Learning quantum kernels with continuous-variable optical circuits

Abstract Kernel-based learning methods are central to classical machine learning, yet their quantum counterparts have largely remained limited to fixed circuit constructions. Most existing approaches aim to improve the performance of kernel-based support vector machines by heuristically tailoring kernel functions to specific tasks. In this work, we introduce a principled framework for adapting quantum kernels by translating two well-established classical techniques—the Fisher criterion and quasi-conformal kernel transformations—into the setting of continuous-variable (CV) quantum circuits. We show that the classical Fisher criterion is equivalent to maximizing the Hilbert–Schmidt distance in the quantum feature space induced by a quantum circuit. We further demonstrate that quasi-conformal kernel transformations admit a direct and natural implementation at the circuit level, realized by augmenting the original quantum circuit with additional modes. Using experimentally accessible optical circuits composed solely of displacement and squeezing operations, we numerically demonstrate that each method independently leads to a systematic enhancement of learning performance. Taken together, our theoretical and numerical results establish a concrete route for importing powerful classical kernel-learning strategies into quantum machine learning, and position CV quantum optics as a flexible and experimentally viable platform for adaptive, task-driven quantum kernel design.