Methods of finding a PDA
If you can find a PDA for the language $L_0 = \{ 0^i 1^j : i &lt; j \}$, then you can probably find a PDA for your language as well, which is just $L_0^2$. Hence I suggest the following plan:


Figure out a PDA for $L_0$.
Figure out a general construction that, given PDAs for languages $L_1,L_2$, outputs a PDA for their concatenation $L_1 L_2$.
Apply the construction to the PDA you found in the first step.