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On the exact quantum query complexity of MOD and EXACT functions

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Abstract

In this paper, we consider the exact quantum query complexity of two fundamental symmetric functions. 1) MOD nm , which calculates the Hamming weight of an n-bit string modulo m; 2) EXACT nk,l ;, which determines if the Hamming weight of an n-bit string is exactly k or l. Although these two symmetric functions have received considerable attention, their exact quantum query complexities have not been fully characterized. Specifically, our results are as follows:

  1. 1)

    We design an optimal quantum query algorithm to compute MOD nm exactly and thus provide a tight characterization of its exact quantum query complexity, which settles a previous conjecture. Based on this algorithm, we demonstrate that a broad class of symmetric functions is not evasive in the quantum model, i.e., there exist quantum algorithms to compute these functions exactly when the number of queries is less than their input size.

  2. 2)

    By proposing a quantum algorithm that utilizes the minimum number of queries to compute EXACT nk,l exactly for some specific values of k and l, we give a tight characterization of its exact quantum query complexity in these scenarios.

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Acknowledgements

We would like to thank Lvzhou Li and Jingquan Luo for pointing out a flaw in Algorithm 1 in an early version of this manuscript. We would like to thank the anonymous reviewers for their advice on improving the article. This research was supported by the National Natural Science Foundation of China (Grant Nos. 62332009, 12347104, and 61972191) and the Innovation Program for Quantum Science and Technology (2021ZD0302901).

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Correspondence to Penghui Yao or Zekun Ye.

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Competing interests The authors declare that they have no competing interests or financial conflicts to disclose.

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Penghui Yao is currently an associate professor in the Department of Computer Science and Technology at Nanjing University, China. He received his PhD degree from the Centre for Quantum Technologies, National University of Singapore, Singapore in 2013. His research interests include quantum complexity theory, quantum information theory, quantum distributed computing, Fourier analysis in theoretical computer science and quantum computation, and derandomization.

Zekun Ye is currently a PhD student in the Department of Computer Science and Technology at Nanjing University, China. His research interests include quantum algorithms, quantum query complexity, and quantum communication complexity.

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Yao, P., Ye, Z. On the exact quantum query complexity of MOD and EXACT functions. Front. Comput. Sci. 19, 194901 (2025). https://doi.org/10.1007/s11704-024-3770-4

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