Parity-query realization of the logarithmic lower bound for superlinear query budgets

Construct a single initial encoding on \(O(n)\) fixed bits together with causal single-parity queries that preserve the information-hiding and population-mean properties of the Lyu–Talwar adaptive attack, and establish failure probability strictly greater than \(1/3\), thereby proving an \(\Omega_\alpha(\sqrt{n}\log n)\) fixed-accuracy lower bound for adaptive Pauli parity prediction with \(M=\lceil n^\alpha\rceil\) queries for every fixed \(\alpha>1\).

Background

For general bounded queries, the cited Lyu–Talwar construction yields an $\Omega_\alpha(\sqrt n\log n)$ lower bound for $M=\lceil n^\alpha\rceil$, but its failure probability is only an unspecified positive constant and therefore does not directly contradict the paper’s required $2/3$ success guarantee.

The paper explains that transferring this result to Pauli prediction would require one fixed $O(n)$-bit encoding, single-parity queries at every stage, a causal query rule, and preservation of the attack’s information restriction and population means. Achieving failure probability greater than $1/3$ would close the logarithmic interval between the known $\Omega(\sqrt n)$ lower bound and the $O(\sqrt n\log n)$ upper bound for superlinear query budgets.