We consider the problem of implementing mutually unbiased bases (MUB) for a polarization qubit with only one wave plate, the minimum number of wave plates. We show that one wave plate is sufficient to realize two MUB as long as its phase shift (modulo 360°) ranges between 45° and 315°. It can realize three MUB (a complete set of MUB for a qubit) if the phase shift of the wave plate is within [111.5°, 141.7°] or its symmetric range with respect to 180°. The systematic error of the realized MUB using a third-wave plate (TWP) with 120° phase is calculated to be a half of that using the combination of a quarter-wave plate (QWP) and a half-wave plate (HWP). As experimental applications, TWPs are used in single-qubit and two-qubit quantum state tomography experiments and the results show a systematic error reduction by 50%. This technique not only saves one wave plate but also reduces the systematic error, which can be applied to quantum state tomography and other applications involving MUB. The proposed TWP may become a useful instrument in optical experiments, replacing multiple elements like QWP and HWP.