2020/06/23 by Shalev Ben-David, Andrew M. Childs, András Gilyén +3
Physics and Astronomy · Computer Science · #quant-ph #cs.CC
paper · pdf · doi:10.1109/focs46700.2020.00066
published as Proceedings of the 61st IEEE Symposium on Foundations of Computer Science (FOCS 2020), pp. 649-660 (2020) · 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
arxiv created 2020/06/23 · arxiv updated 2021/03/09
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).