Shows that the universal bound R(f)=O((1+Q(f))^4) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.
Manuscript
A Nearly Quartic Separation Between Randomized and Quantum Query Complexity
October 5, 2026 · source · cite as \cite{OAI:A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026}
We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as \mathrm R(f) and \mathrm Q(f), the examples rule out every universal bound \mathrm R(f)=O((1+\mathrm Q(f))^\alpha) with α < 4. Thus the known quartic upper bound has the optimal exponent, disproving the conjectured cubic bound. Both complexities count worst-case bit queries with error at most 1/3 on every input.
Result 284 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md