerged when

scientists were pondering the fundamental limits of computation. They understood that if technology continued to abide by Moore's Law, then the continually shrinking size of circuitry packed onto silicon chips would eventually reach a point where individual elements would be no larger than a few atoms. Here a problem arose because at the atomic scale the physical laws that govern the behavior and properties of the circuit are inherently quantum mechanical in nature, not classical. This then raised the question of whether a new kind of computer could be devised based on the principles of quantum physics. Feynman was among the first to attempt to provide an answer to this question by producing an abstract model in 1982 that showed how a quantum system could be used to do computations. He also explained how such a machine would be able to act as a simulator for quantum physics. In other words, a physicist would have the ability to carry out experiments in quantum physics inside a quantum mechanical computer. Later, in 1985, Deutsch realized that Feynman's assertion could eventually lead to a general purpose quantum computer and published a crucial theoretical paper showing that any physical process, in principle, could be modeled perfectly by a quantum computer. Thus, a quantum computer would have capabilities far beyond those of any traditional classical computer. After Deutsch published this paper, the search began to find interesting applications for such a machine. Unfortunately, all that could be found were a few rather contrived mathematical problems, until Shor circulated in 1994 a preprint of a paper in which he set out a method for using quantum computers to crack an important problem in number theory, namely factorization. He showed how an ensemble of mathematical operations, designed specifically for a quantum computer, could be organized to enable a such a machine to factor huge numbers extremely rapidly, much faster than is possible on conventional computers. With this breakthrough, quantum computing transformed from a mere academic curiosity directly into a national and world interest. 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 still remain that prevent us from "just building one," or more precisely, building a quantum computer that can rival today's modern digital computer. Among these difficulties, error correction, decoherence, and hardware architecture are probably the most formidable. 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 led by Raymond Laflamme 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. 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 qubit's 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. Currently, research in quantum error correction continues with groups at Caltech (Preskill, Kimble), Microsoft, Los Alamos, and elsewhere. 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. Seth Lloyd of MIT is currently a prominent researcher in quantum hardware. 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.

Future Outlook At present, quantum computers and quantum information technology remains in its pioneering stage. At this very moment obstacles are being surmounted that will provide the knowledge needed to thrust quantum computers up to their rightful position as the fastest computational machines in existence. Error correction has made promising progress to date, nearing a point now where we may have the tools required to build a computer robust enough to adequately withstand the effects of decoherence. Quantum hardware, on the other hand, remains an emerging field, but the work done thus far suggests that it will only be a matter time before we have devices large enough to test Shor's and other quantum algorithms. Thereby, quantum computers will emerge as the superior computational devices at the very least, and perhaps one day make today's modern computer obsolete. Quantum computation has its origins in highly specialized fields of theoretical physics, but its future undoubtedly lies in the profound effect it will have on the lives of all mankind.