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The proof uses the comparison theorem of Diaconis and Saloff-Coste and our previous results.\n  Let $2n$ be the number of spins. We prove that the gap of the Fredkin quantum spin chain Hamiltonian [6, 20], is $\\Theta(n^{-c})$ with $c\\ge2$. 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The proof uses the comparison theorem of Diaconis and Saloff-Coste and our previous results.\n  Let $2n$ be the number of spins. We prove that the gap of the Fredkin quantum spin chain Hamiltonian [6, 20], is $\\Theta(n^{-c})$ with $c\\ge2$. 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