Abstract Quantum Automata as Formal Models of Quantum Information Processing Systems

Abstract

Nowadays, quantum computation is considered as a perspective way to overcome the computational complexity barrier. Development of a quantum programming technology requires to build theoretical background of quantum computing similarly to the classical computability theory and the classical computational complexity theory. The challenge to develop such theoretical background was posed by Yu.I. Manin. An attempt to build a mathematically rigorous model for quantum information processing systems in compliance with the concept of Yu.I. Manin is presented in the chapter. The attempt carries out by identifying elementary constituents of quantum computational processes. They are called quantum actions and their properties are studied in the chapter. In particular, the equivalence criterion of quantum actions in terms of their generating operators has been found; the special class of quantum actions has been characterised in terms of generating operators too. This class is formed by quantum actions leading to the collapse of quantum states. Further in the chapter, the mathematical model of quantum information processing systems. It is defined as an ensemble of interacting quantum actions on the common memory. The term ”abstract quantum automata” is introduced to denote such model. At the end of the chapter models of some important quantum information processing systems are presented.