The multidisciplinary field of quantum computing strives to exploit some of the uncanny aspects of quantum mechanics to expand our computational horizons. Quantum Computing for Computer Scientists takes readers on a tour of this fascinating area of cutting-edge research. Written in an accessible yet rigorous fashion, this book employs ideas and techniques familiar to every student of computer science. The reader is not expected to have any advanced mathematics or physics background. After presenting the necessary prerequisites, the material is organized to look at different aspects of quantum computing from the specific standpoint of computer science. There are chapters on computer architecture, algorithms, programming languages, theoretical computer science, cryptography, information theory, and hardware. The text has step-by-step examples, more than two hundred exercises with solutions, and programming drills that bring the ideas of quantum computing alive for today's computer science students and researchers.

Hidden variables are extra components added to try to banish counterintuitive features of quantum mechanics. We start with a quantum-mechanical model and describe various properties that can be asked of a hidden-variable model. We present six such properties and a Venn diagram of how they are related. With two existence theorems and three no-go theorems (EPR, Bell, and Kochen-Specker), we show which properties of empirically equivalent hidden-variable models are possible and which are not. Formally, our treatment relies only on classical probability models, and physical phenomena are used only to motivate which models to choose. IntroductionBegun by von Neumann [35, 1932], the hidden-variable program in quantum mechanics (QM) adds extra ("hidden") ingredients in order to try to banish some of the counterintuitive features of QM. These features are: (i) the probabilistic nature of quantum behavior, (ii) the possibility of so-called non-local effects between widely separated particles, and (iii) the idea of an intrinsic dependence between the observer of a QM system and the properties of the system itself.Hidden-variable theories aim to remove these strange aspects of QM by building more "complete" models (in the terminology of Einstein-Podolsky-Rosen [16, 1935]). The completed models should agree with the predictions of QM, but exhibit one or more of the desired properties of: (i) determinism, (ii) locality, and (iii) independence.Can such models actually be built? The famous "no-go" theorems of QM show that there are severe limitations to what can be done. But it is also true that certain combinations of properties are possible.

We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are "essentially" the same program. The set of all equivalence classes is the category of all algorithms. In order to explore these ideas, the set of primitive recursive functions is considered. Each primitive recursive function can be described by many labeled binary trees that show how the function is built up. Each tree is like a program that shows how to compute a function. We give relations that say when two such trees are "essentially" the same. An equivalence class of such trees will be called an algorithm. Universal properties of the category of all algorithms are given.

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory.

Many different programs are the implementation of the same algorithm. The collection of programs can be partitioned into different classes corresponding to the algorithms they implement. This makes the collection of algorithms a quotient of the collection of programs. Similarly, there are many different algorithms that implement the same computable function. The collection of algorithms can be partitioned into different classes corresponding to what computable function they implement. This makes the collection of computable functions into a quotient of the collection of algorithms. Algorithms are intermediate between programs and functions: Programs Algorithms Functions. Galois theory investigates the way that a subobject sits inside an object. We investigate how a quotient object sits inside an object. By looking at the Galois group of programs, we study the intermediate types of algorithms possible and the types of structures these algorithms can have.

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