Book review on quantum computation and quantum information

In a space of less than ten years, a cadre of inspired researchers from theoretical and experimental physics, computer science, and mathematics have been incessantly at work building an entirely new body of knowledge about the information processing capabilities of quantum systems. The amount of wisdom accumulated in these short years already presses at the limits of what one mind can reasonably apprehend. But many of us have felt from the beginning that our subject possesses an underlying intellectual structure of great power and clarity, suÆcient to hold and display even the weighty mass of knowledge that we have accumulated, and to provide a framework for future developments. Nielsen and Chuang have tackled the job of drawing the blueprint of this structure. This in itself has been a Herculean task: the building is extensive, and most of it has never been mapped out before. Their plan has been rigorous and original. More than any of the previous attempts, this book has identi ed the essential foundations of quantum information theory with a clarity that has even, in a few cases, permitted the authors to obtain some original results and point toward new research directions. The plan of the book weaves together topics of physical theory and information theory. Nielsen and Chuang start with basic introductions to quantum theory and to the basic elements of theoretical computer science. For the physicist, the quantum mechanics will be elementary, but it is deceptively so: for example, we get a careful discussion here of the tensor product structure of Hilbert space, a topic that is crucial to quantum information theory, but is usually barely mentioned (and then forgotten and misunderstood) in conventional quantum theory texts. The book then moves on to discuss quantum circuits and quantum algorithms, physical realizations of quantum computers, decoherence and quantum error correction, and nally quantum entropies and quantum channel theory. This book is no mere compendium of standard results. In several instances, the authors delve very deeply into a problem, and come up with results that appear nowhere else in the literature. An example of this is their treatment of quantum gate universality. Most repertoires of quantum gates can be used to approximate, with arbitrarily high accuracy, any unitary transformation on a set of qubits. This result is known as the \Solovay-Kitaev theorem," in honor of the workers who reported (but did not entirely publish) results on this problem. The present treatment is the most complete, and certainly the most clear,