We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllabil-ity of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.