In a sense, theoretical computer science is uniquely qualified to study quantum computing. After all, Alan Turing and the other founders of theoretical computer science studied formal computation long before engineers actually produced a real-life computer. At present, large-scale quantum computers are not a reality yet. Nevertheless, the theoretical analysis of quantum computability and complexity is well on its way.

In Section 8.1, we start with a quick review of some of the basics of deterministic and nondeterministic Turing machines and the complexity classes that they engender. However, we shall discuss them in a way that is easily generalizable for our purposes. Section 8.2 moves on to probabilistic Turing machines and their zoo of complexity classes. Our main objective is found in Section 8.3, where we meet quantum Turing machines and their complexity classes. We shall also state some basic theorems and ideas about quantum computation.

DETERMINISTIC AND NONDETERMINISTIC COMPUTATIONS

Theoretical computer science deals with the question, “What is computable?” We must immediately qualify the question: “computable according to which model of computation?” It turns out that if we omit the question of efficiency, all sufficiently complicated formal models of computation can simulate each other. However, in order to fix our ideas and notation, we have to stick with one and work with it.