Advantage of Sample Complexity in Quantum PAC Learning Requires Inverse Access to State-Preparation Unitaries

Natsuto Isogai, Satoshi Yoshida, Mio Murao

Whether quantum computation can reduce the amount of data sampled from an unknown probability distribution required to learn a prediction rule is a fundamental question in quantum machine learning. Quantum PAC learning studies this question using quantum data as a quantum state whose squared amplitudes encode the unknown distribution from which classical learning data are sampled. With only copies of such quantum data, the optimal worst-case sample complexity asymptotically matches that of classical PAC learning. In contrast, access to both a state-preparation unitary for this state and its inverse can improve the query-complexity dependence on the accuracy parameter in realizable learning. However, it has remained unclear whether forward-only access allows such an improvement. In this work, taking the worst case over compatible state-preparation unitaries and their finite ambient dimensions, we show that the optimal forward-only query complexities of realizable and agnostic learning are, respectively, $Θ((d+\log(1/δ))/\varepsilon)$ and $Θ((d+\log(1/δ))/\varepsilon^2)$, where $d$ is the VC dimension of the concept class, $\varepsilon$ the accuracy parameter, and $δ$ the failure…