An Optimal Separation of Randomized and Quantum Query Complexity

We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order sum to at most , where is the number of variables, is the tree depth, and is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with , becoming trivial already at . As an application, we obtain, for every integer , a partial Boolean function on bits that has bounded-error quantum query complexity at most and randomized query complexity . This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: versus for any [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of versus for bounded-error quantum versus randomized communication complexity for any . The best previous separation was polynomially weaker: versus (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).