This paper explores a fine-grained version of the Watrous conjecture, including the randomized and quantum algorithms with success probabilities arbitrarily close to $1/2$. Our contributions include the following: i) An analysis of the optimal success probability of quantum and randomized query algorithms of two fundamental partial symmetric Boolean functions given a fixed number of queries. We prove that for any quantum algorithm computing these two functions using $T$ queries, there exist randomized algorithms using $\mathsf{poly}(T)$ queries that achieve the same success probability as the quantum algorithm, even if the success probability is arbitrarily close to 1/2. ii) We establish that for any total symmetric Boolean function $f$, if a quantum algorithm uses $T$ queries to compute $f$ with success probability $1/2+\beta$, then there exists a randomized algorithm using $O(T^2)$ queries to compute $f$ with success probability $1/2+\Omega(\delta\beta^2)$ on a $1-\delta$ fraction of inputs, where $\beta,\delta$ can be arbitrarily small positive values. As a corollary, we prove a randomized version of Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the regime where the success probability of algorithms can be arbitrarily close to 1/2. iii) We present polynomial equivalences for several fundamental complexity measures of partial symmetric Boolean functions. Specifically, we first prove that for certain partial symmetric Boolean functions, quantum query complexity is at most quadratic in approximate degree for any error arbitrarily close to 1/2. Next, we show exact quantum query complexity is at most quadratic in degree. Additionally, we give the tight bounds of several complexity measures, indicating their polynomial equivalence.

翻译：本文探讨了Watrous猜想的细粒度版本，涵盖成功概率可任意接近$1/2$的随机化算法和量子算法。我们的贡献包括：i) 对于两个基本部分对称布尔函数，在给定查询次数下，分析了量子与随机查询算法的最优成功概率。我们证明，对于使用$T$次查询计算这两个函数的任何量子算法，存在使用$\mathsf{poly}(T)$次查询的随机化算法，能够达到与量子算法相同的成功概率，即使成功概率可任意接近1/2。ii) 我们证明：对于任意全对称布尔函数$f$，若某量子算法使用$T$次查询以成功概率$1/2+\beta$计算$f$，则存在某随机化算法使用$O(T^2)$次查询，在$1-\delta$比例输入上以成功概率$1/2+\Omega(\delta\beta^2)$计算$f$，其中$\beta,\delta$可为任意小的正值。作为推论，我们在算法成功概率可任意接近1/2的设定下，证明了全对称布尔函数Aaronson-Ambainis猜想的随机化版本。iii) 我们给出了部分对称布尔函数若干基本复杂度度量的多项式等价性。具体而言，我们首先证明：对于特定部分对称布尔函数，当错误概率可任意接近1/2时，量子查询复杂度至多为近似度的二次方。其次，我们证明精确量子查询复杂度至多为度的二次方。此外，我们给出了若干复杂度度量的紧界，表明其多项式等价性。