The One-Way Communication Complexity of Subgroup Membership

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 y ∈ G. Alice is permitted to send a single message to Bob, after which he must decide if his input y is an element of H. We establish the following bounds on the classical communication complexity of this problem in the bounded-error setting: 1. The problem can be solved with O(log |G|)-bit communication with the promise that H is normal. 2. The problem can be solved with O(dmax · log |G|)-bit communication, where dmax is the maximum degree of an irreducible complex representation of G. 3. For any prime p not dividing |G|, the problem can be solved with O(log |G|+ dmax · log p)-bit communication, where dmax is the maximum degree of an irreducible Fp-representation of G.