In this work, we propose a parameterised quantum circuit learning approach to point set matching problem. In contrast to previous annealing-based methods, we propose a quantum circuit-based framework whose parameters are optimised via descending the gradients w.r.t a kernel-based loss function. We formulate the shape matching problem into a distribution learning task; that is, to learn the distribution of the optimal transformation parameters. We show that this framework is able to find multiple optimal solutions for symmetric shapes and is more accurate, scalable and robust than the previous annealing-based method. Code, data and pre-trained weights are available at the project page: \href{https://hansen7.github.io/qKC}{https://hansen7.github.io/qKC}