Quantum and Probabilistic Computers Rigorously Powerful than Traditional Computers, and Derandomization

Abstract: In this paper, we extend the techniques used in our previous work to show that there exists a probabilistic Turing machine running within time $O(n<sup>k)$ for all $k\in\mathbb{N}_1$ accepting a language $L_d$ that is different from any language in $\mathcal{P}$, and then further to prove that $L_d\in\mathcal{BPP}$, thus separating the complexity class $\mathcal{BPP}$ from the class $\mathcal{P}$ (i.e., $\mathcal{P}\subsetneqq\mathcal{BPP}$). Since the complexity class $\mathcal{BQP}$ of {\em bounded error quantum polynomial-time} contains the complexity class $\mathcal{BPP}$ (i.e., $\mathcal{BPP}\subseteq\mathcal{BQP}$), we thus confirm the widespread-belief conjecture that quantum computers are {\em rigorously more powerful} than traditional computers (i.e., $\mathcal{P}\subsetneqq\mathcal{BQP}$). We further show that (1): $\mathcal{P}\subsetneqq\mathcal{RP}$; (2): $\mathcal{P}\subsetneqq{\rm co-}\mathcal{RP}$; (3): $\mathcal{P}\subsetneqq\mathcal{ZPP}$. Previously, whether the above relations hold or not were long-standing open questions in complexity theory. Meanwhile, the result of $\mathcal{P}\subsetneqq\mathcal{BPP}$ shows that {\em randomness} plays an essential role in probabilistic algorithm design. In particular, we go further to show that (1): The number of random bits used by any probabilistic algorithm that accepts the language $L_d$ can not be reduced to $O(\log n)$; (2): There exists 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 $k(n)=O(\log n)$.