# Symmetries, graph properties, and quantum speedups

### APA

Podder, S. (2020). Symmetries, graph properties, and quantum speedups. Perimeter Institute. https://pirsa.org/20120020

### MLA

Podder, Supartha. Symmetries, graph properties, and quantum speedups. Perimeter Institute, Dec. 09, 2020, https://pirsa.org/20120020

### BibTex

@misc{ pirsa_PIRSA:20120020, doi = {10.48660/20120020}, url = {https://pirsa.org/20120020}, author = {Podder, Supartha}, keywords = {Quantum Information}, language = {en}, title = {Symmetries, graph properties, and quantum speedups}, publisher = {Perimeter Institute}, year = {2020}, month = {dec}, note = {PIRSA:20120020 see, \url{https://pirsa.org}} }

Supartha Podder University of Ottawa

## 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). Based on: arxiv:2006.12760