Symmetries, graph properties, and quantum speedups
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 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).
Keywords
Cite
@article{arxiv.2006.12760,
title = {Symmetries, graph properties, and quantum speedups},
author = {Shalev Ben-David and Andrew M. Childs and András Gilyén and William Kretschmer and Supartha Podder and Daochen Wang},
journal= {arXiv preprint arXiv:2006.12760},
year = {2021}
}
Comments
46 pages. Subsumes arXiv:2001.09642 and arXiv:2001.10520; adds a characterization of permutation groups with speedup and an exponential speedup for adjacency-list graph property testing