Characterizing Projected Quantum Kernel Learning via Data-to-Readout Interfaces

Quantum kernel methods map classical data into high-dimensional quantum state spaces, but projected learners access only a prescribed readout, leaving unclear how circuit structure determines the information available for learning. We introduce the data-to-readout interface to characterize this transfer. For linear projected quantum kernels, we show that the reachable readout dimension is exactly determined by the transfer channel restricted to the encoder span, while the intrinsic transfer dimension is bounded by the squared operator Schmidt rank of the fixed quantum process. Controlled interfaces further admit an exact factorization into input-dependent probabilities and fixed readout geometry. This factorization characterizes the population-kernel spectrum and effective dimension, determines when widening the interface provides additional target-relevant information, and yields finite-sample and measurement guarantees. We further show that statistical simplicity does not imply classical tractability: under standard complexity and cryptographic assumptions, even low-width interfaces can remain classically hard to evaluate, and a logarithmic-width construction yields a quantum–classical learning separation. Experiments on controlled circuits with up to 1000 qubits and on disordered transverse-field Ising ground states validate the structural bounds, target-alignment predictions, and finite-measurement trade-offs.