The aim of this paper is to derive a stable and efficient scheme for solving the one-dimensional time-fractional nonlinear Schrödinger equation set in an unbounded domain. We first derive absorbing boundary conditions for the fractional system by using the unified approach introduced in [47, 48] and a linearization procedure. Then, the initial boundary-value problem for the fractional system with ABCs is discretized, a stability analysis is developed and the error estimate O(h 2 +τ) is stated. To accelerate the L1-scheme in time, a sum-of-exponentials approximation is introduced to speed-up the evaluation of the Caputo fractional derivative. The resulting algorithm is highly efficient for long time simulations. Finally, we end the paper by reporting some numerical simulations to validate the properties (accuracy and efficiency) of the derived scheme.