Answer
Zero.
Work Step by Step
We know that the work done by gravity on an object is given by
$$W=mgy_f-mgy_i=mg\Delta y$$
And since the object returns to its initial position, $\Delta y=0$
$$W =mg\Delta y=mg(0)=\color{red}{\bf 0}\;\rm J$$
Therefore, the work done by the gravity on this particle is zero.
