Quantum Turing Machines: Local Transition, Preparation, Measurement, and Halting

The Church-Turing thesis 2 states that to be computable is to be computable by a Turing machine and the modern discipline in computational complexity theory states that to be efficiently computable is to be computable by a Turing machine within polynomial steps in the length of the input data. However, Feynman pointed out that a Turing machine cannot simulate a quantum mechanical process efficiently and suggested that a computing machine based on quantum mechanics might be more powerful than Turing machines. Deutsch introduced quantum Turing machines and quantum circuits for establishing the notion of quantum algorithm exploiting “quantum parallelism”. A different approach to quantum Turing machines was investigated earlier by Benioff. Bernstein and Vazirani instituted quantum complexity theory based on quantum Turing machines and showed constructions of universal quantum Turing machines. Yao showed that a computation by a quantum Turing machine can be simulated efficiently by a quantum circuit. Deutsch’s idea of quantum parallelism was realized strikingly by Shor, who found efficient quantum algorithms for the factoring problem and the discrete logarithm problem, for which no efficient algorithms have been found for classical computing machines. The purpose of this paper is to discuss foundations of quantum Turing machines and to propose a computational protocol for quantum Turing machines. A precise formulation of quantum Turing machines is given along with Deutsch’s formulation and the computational protocol is discussed for the preparation and the measurement of quantum Turing machines. The characterization of the transition functions of quantum Turing machines is also discussed. Deutsch required that the transition function should be determined