Solving partial differential equations seems to be a useful application of quantum computing because the solutions usually involve doing linear algebra on large sparse matrices which can be encoded into quantum operations. However, for several PDE's it is unclear if quantum computing can provide an advantage in solving the end-to-end problem. In this proposal, we discuss an end-to-end quantum algorithm for the diffusion eigenvalue problem, a type of second-order elliptic PDE. We then discuss potential for quantum advantage, as well as other PDE's to analyze in which one may see an advantage.

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