量子时序机等价性的多项式时间算法
A Polynomial-Time Algorithm for the Equivalence between Quantum Sequential Machines
摘要：
量子顺序机（QSM）是随机顺序机（SSM）的量子版本。最近，我们证明了分别具有n_1和n_2态的两个QSM M_1和M_2是等价的当且仅当它们是（n_1+n_2）^2-等价的（Theoretical Computer Science 358（2006）65-74）。然而，使用该结果来检查等效性可能需要指数期望时间。在本文中，我们考虑的时间复杂性，决定之间的等价性QSM和相关问题。主要结果如下：（1）提出了一个判定QSM等价的多项式时间算法，当两个QSM不等价时，该算法将产生一个长度不超过（n_1+n_2）^2的输入输出对。(2)利用Moore和Crutchfield的方法（Theoretical Computer Science 237（2000）275-306），我们将QSM之间等价的界从（n_1+n_2）^2提高到n_1^2+n_2^2-1. (3)我们给出了两个分别具有n_1和n_2态的MO-1 QFA等价的充要条件是它们是（n_1+n_2）^2-等价的，并进一步得到了判定两个MO-1 QFA等价的多项式时间算法. (4)我们提供了一个反例表明，Koshiba的方法来解决决定MM-1 QFA之间的等价性的问题可能是无效的，因此这个问题是开放的。
英文摘要：
Quantum sequential machines (QSMs) are a quantum version of stochastic sequential machines (SSMs). Recently, we showed that two QSMs M_1 and M_2 with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent (Theoretical Computer Science 358 (2006) 65-74). However, using this result to check the equivalence likely needs exponential expected time. In this paper, we consider the time complexity of deciding the equivalence between QSMs and related problems. The main results are as follows: (1) We present a polynomial-time algorithm for deciding the equivalence between QSMs, and, if two QSMs are not equivalent, this algorithm will produce an input-output pair with length not more than (n_1+n_2)^2. (2) We improve the bound for the equivalence between QSMs from (n_1+n_2)^2 to n_1^2+n_2^2-1, by employing Moore and Crutchfield's method (Theoretical Computer Science 237 (2000) 275-306). (3) We give that two MO-1QFAs with n_1 and n_2 states, respectively, are equivalent iff they are (n_1+n_2)^2--equivalent, and further obtain a polynomial-time algorithm for deciding the equivalence between two MO-1QFAs. (4) We provide a counterexample showing that Koshiba's method to solve the problem of deciding the equivalence between MM-1QFAs may be not valid, and thus the problem is left open again.
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