The query model has generated considerable interest in both classical and quantum computing communities. Typically, quantum advantages are demonstrated by showcasing a quantum algorithm with a better query complexity compared to its classical counterpart. Exact quantum query algorithms play a pivotal role in developing quantum algorithms. For example, the Deutsch-Jozsa algorithm demonstrated exponential quantum advantages over classical deterministic algorithms. As an important complexity measure, exact quantum query complexity describes the minimum number of queries required to solve a specific problem exactly using a quantum algorithm. In this paper, we consider the exact quantum query complexity of the following two $n$-bit symmetric functions: $\text{MOD}_m^n(x) = |x| \bmod m$ and $$ \text{EXACT}_{k,l}^n(x) = \begin{cases} 1, &\text{if }|x| \in \{k,l\}, \\ 0, &\text{otherwise}, \end{cases} $$ where $|x|$ is the number of $1$'s in $x$. Our results are as follows: \begin{itemize} \item We present an optimal quantum algorithm for computing $\Mod_m^n$, achieving a query complexity of $\lceil n(1-\frac{1}{m}) \rceil$ for $1 < m \le n$. This settles a conjecture proposed by Cornelissen, Mande, Ozols and de Wolf (2021). Based on this algorithm, we show the exact quantum query complexity of a broad class of symmetric functions that map $\B^n$ to a finite set $X$ is less than $n$. \item When $l-k \ge 2$, we give an optimal exact quantum query algorithm to compute $\text{EXACT}_{k,l}^n$ for the case $k=0$ or $k=1,l=n-1$. This resolves the conjecture proposed by Ambainis, Iraids and Nagaj (2017) partially. \end{itemize}

翻译：查询模型在经典计算和量子计算领域都引起了广泛关注。通常，量子优势通过展示一个比其经典对应算法具有更好查询复杂度的量子算法来体现。精确量子查询算法在量子算法的发展中起着关键作用。例如，Deutsch-Jozsa算法比经典确定性算法表现出指数级的量子优势。作为一个重要的复杂度度量，精确量子查询复杂度描述了使用量子算法精确求解特定问题所需的最少查询次数。本文中，我们研究了以下两个$n$比特对称函数的精确量子查询复杂度：$\text{MOD}_m^n(x) = |x| \bmod m$ 和 $$ \text{EXACT}_{k,l}^n(x) = \begin{cases} 1, &\text{if }|x| \in \{k,l\}, \\ 0, &\text{otherwise}, \end{cases} $$ 其中 $|x|$ 是 $x$ 中 $1$ 的个数。我们的结果如下： \begin{itemize} \item 我们提出了一种计算 $\Mod_m^n$ 的最优量子算法，当 $1 < m \le n$ 时，实现了 $\lceil n(1-\frac{1}{m}) \rceil$ 的查询复杂度。这解决了 Cornelissen、Mande、Ozols 和 de Wolf (2021) 提出的猜想。基于该算法，我们证明了一类将 $\B^n$ 映射到有限集 $X$ 的广泛对称函数的精确量子查询复杂度小于 $n$。 \item 当 $l-k \ge 2$ 时，针对 $k=0$ 或 $k=1,l=n-1$ 的情况，我们给出了计算 $\text{EXACT}_{k,l}^n$ 的最优精确量子查询算法。这部分解决了 Ambainis、Iraids 和 Nagaj (2017) 提出的猜想。 \end{itemize}