Fock-Label Moment Recovery and Coherent Quantum Sample Separation in High-Frequency Gaussian Ridge Learning

Abstract Basis transforms are a basic mechanism by which quantum algorithms extract hidden structure. For standard Gaussian inputs, Hermite functions and Fock states provide the spectral representation matched to the input measure. Recent work introduced an efficient quantum Hermite transform (QHT) and pointed to its possible uses in property testing and learning, but a clear model showing how QHT yields a provable readout advantage in a concrete Gaussian learning task has been missing. We introduce a high-frequency Gaussian ridge learning model. The target function is controlled by a small number of hidden directions, and binary labels are generated by passing a high-frequency periodic score through a weak noisy classification channel. Given coherent binary samples, the quantum procedure applies QHT to each copy, measures Fock labels, and uses shell postselection, low-order moment estimation, and tensor decomposition to recover the hidden-direction energy structure. We prove that this recovery uses polynomially many coherent sample calls and quantum gates. For the same candidate family and recovery target, any classical general statistical query (SQ) algorithm with inverse-polynomial tolerance requires exponentially many queries. This gives an exponential query separation between coherent quantum sample access and classical general SQ access. The result promotes QHT from a Gaussian spectral-transform primitive to a readout mechanism for learning, while making explicit the access models and recovery target underlying the separation.