AI 中文总结
利用图论方法，证明量子有限自动机在语境性下相对于经典自动机具有指数级、抗噪声的记忆优势，并给出基于排他性图的承诺问题。
AI 中文摘要
量子语境性被广泛认为是量子技术背后的一种重要的非经典资源，然而阐明其转化为无条件计算优势的精确机制仍然是一个持续的挑战。我们展示了量子有限自动机从语境性的图论方法中获得的指数级、抗噪声的记忆优势。我们在一个排他性图$G$上定义了一个承诺问题，对于该问题，任何经典确定性自动机都充当一个非语境隐变量模型，需要至少$N=\chi(G)$个状态，其中$\chi(G)$是图的色数。相比之下，通过利用我们称之为\textit{表征语境性}的结构现象，一个QFA使用最多$d=\xi(G)+1$维的记忆来解决该任务，其中$\xi(G)$是图的正交秩。对于布尔正交图，这种分离呈指数级缩放（$d=\mathcal O(n)$对$N=2^{\Omega(n)}$）。关键的是，这种记忆优势在退极化噪声和相干噪声下都保持$\mathcal{O}(1)$的阈值。
英文摘要
Quantum contextuality is a vital non-classical resource, yet illuminating the precise mechanisms through which it enables unconditional computational advantages remains a challenge. We translate graph-theoretic formulations of contextuality into an unconditional quantum memory advantage for formal language recognition. We define a promise problem on an exclusivity graph $G$ where any classical finite automaton respecting exclusivity requires $N \ge χ(G)$ memory states, whereas a QFA requires a memory of dimension $d = ξ(G)$. The gap between these bounds isolates a structural, information-theoretic incompatibility between classical and quantum descriptions that we term \textit{representational contextuality}. For Boolean orthogonality graphs, this exacts an exponential classical memory penalty ($d=\mathcal{O}(n)$ vs $N=2^{Ω(n)}$). Finally, we demonstrate a sharp algorithmic phase transition: allowing the classical machine a finite confusability of mutually exclusive events reduces this exponential classical memory cost to $\mathcal{O}(n)$.
Comments12 pages, 4 figures total (includes Supplemental Material). v2: Main results tightened and extended to include bounded-error probabilistic automata and entropic bounds
