On the exact quantum query complexity of $\text{MOD}_m^n$ and $\text{EXACT}_{k,l}^n$

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: i) We present an optimal quantum algorithm for computing $\text{MOD}_m^n$, 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 $\{0,1\}^n$ to a finite set $X$ is less than $n$. ii) 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.