Complexity of Simon's problem in classical sense
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Simon's problem is a standard example of a problem that is exponential in classical sense, while it admits a polynomial solution in quantum computing. It is about a function f for which it is given that a unique non-zero vector s exists for which f(x) = f(x ⊕ s) for all x, where ⊕ is the exclusive or operator. The goal is to find s. The exponential lower bound for the classical sense assumes that f only admits black box access. In this paper we investigate classical complexity when f is given by a standard representation like a circuit. We focus on finding the vector space of all vectors s for which f(x) = f(x ⊕ s) for all x, for any given f. Two main results are: (1) if f is given by any circuit, then checking whether this vector space contains a non-zero element is NP-hard, and (2) if f is given by any ordered BDD, then a basis of this vector space can be computed in polynomial time.
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