Simons problem is one of the most important problems demonstrating the power of quantum computers, which achieves a large separation between quantum and classical query complexities. However, Simons discussion on his problem was limited to bounded-error setting, which means his algorithm can not always get the correct answer. Exact quantum algorithms for Simons problem have also been proposed, which deterministically solve the problem with O(n) queries. Also the quantum lower bound Omega(n) for Simons problem is known. Although these algorithms are either complicated or specialized, their results give an O(n) versus Omega(sqrt{2^{n}}) separation in exact query complexities for Simons problem (Omega(sqrt{2^{n}}) is the lower bound for classical probabilistic algorithms), but it has not been proved whether this separation is optimal. In this paper, we propose another exact quantum algorithm for solving Simons problem with O(n) queries, which is simple, concrete and does not rely on special query oracles. Our algorithm combines Simons algorithm with the quantum amplitude amplification technique to ensure its determinism. In particular, we show that Simons problem can be solved by a classical deterministic algorithm with O(sqrt{2^{n}}) queries (as we are aware, there were no classical deterministic algorithms for solving Simons problem with O(sqrt{2^{n}}) queries). Combining some previous results, we obtain the optimal separation in exact query complexities for Simons problem: Theta({n}) versus Theta({sqrt{2^{n}}}).
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