The one-way communication complexity of group membership
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Author(s)
Le Gall, Francois
Tani, Seiichiro
Russell, Alexander
Aaronson, Scott
Date Issued
February 2009
Citation
Aaronson, Scott, Francois Le Gall, Alexander Russell and Seiichiro Tani. "The One-Way Communication Complexity of Group Membership."
Version
Original manuscript
Abstract
This paper studies the one-way communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input,
a subgroup H of a finite group G; Bob receives an element x ∈ G. Alice is permitted to send a single
message to Bob, after which he must decide if his input x is an element of H. We prove the following upper bounds on the classical communication complexity of this problem in the bounded-error setting: 1. The problem can be solved with O(log|G|) communication, provided the subgroup H is normal.
2. The problem can be solved with O(d[subscript max] · log|G|) communication, where d[subscript max] is the maximum of
the dimensions of the irreducible complex representations of G.
3. For any prime p not dividing |G|, the problem can be solved with O(d[subscript max] · log p) communication,
where d[subscript max] is the maximum of the dimensions of the irreducible Fp-representations of G.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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