In this paper, a sliding mode control strategy is proposed for single-phase grid-connected three-level neutral-point-clamped quasi-z-source inverters with constant switching frequency. While dc-side variables are controlled by using simple boost control, ac-side variables are controlled via SMC method. The sliding surface function is passed through a boundary layer so as to eliminate chattering and achieve a continuous modulating signal. The switching signals are then obtained by using traditional sinusoidal pulse width modulation which leads to constant switching frequency. The proposed SMC strategy offers many advantages such as fast dynamic response, zero grid current error, simple implementation, robustness to parameter variations and constant switching frequency. In addition, it does not need an active damping method for damping the oscillations arising due the LCL filter resonance. The effectiveness of the proposed strategy is verified by computer simulations under steady-state and transient conditions.