Quantum Computing Since Democritus, Scott Aaronson, Cambridge U. Press, 2013. $39.99 paper (370 pp.). ISBN 978-0-521-19956-8
Scott Aaronson’s Quantum Computing Since Democritus is lively, casual, and clearly informed by the author’s own important work. In that respect it resembles George Gamow’s One Two Three … Infinity: Facts and Speculations of Science (Viking Press, 1947). Aaronson’s book might similarly be a revelation to potential scientists. But unlike Gamow’s classic, this book touches on profound issues, subtle questions, and debates that have not been—perhaps can’t be—resolved. They range from the meaning of the formal statement of the Zermelo–Fraenkel axioms for set theory, to conflicting notions of the interpretation of quantum mechanics, all the way to the nature of consciousness. In short, Quantum Computing Since Democritus is intended to be popular, but not that popular.
The book consists of 22 short chapters, each covering one or two related topics. Later chapters require some understanding of the early material and include occasional exercises that run the gamut from merely amusing to aspects of serious research problems. For those who know the subjects covered, the book will be entertaining; for those who don’t, it will be a good introduction to the main ideas of quantum computing and to some of the related controversies.
The early chapters are basic and instructional; the later ones move quickly and are more advanced, even philosophical. Several things are done extremely well. One example is the treatment of hidden-variable theories in chapter 12; I was surprised to see them presented so briefly and yet clearly.
Despite its better features, this book covers too much territory to be used as a textbook. In my view, the author introduces many topics because he needs to develop a context in which his arguments about the nature of the physical universe can be clearly stated. I think his aim is to build on John Wheeler’s ideas and argue that the definition and meanings of “information” and “computational complexity” are as fundamental to comprehending and explaining the universe as are the foundations of physics and mathematics.
I have only one serious reservation, which has to do with the author’s claim that mathematicians like complication because it makes things more interesting. That point of view, I think, leads to some difficulties. One such difficulty is born in the early chapters, where Aaronson introduces the notion of superposition of states and the axioms of set theory—including the axiom of choice (AC), its special role, and the puzzles it raises. However, he neglects to mention the important connection between AC and issues raised by attempts to have a completely satisfying theory of “measure” or “probability.” In particular, paradoxes raised by AC are one of the reasons that “perfectly random” can’t be defined in a way consistent with a rigorous definition of probability via measure theory.
In his discussion of probability, Aaronson rules out measure theory because all of his probability spaces are finite. However, I think that raises the question of what we actually mean when we say that the single qubit superposition, when observed, gives either zero or one with probability 1⁄2. What’s the precise meaning of “probability” here? Do we mean that making the observation yields a perfectly random choice of zero or one?
What, then, would we expect if we could make such an observation one trillion times, thus generating the beginning of a perfectly random sequence? Or, since we can’t really define “random sequence,” do we mean something like measure theory? I suspect that one is forced to take the definition of “random” to be “outcome of an observation, or measurement, of a single qubit whose state is a superposition of two basis states.” I have no objection to that definition, just as I have no objection to the second law of thermodynamics. But the notions of probability and random must be made explicit.
My objections should not be taken as negative. In fact, they are evidence that Quantum Computing Since Democritus is stimulating. It should prove valuable to anyone interested in computational complexity, quantum mechanics, and the theory of quantum computing.
Francis Sullivan ([email protected]) is the director of the Center for Computing Sciences at the Institute for Defense Analyses in Bowie, Maryland. His research interests are algorithm design and the theory and application of Monte Carlo techniques to a wide variety of scientific problems.
