Quantum kernel methods for solving regression problems and differential equations

We propose several approaches for solving regression problems and differential equations (DEs) with quantum kernel methods. compose models as weighted sums of functions, where variables are encoded using feature maps model derivatives represented automatic differentiation circuits. While previously methods primarily targeted classification tasks, here we consider their applicability to based on available data constraints. use two strategies approach these problems. First, devise a mixed trial solution by kernel-based which is trained minimize loss specific constraints or datasets. Second, support vector that accounts the structure equations. The developed capable both linear nonlinear systems. Contrary prevailing hybrid variational parametrized circuits, perform training weights classically. Under certain conditions this corresponds convex optimization problem, can be solved provable convergence global optimum model. proposed also favor hardware implementations, only uses evaluated Gram matrices, but require quadratic number function evaluations. highlight trade-offs when comparing our those circuits such recently differentiable approach. offer potential enhancement through rich representations power maps, start quest towards provably trainable DE solvers.