Simulations of Quantum Turing Machines by Quantum Multi-Stack Machines

As was well known, in classical computation, Turing machines, circuits, multi-stack machines, and multi-counter machines are equivalent, that is, they can simulate each other in polynomial time. In quantum computation, Yao [11] first proved that for any quantum Turing machines M , there exists quantum Boolean circuit (n, t)-simulating M , where n denotes the length of input strings, and t is the number of move steps before machine stopping. However, the simulations of quantum Turing machines by quantum multistack machines and quantum multi-counter machines have not been considered, and quantum multi-stack machines have not been established, either. Though quantum counter machines were dealt with by Kravtsev [6] and Yamasaki et al. [10], in which the machines count with 0,±1 only, we sense that it is difficult to simulate quantum Turing machines in terms of this fashion of quantum computing devices, and we therefore prove that the quantum multicounter machines allowed to count with 0,±1,±2, . . . ,±n for some n > 1 can efficiently simulate quantum Turing machines. So, our mail goals are to establish quantum multi-stack machines and quantum multicounter machines with counts 0,±1,±2, . . . ,±n and n > 1, and particularly to simulate quantum Turing machines by these quantum computing devices. The major technical contributions of this article are stated as follows: (i) We define quantum multi-stack machines (abbr. QMSMs) by generalizing a kind of quantum pushdown automata (abbr. QPDAs) from one-stack to multi-stack, and the wellformedness (abbr. W-F) conditions for characterizing the unitary evolution of the QMSMs are presented. (ii) By means of QMSMs we define quantum multi-counter machines (abbr. QMCMs) ∗Email-address: [email protected] (D. Qiu).