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Perspectives in Computation

Computation is the process of applying a procedure or algorithm to the solution of a mathematical problem. Mathematicians and physicists have been occupied for many decades pondering which problems can be solved by which procedures, and, for those that can be solved, how this can most efficiently be done. In recent years, quantum mechanics has augmented our understanding of the process of computation and of its limitations.

Perspectives in Computation covers three broad topics: the computation process and its limitations, the search for computational efficiency, and the role of quantum mechanics in computation. The emphasis is theoretical; Robert Geroch asks what can be done, and what, in principle, are the limitations on what can be done? Geroch guides readers through these topics by combining general discussions of broader issues with precise mathematical formulations—as well as through examples of how computation works.

Requiring little technical knowledge of mathematics or physics, Perspectives in Computation will serve both advanced undergraduates and graduate students in mathematics and physics, as well as other scientists working in adjacent fields.

208 pages | 6 x 9 | © 2009

Chicago Lectures in Physics

Physical Sciences: Experimental and Applied Physics, Theoretical Physics

Reviews

"A short, beautiful set of seminar lecture notes for physics graduate students on the theory of computing with an emphasis on the flowering field of quantum computing. Perspectives in Computation is not an encyclopedic treatment; the book’s references to the literature are sparse. Rather, it is a carefully constructed single story line presented with outstanding clarity. It contains few equations but many carefully conceived logical arguments. . . . The book is an eccentric and rewarding tour de force."

William H. Press | Physics Today

Table of Contents

1          Introduction
2          Characters and Strings
3          Problems
4          Computability
5          Turing Machines
6          Noncomputable Problems
7          Noncomputable Numbers
8          Formal Mathematics
9          Difficulty Functions
10        Difficult Problems; Best Algorithms
11        A Language for Efficiency
12        Are There Better Languages?
13        Probabilistic Computing
14        Quantum Mechanics
15        Grover Construction
16        Grover Construction: Six Issues
            16.1     Initial State
            16.2     Final Observation on Hin
            16.3     Building the Operator W
            16.4     Building the Operator V
            16.5     Errors
            16.6     What Is the Problem?
17        Quantum-Assisted Computing
18        Quantum-Assisted Computability
19        Quantum-Assisted Difficulty Functions
20        Quantum-Assisted Efficiency I
21        Quantum-Assisted Efficiency II
22        Conclusion
            References
            Index

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