Two quantum finite automata are equivalent if for all input string $\omega$ over the input alphabet the two automata accept $\omega$ with equal probability. In [Theoret. Comput. Sci. 410 (2009) 3006-3017], it was shown that a $k_1$-letter QFA $\mathcal{A}_1$ and a $k_2$-letter QFA $\mathcal{A}_2$ over $\Sigma=\{\sigma\}$, are equivalent if and only if they are $(n_1+n_2)^4+k-1$-equivalent where $n_i$ is the number of states of $\mathcal{A}_i$, $i=1,2$, and $k=\max\{k_1,k_2\}$. In this letter, we improve the above upper-bound to $(n_1^2+n_2^2-1)+k$. This also answers an open problem of Qiu et al. [Acta Informatica 48 (2011) 271-290]. Further, we show that, in the case of $\Sigma=\{\sigma_1,...,\sigma_t\}$ with $2\leq t<\infty$, there exists an integer $z$ such that $\mathcal{A}_1$ and $\mathcal{A}_2$ are equivalent if and only if they satisfy $z$-equivalent.

翻译：两个量子有限自动机等价，当且仅当对于输入字母表上的所有输入字符串$\omega$，两个自动机以相等概率接受$\omega$。在文献[Theoret. Comput. Sci. 410 (2009) 3006-3017]中，证明了对于$\Sigma=\{\sigma\}$上的$k_1$-字母QFA $\mathcal{A}_1$和$k_2$-字母QFA $\mathcal{A}_2$，它们等价当且仅当它们是$(n_1+n_2)^4+k-1$等价的，其中$n_i$是$\mathcal{A}_i$的状态数，$i=1,2$，且$k=\max\{k_1,k_2\}$。在本文中，我们将上述上界改进为$(n_1^2+n_2^2-1)+k$。这同时也回答了Qiu等人在文献[Acta Informatica 48 (2011) 271-290]中提出的一个开放问题。此外，我们证明，在$\Sigma=\{\sigma_1,...,\sigma_t\}$且$2\leq t<\infty$的情况下，存在一个整数$z$，使得$\mathcal{A}_1$和$\mathcal{A}_2$等价当且仅当它们满足$z$等价。