We fully determine the communication complexity of approximating matrix rank, over any finite field F. We study the most general version of this problem, where$0\leqslant r < R\leqslant n$are given integers and Alice and Bob need to determine whether their respective matrices$A, B\in \mathbb{F}^{n\times n}$satisfy rk$(A+B)=r$versus rk$(A+B)=R$. We show that this problem has commu-nication cost$\Omega(r^{2}\log\vert \mathbb{F}\vert)$, which is optimal. Our lower bound holds even for quantum protocols and even for error probability$\frac{1}{2}(1-\vert \mathbb{F}\vert ^{-r/3})$, which too is optimal because this problem has a two-bit classical protocol with error$\frac{1}{2}(1-\vert \mathbb{F}\vert ^{-\Theta(r)})$. Prior to our work, lower bounds were known only for constant-error protocols and only for consecutive integers$r$and$R$, with no implication for the approximation of matrix rank. We also settle an analogous question for subspaces, where Alice has a subspace$S$, Bob has a subspace$T$) and they need to approximate the dimension of the subspace$S+T$generated by$S$and$T$(equivalently, approximate the dimension of$S\cap T$). As an application, we obtain an$\Omega(n^{2}\log\vert \mathbb{F}\vert)/k$memory lower bound for any streaming algorithm with$k$passes that approximates the rank of an input matrix$M\in \mathbb{F}^{n\times n}$within a factor of$\sqrt{2}-\delta$, for any$\delta > 0$. Our result is an exponential improvement in$k$over previous work.

Abstract. We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order [Formula: see text] sum to at most [Formula: see text], where [Formula: see text] is the number of variables, [Formula: see text] is the tree depth, and [Formula: see text] 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 [Formula: see text], becoming trivial already at [Formula: see text]. As an application, we obtain, for every integer [Formula: see text], a partial Boolean function on [Formula: see text] bits that has bounded-error quantum query complexity at most [Formula: see text] and randomized query complexity [Formula: see text]. 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: [Formula: see text] versus [Formula: see text] for any [Formula: see text] [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 [Formula: see text] versus [Formula: see text] for bounded-error quantum versus randomized communication complexity for any [Formula: see text]. The best previous separation was polynomially weaker: [Formula: see text] versus [Formula: see text] (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]).

We prove that for every decision tree, the absolute values of the Fourier coefficients of given order t≥1 sum to at most (cd/t)t/2(1+logn)(t−1)/2, where n is the number of variables, d is the tree depth, and c>0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with t, becoming trivial already at t=√d.