On the exact quantum query complexity of MOD_m^n and EXACT_k,l^n
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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 MOD_m^n:{0,1}^n →{0,...,m-1} and EXACT_k,l^n:{0,1}^n →{0,1}, which are defined as MOD_m^n(x) = |x| m and EXACT_k,l^n(x) = 1 iff |x| ∈{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 MOD_m^n, achieving a query complexity of ⌈ n(1-1/m) ⌉ for 1 < m ≤ 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 ≥ 2, we give an optimal exact quantum query algorithm to compute 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.
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