Symmetries, graph properties, and quantum speedups

TitleSymmetries, graph properties, and quantum speedups
Publication TypeJournal Article
Year of Publication2020
AuthorsBen-David, S, Childs, AM, Gilyen, A, Kretschmer, W, Podder, S, Wang, D
JournalTo appear in Proceedings of the 61st Annual Symposium on Foundations of Computer Science (FOCS 2020)
Date Published6/23/2020

Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup?
In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups.
In contrast, in the adjacency list model for bounded-degree graphs (where graph symmetry is manifested differently), we exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).