Zha+26 introduce quantum oracle sketching (QOS), a streaming algorithm that constructsan approximate phase oracle of a Boolean function f: N → 0, 1 from M =Θ (NQ2/ε) classical samples and then executes any quantum query algorithm of complexityQ on the sketched oracle. Combined with an interferometric variant of classical shadowtomography, QOS yields exponential separations in machine size between quantum and classicallearners for linear systems, classification, and dimensionality reduction. The argumentis information–theoretic and unconditional, depending only on quantum mechanics. This work offers a rigorous critique and constructive extension. We (i) sharpen the diamond–norm guarantee using non–commutative matrix Bernstein, recovering the linear N/Mbias scaling without the spurious logarithmic factor and with explicit constants; (ii) recastthe optimal Q2 scaling as a quantum Cram´er–Rao bound on the symmetric–logarithmic–derivative (SLD) Fisher information of the random oracle channel; (iii) generalize QOSto non–uniform priors by showing that sample complexity is governed by the effective dimensionNeff = ∥p∥21/2, allowing genuine improvements when data are heavy–tailed; (iv) replacethe hard spectral–gap assumption in PCA by a polynomial–filter analysis that pays onlyquadratically in the soft gap; (v) building on the parallel manuscript of Sep26, extendQOS to non–linear kernel learning via random Fourier features, deriving the explicit samplecomplexity for shift–invariant kernels via an independent route and connecting it to the information–geometric framework of (ii) and the effective–dimension framework of (iii) ; (vi) provethat QOS is structurally compatible with R´enyi differential privacy, achieving a per–sampleprivacy budget that scales as ε2DP = Θ (1/ (NM) ) ; (vii) analyse a hybrid scheme that pre—composes QOS with a Johnson–Lindenstrauss embedding, isolating the regimes in which thequantum advantage survives classical compression; (viii) building further on Sep26, give amatrix Bernstein proof of the QOS noise threshold pg ≲ ε2/ (NQ2) with explicit constants, and pose the matching tight–threshold question as an open problem alongside the recentnoisy–random–circuit obstruction Aha+23. We close with a candid critique of the resourceaccounting underlying the “60–logical–qubit” demonstration and a list of open problems insoft–gap, non–Hermitian, and continuous–domain QOS.
Alfredo Sepulveda-Jimenez (Sat,) studied this question.