Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem

TL;DR

This paper leverages Huang's recent breakthrough to establish tight bounds relating degree, approximate degree, and quantum complexity of Boolean functions, with implications for quantum query complexity and graph property testing.

Contribution

It proves the quadratic relation between degree and approximate degree, and the quartic relation between deterministic and quantum query complexities, extending Huang's results and resolving related conjectures.

Findings

Degree of f is at most quadratic in approximate degree.

Deterministic query complexity is at most quartic in quantum query complexity.

Quantum query complexity of nontrivial monotone graph properties is linear in n.

Abstract

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, $\bullet \quad \mathrm{deg}(f) = O(\widetilde{\mathrm{deg}}(f)^2)$: The degree of $f$ is at most quadratic in the approximate degree of $f$. This is optimal as witnessed by the OR function. $\bullet \quad \mathrm{D}(f) = O(\mathrm{Q}(f)^4)$: The deterministic query complexity of $f$ is at most quartic in the quantum query complexity of $f$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa--Karp--Rosenberg conjecture. We show that if $f$ is a nontrivial monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $\mathrm{Q}(f)=\Omega(n)$, which is also optimal. We also show that the approximate degree of any read-once formula on $n$ variables is $\Theta(\sqrt{n})$.