# Classical Lower Bounds from Quantum Upper Bounds

Shalev Ben-David,
Adam Bouland,
Ankit Garg,
Robin Kothari

October 2018

### Abstract

We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem.

We show that for any function f, the approximate degree of computing the OR of n copies of f is Ω(√n) times the approximate degree of f, which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013.

We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate γ2 norm) of F:X×Y→{0,1} grows by a factor of Ω~(√n) when we take the OR of n copies of F, which is also essentially optimal. As a corollary, we give a new proof of Razborov’s celebrated Ω(√n) lower bound on the quantum communication complexity of the disjointness problem.

Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.

Publication

*Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS)*