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 $\text{MOD}_m^n:\{0,1\}^n \rightarrow \{0,...,m-1\}$ and $\text{EXACT}_{k,l}^n:\{0,1\}^n \rightarrow \{0,1\}$, which are defined as $\text{MOD}_m^n(x) = |x| \bmod m$ and $ \text{EXACT}_{k,l}^n(x) = 1$ iff $|x| \in \{k,l\}$, 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.

翻译：查询模型在经典计算和量子计算社区中均引起了广泛关注。通常，量子优势通过展示比经典对应算法具有更好查询复杂度的量子算法来体现。确切量子查询算法在量子算法发展中扮演着关键角色。例如，Deutsch-Jozsa算法相较于经典确定性算法展示了指数级的量子优势。作为重要的复杂度度量，确切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小查询次数。本文研究了以下两个$n$比特对称函数的确切量子查询复杂度：$\text{MOD}_m^n:\{0,1\}^n \rightarrow \{0,...,m-1\}$和$\text{EXACT}_{k,l}^n:\{0,1\}^n \rightarrow \{0,1\}$，其定义分别为$\text{MOD}_m^n(x) = |x| \bmod m$，以及$\text{EXACT}_{k,l}^n(x) = 1$当且仅当$|x| \in \{k,l\}$，其中$|x|$表示$x$中1的个数。我们的结果如下：i) 我们提出了计算$\text{MOD}_m^n$的最优量子算法，对于$1 < m \le n$，达到了$\lceil n(1-\frac{1}{m}) \rceil$的查询复杂度。这解决了Cornelissen、Mande、Ozols和de Wolf（2021年）提出的一个猜想。基于该算法，我们证明了一类将$\{0,1\}^n$映射到有限集$X$的广泛对称函数的确切量子查询复杂度小于$n$。ii) 当$l-k \ge 2$时，我们针对$k=0$或$k=1,l=n-1$的情况，给出了计算$\text{EXACT}_{k,l}^n$的最优确切量子查询算法。这部分解决了Ambainis、Iraids和Nagaj（2017年）提出的猜想。