Information Processing beyond Quantum Computation∗ Apoorva Patel Centre for High Energy Physics and Supercomputer Education and Research Centre Indian Institute of Science, Bangalore-560012, India E-mail: [email protected] Abstract– Recent developments in quantum computation have Instructions made it clear that there is a lot more to computation than the con- ventional Boolean algebra. Is quantum computation the most ❄ general framework for processing information? Having gath- Physical Input ✲ ✲ Output ered the courage to go beyond the traditional definitions, we are Device now in a position to answer: Certainly not. The meaning of a mes- sage being “a collection of building blocks” can be explored in a ✻ variety of situations. A generalised computational framework is proposed based on group theory, and it is illustrated with well- Oracles/Look-up Tables known physical examples. A systematic information theoretical Fig. 1. Schematic representation of a computer. approach is yet to be developed in many of these situations. Some Every computation may not use oracles or look-up tables. directions for future development are pointed out. drawn parallel to a given line and passing through a point out- side the given line. Euclid considered one parallel line to be I. MOTIVATION the self-evident answer, but could not prove that. So he in- cluded this property as a postulate in his theory, even though The silicon transistor was invented about half a century ago. it was quite different from the first four which defined basic Since then the semiconductor technology has grown at a rapid components of geometry. This state of affairs no doubt trou- pace to pervade almost all aspects of our lives. This growth bled mathematicians. For many centuries, they tried to deduce has been so explosive—doubling the number of transistors on the fifth postulate from the other four, and failed. Only after a chip every 18-24 months according to Moore’s law—that two thousand years, they found the courage to discard this many choices made in constructing the theoretical framework fifth postulate, which resulted in development of curved space of computer science (see for example, [1]) were almost forgot- geometry. We now know that a surface of positive curvature ten. Computer architecture became essentially synonymous allows no parallel lines, while a surface of negative curvature with digital electronic circuits implementing Boolean opera- allows an infinite number of parallel lines (if two parallel lines tions, pushing aside other competing models. Developments can be drawn passing through a point, there will be an infinite in quantum computation during the past decade have led us number of parallel lines in between). to question this attitude, and brought in focus the fact that When the concepts of a subject are generalised, it is quite there is much more to information theory than just Boolean common that definitions are extended, new terminology is in- logic. The concept of what is computable and what is not vented, and meanings of some old words change. (For exam- has not changed, but the criteria determining how efficiently ple, a line in a curvedspace has to be interpretedas a geodesic, a computational task can be implemented have been altered. arXiv:quant-ph/0306158v2 13 Dec 2004 i.e. the shortest curve connecting two points.) Such a situa- The reason behind this change is that some of the implicit as- tion can create unnecessary confusion, while developing new sumptions of theoretical computer science are too restrictive, methodology. To guard against it, words used in technical when compared to physically realisable models. Computa- contexts should be interpreted only in technical terms, and not tional power of a framework can be enhanced by discarding confused with their common language meanings. I begin by such unnecessary assumptions. My aim in this article is to describing my interpretation of some of the technical words describe possible directions of expansion for generalised in- that appear in information theory. formation theory, based on many physical examples we en- counter in the world around us. To obtain a perspective of how profoundlya subject can ex- II. TECHNICAL CONCEPTS pand up on modification of its fundamental postulates, it is worthwhile to look at what happened in geometry. Euclid’s formulation of planar geometry had five postulates. The fifth Information theory deals with two broad areas, communi- of these postulates concerned the number of lines that can be cation and computation. Communication is quite simple—the receiver gets whatever is sent by the sender. There is no pro- cessing in the middle, unless there is some unexpected noise. Computation is more complex—the input is intentionally ma- ∗Keynote talk presented at the First World Congress on Lateral Computing nipulated by an external agency to produce an output. (A (WCLC 2004), Bangalore, December 2004, quant-ph/0306158. schematic representation of a computer is shown in Fig. 1.) 2 The words data, information and knowledge often appear in the word and the object is established in the baby’s mind, we discussions of communicationand computation. I assign them can reduce the physical dependence—point at the book from specific meanings as explained below. afar, show a picture of it, and ultimately just utter or write the Data list physical properties of a system. They describe a word.) In case of the most primitive (or low level) messages, particular realisation of the physical system, amongst its many there is no luxury of abstract languages—the only language possible states. Data are often obtained by experimental ob- that exists is the one labeled by physical properties. In such servations of the system, and generally provide the starting cases, the physical objects that carry the message have to con- point of a computational process. I stress that data are always vey the information as well as its interpretation to the receiver. firmly rooted in physical characteristics, and should not be Once again, an optimal language can only be designed if the separated from them. available physical means are known. Information is the abstract mathematical property obtained To summarise, data is not information and information is by detaching all the physical characteristics from data. It just not knowledge. We have instead, becomes a measure of the number of possible states of the Information = Data - Physical Realisation , system [2]. This mathematical abstraction proves to be very Knowledge = Information + Interpretation . useful, because in dealing with information, at no stage one Abstract information theory does not tell us what physical re- has to worry about where the information came from or what alisation would be appropriate for a particular message, nor it means. The physical realisation of information may change does it tell us the best way of implementing a computational according to the convenience of the task to be carried out. task. To make such choices, we must look at the physical re- (For example, our electronic computers compute using elec- sources available, i.e. analyse the type of information and not trical signals, but store the results on the disk using magnetic the amount of information. signals; the former realisation is suitable for fast processing, The number of fundamental physical interactions is rather while the latter is suitable for long term storage.) Abstract small, and that limits the possible physical realisations of a information theory also allows manipulation of information computer. A variety of computational schemes can still be without going into nitty-gritty of its meaning, e.g. compress created, however, by combining the fundamental ingredients data, quantify error rate, devise codes, and so on. in different ways. To get an idea about the multitude of phys- While extraction of abstract information from data allows ical resources that can be used to process information, it is in- one to formulate precise mathematical rules for its systematic structive to look at biological systems. Over billions of years, analysis, the abstraction also brings in a limitation. Though evolution has had plenty of time (which we do not and can- the manipulations of information can be defined as mathe- not have) for experimentation with a wide range of physical matical algorithms, they have to be implemented by physical systems. Let us look at an example. devices. In order to manipulate information, one must map information to physical properties, and the types of manip- ulations that can be carried out are limited by the types of physical devices available. (For example, we use various pro- III. AN EXAMPLE FROM BIOLOGY gramming languages to implement mathematical algorithms on a computer. On the other hand, the electronic computer Consider the following biological communication system hardware responds only to voltages and currents. So a whole devoid of human involvement, to get a feeling for the wide ap- hierarchy of translation machinery is constructed, involving plicability of information theory. A plant attracts an insect to compilers and operating systems, to convert the algorithms to its flower. How does this take place? The flower releases cer- binary machine codes and then map them on to off/on states of tain fragrant molecules which are received by the insect. How silicon transistors.) An important consequence of this physi- does the insect decide which molecule is fragrant and which cal dependence is that the efficiency of a computational task is not? That information is encoded in the 3-dimensional cannot be determined solely by its mathematical algorithm— atomic structure of the molecule, which determines how it the efficiency depends on the algorithm as well as on the prop- will bind to the smell-receptors of the insect. How does the erties of the physical device that implements it. plant know which molecule to release, and the insect know The role of physical properties is also inevitable in adding which molecule to look for? That information resides in their a sense of purpose to information, and converting it to knowl- genomes, which have evolved together for millions of years edge.