We propose a scheme to perform arbitrary unitary operations on a single electron-spin qubit in a quantum dot. The design is solely based on the geometrical phase that the qubit state acquires after a cyclic evolution in the parameter space. The scheme is utilizing ultrafast linearly-chirped pulses providing adiabatic excitation of the qubit states and the geometric phase is fully controlled by the relative phase between pulses. The analytic expression of the evolution operator for the electron spin in a quantum dot, which provides a clear geometrical interpretation of the qubit dynamics is obtained. Using parameters of InGal\l/GaN, GaN/AlN quantum dots we provide an estimate for the time scale of the qubit rotations and parameters of the external fields. (C) 2012 Elsevier Ltd. All rights reserved.