Simulations of Quantum Turing Machines by Quantum Multi-Stack Machines

TL;DR

This paper introduces quantum multi-stack and multi-counter machines with extended counting capabilities and demonstrates their ability to efficiently simulate quantum Turing machines, expanding the understanding of quantum computational models.

Contribution

The paper establishes the models of quantum multi-stack and multi-counter machines with counts beyond ±1 and proves their efficiency in simulating quantum Turing machines.

Findings

Quantum multi-counter machines with counts up to ±n can simulate quantum Turing machines.

Quantum multi-stack machines are formally defined and analyzed.

Simulation efficiency of quantum Turing machines by these models is demonstrated.

Abstract

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 multi-stack 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 {\it et al.} [10], in which the machines count with $0,\pm 1$ 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 multi-counter machines allowed to count with $0,\pm 1,\pm 2,...,\pm n$ for some $n>1$ can efficiently simulate quantum Turing machines. Therefore, our mail goals are to establish quantum multi-stack machines and quantum multi-counter machines with counts $0,\pm 1,\pm 2,...,\pm n$ and $n>1$, and particularly to simulate quantum Turing machines by these quantum computing devices.