QIP2021 | Symmetries, graph properties, and quantum speedups (William Kretschmer)

Authors: Shalev Ben-David, Andrew M. Childs, András Gilyén, William Kretschmer, Supartha Podder and Daochen Wang
Affiliations: University of Waterloo | University of Maryland | California Institute of Technology | University of Texas at Austin | University of Ottawa | University of Maryland

Abstract
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 graphswhere graph symmetry is manifested differentlywe 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).

Get entangled with us!