Result 284, Mathematical physics

The optimal quartic separation between randomized and quantum queries

Shows that the universal bound R(f)=O((1+Q(f))4)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.

Disproof or counterexample

The bigger picture

Why it matters

How much can quantum computation reduce the number of input bits an algorithm must inspect? The manuscript reports examples showing that the known fourth-power limit on this advantage cannot be replaced by any smaller power.

What changes?

The examples are total Boolean functions: rules assigning one output bit to every input bit string of a given length. Randomized and quantum query complexities count worst-case input-bit queries, with error at most one-third on every input and unrestricted computation between queries. The known universal bound puts randomized complexity at most a constant times the fourth power of one plus quantum complexity. The manuscript claims examples ruling out every smaller exponent, including the conjectured exponent three.

What does that help mathematicians do?

The claimed separation would rule out any general method guaranteeing a cubic randomized query bound from quantum query complexity. Because these functions are defined on every input, the obstruction does not depend on promises excluding certain inputs. Researchers would therefore have a sharp limit on the exponent achievable in a universal comparison. This does not assert an exact quartic separation without smaller-order losses.

Are there practical applications?

Its immediate value is foundational for quantum algorithms: it calibrates how large an advantage can be when accessing input bits is the measured resource. This is not a demonstrated running-time improvement. Since computation between queries is unrestricted, translating the separation into an efficient implementation would require further analysis.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Nearly Quartic Separation Between Randomized and Quantum Query Complexity

October 5, 2026 19 pages

We construct total Boolean functions with a nearly quartic separation between bounded-error randomized and quantum query complexity. Writing these complexities as R(f)\mathrm R(f) and Q(f)\mathrm Q(f), the examples rule out every universal bound R(f)=O((1+Q(f))α)\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.

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Cite (BibTeX) 
@misc{OAI:A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026,
  author = {{OpenAI}},
  title = {{A Nearly Quartic Separation Between Randomized and Quantum Query Complexity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf}{OAI:A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.