How to design a circuit that cycles
A few hints:


Your first and third outputs are always the same. So there's no need to calculate them separately. Just connect whatever needs to be driven by the third output to the same logic that generates the first output. If there is a substantial load current, you might need buffers to provide it.

Same goes for the second and fourth outputs.
You have 4 states, but you can encode them in various ways. One obvious way is to make the state machine with a two bit counter and use a couple of gates to decode the outputs. 

But it might be even simpler to use a one-hot encoding (google it for more details) using 4 flip-flops. Then you'll find that each of your outputs happens to be equal to the output of one of the flip flops holding the state. A one-hot state machine for this case (simple sequence of 4 states) can be implemented with a shift register (a single IC if you're doing discrete logic), provided you can arrange to reset it into a state you choose.