Improved Approximate Degree Bounds for k-Distinctness

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 ??(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 ??(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)

An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree

While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions. In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.Comment: 12 page