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    • Improved Approximate Degree Bounds for K-Distinctness.

    Nikhil S. Mande,Justin Thaler,Shuchen Zhu

    Electronic Colloquium on Computational Complexity (ECCC)（2020）

    Cited 5|Views66
    Abstract
    An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element Distinctness) is well-understood, there is a polynomial gap between the known upper and lower bounds for all constants k>2. Specifically, the best known upper bound is O(N^{(3/4)-1/(2^{k+2}-4)}) (Belovs, FOCS 2012), while the best known lower bound for k >= 2 is Omega(N^{2/3} + N^{(3/4)-1/(2k)}) (Aaronson and Shi, J.~ACM 2004; Bun, Kothari, and Thaler, STOC 2018). For any constant k >= 4, we improve the lower bound to Omega(N^{(3/4)-1/(4k)}). This yields, for example, the first proof that 4-distinctness is strictly harder than Element Distinctness. Our lower bound applies more generally to approximate degree. As a secondary result, we give a simple construction of an approximating polynomial of degree O(N^{3/4}) that applies whenever k <= polylog(N).
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    Approximation Algorithms,Quantum Error Correction
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    【要点】：本文改进了k-不同性问题的近似度界，为k≥4的情况提供了新的下界，证明了4-不同性问题比元素不同性问题更难，并给出了一个近似多项式的构造方法。
    

    【方法】：作者通过分析k-不同性函数的量子查询复杂性，提出了新的近似度界下界公式，并使用近似多项式的方法来支持其理论。
    

    【实验】：本文主要侧重于理论分析和证明，未提供具体实验。文中提及的数据集名称未明确，但结果为将k-不同性问题的下界从Omega(N^{2/3} + N^{(3/4)-1/(2k)})改进至Omega(N^{(3/4)-1/(4k)})，并为k≤polylog(N)的情况构造了一个度数为O(N^{3/4})的近似多项式。
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