Universal quantum algorithms that prepare arbitrary n-qubit quantum states require O(2n) gate complexity. The complexity can be reduced by considering specific families of quantum states depending on the task at hand. In particular, multipartite quantum states that are invariant under permutations, e.g. Dicke states, have intriguing properties. In this paper, we consider states invariant under cyclic permutations, which we call cyclic states. We present a quantum algorithm that deterministically prepares cyclic states with gate complexity O(n) without requiring any ancillary qubit. Through both analytical and numerical analyses, we show that our algorithm is more efficient than existing ones.