In this paper, we extend the techniques used in our previous work to show that there exists a probabilistic Turing machine running within time $O(n^k)$ for all $k\in\mathbb{N}_1$ accepting a language $L_d$ which is different from any language in $\mathcal{P}$, and then to show that $L_d\in\mathcal{BPP}$, thus separating the complexity classes $\mathcal{P}$ and $\mathcal{BPP}$ (i.e., $\mathcal{P}\subsetneq\mathcal{BPP}$). Since the complexity class of {\em bounded error quantum polynomial-time} $\mathcal{BQP}$ contains the complexity class $\mathcal{BPP}$, i.e., $\mathcal{BPP}\subseteq\mathcal{BQP}$, we thus obtain the result that quantum computers are {\em rigorously powerful than} traditional computers. Namely, $\mathcal{P}\subsetneq\mathcal{BQP}$. We further show that (1). $\mathcal{P}\subsetneq\mathcal{RP}$; (2). $\mathcal{P}\subsetneq\text{co-}\mathcal{RP}$; (3). $\mathcal{P}\subsetneq\mathcal{ZPP}$. The result of $\mathcal{P}\subsetneq\mathcal{BPP}$ shows that {\em randomness} plays an essential role in probabilistic algorithm design. Specifically, we show that: (1). The number of random bits used by any probabilistic algorithm which accepts the language $L_d$ can not be reduced to $O(\log n)$; (2). There exits no efficient (complexity-theoretic) {\em pseudorandom generator} (PRG) $G:\{0,1\}^{O(\log n)}\rightarrow \{0,1\}^n$; (3). There exists no quick HSG $H:k(n)\rightarrow n$ such that $k(n)=O(\log n)$.

翻译：本文扩展了我们前期工作中的技术，证明存在一个运行时间为$O(n^k)$（对所有$k\in\mathbb{N}_1$）的概率图灵机，其接受的语言$L_d$与$\mathcal{P}$中的任何语言均不同，进而证明$L_d\in\mathcal{BPP}$，从而分离了复杂度类$\mathcal{P}$与$\mathcal{BPP}$（即$\mathcal{P}\subsetneq\mathcal{BPP}$）。由于有界误差量子多项式时间复杂度类$\mathcal{BQP}$包含复杂度类$\mathcal{BPP}$（即$\mathcal{BPP}\subseteq\mathcal{BQP}$），我们得到量子计算机严格强于传统计算机的结果，即$\mathcal{P}\subsetneq\mathcal{BQP}$。我们进一步证明：(1) $\mathcal{P}\subsetneq\mathcal{RP}$；(2) $\mathcal{P}\subsetneq\text{co-}\mathcal{RP}$；(3) $\mathcal{P}\subsetneq\mathcal{ZPP}$。$\mathcal{P}\subsetneq\mathcal{BPP}$的结果表明，随机性在概率算法设计中起着本质作用。具体而言，我们证明：(1) 任何接受语言$L_d$的概率算法所使用的随机比特数无法降至$O(\log n)$；(2) 不存在有效的（复杂度理论意义上的）伪随机生成器（PRG）$G:\{0,1\}^{O(\log n)}\rightarrow \{0,1\}^n$；(3) 不存在满足$k(n)=O(\log n)$的快速HSG $H:k(n)\rightarrow n$。