Adaptive-query security of the permutation-lifted unitary ensemble (conjecture)

Prove that the unitary ensemble obtained by left- and right-multiplying products of exponentials of sums of phased permutations—constructed via lifting (pseudo)random permutations to (pseudo)random unitaries—achieves adaptive security; that is, demonstrate computational indistinguishability from Haar measure against any polynomial-time quantum adversary allowed to make adaptive oracle queries.

Background

The paper constructs unitaries by exponentiating sparse Hermitian matrices formed from phased permutations and proves indistinguishability from Haar for nonadaptive (parallel) queries. The analysis leverages large-N asymptotics and properties of the partition algebra to control correlations of words in permutations.

The authors conjecture that the same ensemble should be secure even against adaptive adversaries, but indicate that new proof ideas—such as a refined notion of independence between different permutation words—would be required.

References

We conjecture our ensemble also gives adaptive security, but we note this would require further proof ideas, such as defining a more refined notion of independence of different words.

— Efficient unitary designs and pseudorandom unitaries from permutations  (2404.16751 - Chen et al., 2024) in Section 1.2 (Proof idea), Large-N limit discussion, footnote