基于安全性检测的广义量子Loop程序终止验证

Abstract

To address the termination problem of the generalized quantum Loop program, in this paper, we first introduce two definitions of termination and almost termination. We show that their expressive power regarding the termination of a program is not sufficient. To compensate for this, we define two other definitions:the nth termination with a specified probability and terminability, and we obtain some conclusions by comparing these four definitions. Secondly, we introduce the definition of the quantum (regular) safety property, which is an important kind of linear-time property of quantum Markov chains. We show that the satisfiability probability of the quantum (regular) safety property is equivalent to the unreachability probability of the termination state in the product of the quantum Markov chains. Then, we discuss a checking technique for the quantum regular safety property. By applying this technique, we reduce the termination problem of the generalized quantum Loop program into the satisfiability problem of the quantum regular safety property, and we propose an algorithm to calculate the probability and the termination state of the nth termination and the terminability. To determine the termination state and the probability of a multi-qubit (single-qubit) quantum system, compound quantum system, and nested quantum system, we derive the probability and the termination state of their nth termination and their terminability, and we present expressions for calculating them. Finally, we prove the necessary and sufficient conditions for the finite-step termination and terminability of the generalized quantum Loop program.