Machine Learning with Continuous Variable Quantum Kernels: A Toolkit — Laura Henderson

Описание: Machine Learning with Continuous Variable Quantum Kernels: A Toolkit — Laura Henderson (Queensland)

The popular qubit framework has dominated recent work on quantum ker- nel machine learning, with results characterising expressivity, learnability and generalisation. As yet, there is no comparative framework to understand these concepts for continuous variable (CV) quantum computing platforms. In this paper we represent CV quantum kernels as holomorphic functions and use this representation to provide several important theoretical insights. We derive a general closed form solution for all CV quantum kernels and show every such kernel can be expressed as the product of Gaussian and polynomial terms. Furthermore, we present quantification of a quantum-classical separation for all quantum kernels via a hierarchical notion of “stellar rank”. We then prove kernels of infinite stellar rank, such as those generated by GKP-state encod- ings, can be approximated arbitrarily well by kernels of finite stellar rank. Finally, we simulate learning with a single-mode displaced Fock state encoding and show that (i) accuracy on our specific task (an annular data set) increases with stellar rank, (ii) for underfit models, accuracy can be improved by in- creasing a bandwidth hyperparameter, and (iii) for noisy data that is overfit, decreasing the bandwidth will improve generalisation but does so at the cost of effective stellar rank and thus quantum advantage.