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

TL;DR

The paper proves the existence of a language $L_d$ that cannot be decided by any polynomial-time deterministic Turing machine but can be decided by a probabilistic Turing machine in time $O(n^k)$ for every $k$, placing $L_d$ in $\mathcal{BPP}$ and implying $\mathcal{P}\subsetneqq\mathcal{BPP}$ and hence $\mathcal{P}\subsetneqq\mathcal{BQP}$. It develops a diagonalization framework against polynomial-time deterministic machines and extends it to establish separations $\mathcal{P}\subsetneqq\mathcal{RP}$, $\mathcal{P}\subsetneqq\mathrm{co-}\mathcal{RP}$, and $\mathcal{P}\subsetneqq\mathcal{ZPP}$, while arguing that randomness remains a necessary resource for efficient computation. The work further argues there is no efficient complexity-theoretic pseudorandom generator with seed length $O(\log t)$ and no quick hitting set generator achieving $k(n)=O(\log n)$, underscoring limits to derandomization. Taken together, these results strengthen the case that randomness contributes real power to probabilistic algorithms and provide new benchmarks for comparing classical probabilistic, quantum, and deterministic computation.

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 $O(n^k)$ 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 $k(n)=O(\log n)$.

Figures (5)

• Figure 1: Cantor pairing function
• Figure 2: The probabilistic transitions of $M_0$ and the corresponding probabilities
• Figure 3: The transitions of states of $M_0$ and corresponding probabilities by setting $\lambda=\frac{2}{3}$
• Figure 4: The transitions of states and corresponding probabilities for $\widetilde{L_d}\in\mathcal{RP}$
• Figure 5: The transitions of states of $M_0$ and corresponding probabilities for $\widehat{L_d}\in\text{co-}\mathcal{RP}$

Theorems & Definitions (55)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Remark 1.1
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9