Symmetries, graph properties, and quantum speedups

TL;DR

This paper investigates how symmetries in functions and graphs influence the potential for quantum speedups, showing that certain symmetries limit speedups to polynomial, while others allow exponential advantages.

Contribution

It characterizes the types of symmetries that restrict quantum speedups and identifies the unique permutation groups that prevent super-polynomial advantages.

Findings

Hypergraph symmetries limit quantum speedups to polynomial.

Certain permutation groups are the only ones preventing super-polynomial speedups.

An exponential quantum speedup is demonstrated in a specific graph property testing problem.

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).