  We introduce an efficient variational hybrid quantum-classical algorithm
designed for solving Caputo time-fractional partial differential equations. Our
method employs an iterable cost function incorporating a linear combination of
overlap history states. The proposed algorithm is not only efficient in time
complexity, but has lower memory costs compared to classical methods. Our
results indicate that solution fidelity is insensitive to the fractional index
and that gradient evaluation cost scales economically with the number of time
steps. As a proof of concept, we apply our algorithm to solve a range of
fractional partial differential equations commonly encountered in engineering
applications, such as the sub-diffusion equation, the non-linear Burgers'
equation and a coupled diffusive epidemic model. We assess quantum hardware
performance under realistic noise conditions, further validating the practical
utility of our algorithm.
