How symmetric is too symmetric for large quantum speedups?
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Suppose a Boolean function f is symmetric under a group action G acting on the n bits of the input. For which G does this mean f does not have an exponential quantum speedup? Is there a characterization of how rich G must be before the function f cannot have enough structure for quantum algorithms to exploit? In this work, we make several steps towards understanding the group actions G which are "quantum intolerant" in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a "well-shuffling" property of group actions – which happens to be preserved by several natural transformations – implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where G=S_n. Our main application is for graph symmetries: we show that any Boolean function f defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power 6 relationship between its randomized and quantum query complexities, even if f is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).
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