Noisy Quantum Kernel Machines

In the noisy intermediate-scale quantum era, an important goal is the conception of implementable algorithms that exploit the rich dynamics of quantum systems and the high dimensionality of the underlying Hilbert spaces to perform tasks while prescinding from noise-proof physical systems. An emerging class of quantum learning machines is that based on the paradigm of quantum kernels. Here, we study how dissipation and decoherence affect their performance. We address this issue by investigating the expressivity and the generalization capacity of these models within the framework of kernel theory. We introduce and study the effective kernel rank, a figure of merit that quantifies the number of independent features a noisy quantum kernel is able to extract from input data. Moreover, we derive an upper bound on the generalization error of the model that involves the average purity of the encoded states. Thereby we show that decoherence and dissipation can be seen as an implicit regularization for the quantum kernel machines. As an illustrative example, we report exact finite-size simulations of machines based on chains of driven-dissipative quantum spins to perform a classification task, where the input data are encoded into the driving fields and the quantum physical system is fixed. We determine how the performance of noisy kernel machines scales with the number of nodes (chain sites) as a function of decoherence and examine the effect of imperfect measurements.