A new quantum walk model that depends on surface embeddings yields a face-decomposed scattering matrix and a comfortability formula ranking embeddings by genus.
and Kokainis, M., Quadratic speedup for finding marked vertices by quantum walks, Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing 19 (2020) 412–424
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Comfortability of quantum walks on embedded graphs on surfaces
A new quantum walk model that depends on surface embeddings yields a face-decomposed scattering matrix and a comfortability formula ranking embeddings by genus.