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Geological Fluid Dynamics: Sub-Surface Flow and Reactions, by O.M. Phillips

computation, treads the border between physics and mathematics. There is also obvious interest in the relationship to quantum mechanics. The work may be classified as (a) the concept of computability, (b) the efficiency of computation and (c) the relation of quantum mechanics to the computation process. The first collection of topics (the concept of computation) follows a linear sequence: strings–problems– algorithms–solution–computability. The second class (efficiency of computation) includes aspects of computational difficulty and the investigation of a ‘best’ algorithm for problems or, alternatively, an intrinsic difficulty to every problem. Onemajor aspect behind the discussion of computability and difficulty is languagedependence. It is proven that computability is languageindependent but difficulty is not. Thus, the idea of probabilistic computability represents an approach aimed at reducing computability difficulty. The third part relates to quantum assisted computation. As the subject is vast, the author focuses on two critical cases. The first is the translation, into mathematics, of the quantummechanics utilised in computation. The second is the investigation of the contribution of quantum mechanics to efficiency.Here, the fundamental process is the construction of a language reflecting the efficiency of quantum mechanics when used in the laboratory. It is shown that quantum programming does not necessarily solve any otherwise noncomputable problem. The author’s handling of the topic leads to the formulation of a mathematical framework wherein conventional programming solutions can be placed alongside quantum-assisted ones in order to facilitate comparison. Ad hoc, there are indications that, for certain kind of problems, quantum-based algorithms may enhance the efficiency of computation. Because of the ongoing controversy over this latter statement, it is necessary to insert and investigate the lower bounds in the case of difficulty in computing. But such bounds do not exist even for the simplest case. Of vital interest is the theorem stated herein that if quantum mechanics yields any gain in computational efficiency, then that efficiency couldnot be more than exponential. There are also thoughts and discussion regarding the determination ofwhether or not quantumsystems contribute to computational efficiency. The overall sense of this book is one of a philosophical endeavour, a kind of philosophy of computation. This is clear, for instance, in the case of the author’s attempt at an abstract comparison between classical-mechanical-alike systems and quantum-based ones. The obvious advantage of quantum systems is their ability to manipulate numbers of an order m, when n physical systems are considered with m states. The key to this is the tensor product ability, which is feasible in quantum mechanics. The author, as an experienced scientist, explores selected areas throughout the general topology of the topic. The principal foundations in the development of this undertaking are (in sequential order): problem, computability, solution, machine, language, mathematics, functions, difficulty, probabilistic approach, quantum mechanics and quantum mechanics computing. There is also a transposition of the computability subject from the machine-dependent solution to the language dependent, and this is crucial. Of important value in this essay is (a) the Grover construction, which elucidates the advantage of quantum computing regarding computability and (b) the Turing machine of course. Finally the author focuses on the infinite number of problems as well as on the existence of noncomputable problems. There is also no definite relationship in the book to specific hardware elements and requirements. The book is particularly interesting for those engaged in theoretical pursuits involving computation, especially those particular to physics, specifically quantum physics. While the book is well-structured and developed in regard to theoretical computing, it, nevertheless – however much it may be expected – has not essential reference to hardware and applied software. Finally, it is directed at scientists and students (graduates and advanced undergraduates). Anyone in informatics would also benefit greatly from a theoretical/analytical tour of the field of computation, uncovering thus thoughts, views, problems, positions, etc. that had not been previously considered.