In this chapter, the authors discuss a part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. There are, however, a number of questions that the people would like to discuss in more detail such as the question of imperfections. It is analogous to the use of transistors in conventional machines where the people don't properly use all the analog continuum of the behavior of transistors, but just try to run them as saturated on or off digital devices so the logical analysis of the system behavior is easier. Of course the large system obeys quantum mechanics, but it is in interaction with the heat baths and other things that could make it effectively irreversible.
In this chapter, the authors discuss a part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. There are, however, a number of questions that the people would like to discuss in more detail such as the question of imperfections. It is analogous to the use of transistors in conventional machines where the people don't properly use all the analog continuum of the behavior of transistors, but just try to run them as saturated on or off digital devices so the logical analysis of the system behavior is easier. Of course the large system obeys quantum mechanics, but it is in interaction with the heat baths and other things that could make it effectively irreversible.
In this chapter, the authors discuss the part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. A more elaborate circuit is a full adder which takes a carry, c , and adds it to the two lines a and b and has an additional line, d, with a 0 input. Calculations can be done with reversible machines containing only reversible primitives. There are many sources of imperfections in the machine, but the first one people would like to consider is the possibility that the coefficients in the couplings, along the program line, are not exactly equal. The switches are represented by diagonal lines, and in boxes put the other matrices that operate on registers such as the NOT b .
A computer is imagined in which all the parts are simple two-state objects obeying quantum mechanical laws. A time-independent Hamiltonian for such a system is written and analyzed. The computation is done in logically reversible steps. It is used to study possible limitations on computers due to fundamental physical laws. The ideal machine can run ballistically, coasting through the calculation in a short time TB. If there are imperfections due to irregularities in construction or finite temperatures the forward moving calculation is often reversed or scattered. External work must be done to drag the computer forward. The free energy lost in doing a calculation is kT In2 per logical step divided by turn factors. One is the mean free number of steps per scattering, measuring how perfectly the machine is built. The other is the mean time (divided by TB) that the calculation actually takes, which measures how gently you urge it forward.
In the preceding chapter we established that a suitable set of quantum gates, complemented by quantum error correction, allows us to produce a desired multiqubit unitary transformation. This transformation is one of the three steps in a quantum computation; the others, of course, are the preparation of the qubits in their initial state and the measurement of them after the transformation has been implemented. A quantum computation is designed to solve a problem or class of problems. The power of quantum computers is that they can do this, at least for some problems, very much more efficiently and quickly than any conventional computer based on classical logic operations. If we can build a quantum computer then a number of important problems which are currently intractable will become solvable. The potential for greatly enhanced computational power is, in itself, reason enough to study quantum computers, but there is another. Moore’s law is the observation that the number of transistors on a chip doubles roughly every eighteen months. A simple corollary is that computer performance also doubles on the same timescale. Associated with this exponential improvement is a dramatic reduction in the size of individual components. If the pace is to be kept up then it is inevitable that quantum effects will become increasingly important and ultimately will limit the operation of the computer. In these circumstances it is sensible to consider the possibility of harnessing quantum effects to realize quantum information processors and computers. We start with a brief introduction to the theory of computer science, the principles of which underlie the operation of what we shall refer to as classical computers. These include all existing machines and any based on the manipulation of classical bits. The development of computer science owes much to Turing, who devised a simple but powerful model of a computing device: the Turing machine. It its most elementary form, this consists of four elements. (i) A tape for data storage, which acts as a memory. This tape has a sequence of spaces, each of which has on it one of a finite set of symbols. (ii) A processor, which controls the operations of the machine.
Digital computers are built out of circuits that have definite, discrete states: on or off, zero or one, high voltage or low voltage. Engineers go to great lengths to make sure these circuits never settle into some intermediate condition. Quantum-mechanical systems, as it happens, offer a guarantee of discreteness without any engineering effort at all. When you measure the spin orientation of an electron, for example, it is always either "up " or "down, " never in between. Likewise an atom gains or loses energy by making a "quantum jump" between specific energy states, without passing through intermediate energy levels. So why not build a digital computer out of quantum-mechanical devices, letting particle spins or the energy levels of atoms stand for binary units of information? One answer to this "Why not? " question is that you can't avoid building a quantum-mechanical computer even if you try. Since quantum mechanics appears to be a true theory of nature, it governs all physical systems, including the transistors and other components of the computer on your desk. All the same, quantum effects are seldom evident in electronic devices; components and circuits are designed so that the quantum states of many millions of electrons are averaged together, blurring their discreteness. In a quantum computer, the basic working parts would probably have to be individual electrons or atoms, and so another answer to the "Why not?" question is that building such a machine is simply beyond our skills. And
This chapter describes the possibility of simulating physics in the classical approximation, a thing which is usually described by local differential equations. But the physical world is quantum mechanical, and therefore the proper problem is the simulation of quantum physics. A computer which will give the same probabilities as the quantum system does. The present theory of physics allows space to go down into infinitesimal distances, wavelengths to get infinitely great, terms to be summed in infinite order, and so forth; and therefore, if this proposition is right, physical law is wrong. Quantum theory and quantizing is a very specific type of theory. The chapter talks about the possibility that there is to be an exact simulation, that the computer will do exactly the same as nature. There are interesting philosophical questions about reasoning, and relationship, observation, and measurement and so on, which computers have stimulated people to think about anew, with new types of thinking.
Q uantum mechanics has had an enormous technological and societal impact.To grasp this point, it is sufficient to cite the invention of the transistor, perhaps the most remarkable among the countless other applications of quantum mechanics.It is also easy to see the enormous impact of computers on everyday life.The importance of computers is such that it is appropriate to say that we are now living in the information age.This information revolution became possible thanks to the invention of the tran sistor, that is, thanks to the synergy between computer science and quantum physics.Today this synergy offers completely new opportunities and promises exciting advances in both funda mental science and technological application.We are referring here to the fact that quantum mechanics can be used to process and transmit information [1,2].Miniaturization provides us with an intuitive way of under standing why, in the near future, quantum laws will become important for computation.The electronics industry for com puters grows hand-in-hand with the decrease in size of integrated circuits.This miniaturization is necessary to increase computa tional power, that is, the number of floating-point operations per second (flops) a computer can perform.In the 1950's, electronic computers based on vacuum-tube technology were capable of performing approximately 103 floating-point operations per second, while nowadays there exist supercomputers whose power is greater than lOteraflops (1013 flops).As we have remarked, this enormous growth of computational power has been made possi ble owing to progress in m in ia tu riz a tio n , w hich m ay be quantified empirically in Moore's law.This law is the result of a remarkable observation made by Gordon Moore in 1965: the number of transistors on a single integrated-circuit chip doubles approximately every 18 24 months.This exponential growth has not yet saturated and Moore's law is still valid.At the present time the limit is approximately 108 transistors per chip and the typical size of circuit components is of the order of 100 nanometres.Extrapolating Moore's law, one would estimate that around the year 2020 we shall reach the atomic size for storing a single bit of inform atio n .At th a t p o in t, q u a n tu m effects will becom e unavoidably dominant.One should be aware that, besides quantum effects, other fac tors could bring Moore's law to an end.In the first place, there are economic considerations.Indeed, the cost of building fabri cation facilities to m an u factu re chips has also increased exponentially with time.Nevertheless, it is important to under stand the ultimate limitations set by quantum mechanics.Even though we might overcome economic barriers by means of tech nological breakthroughs, quantum physics sets fundamental limitations on the size of the circuit components.The first ques tion under debate is whether it would be more convenient to push References