Quantum adder mod 4

Question

Does anyone has (or is aware of) a solution to Nielsen's QCQI 4.36 exercise: construct a circuit to add two-qubit numbers x and y modulo 4 vs $|x,y\rangle \rightarrow |x, x+y \mod 4\rangle$

Answer 1

The trick is to break down, bit by bit, what the calculation is that you need. You'll see on the right-hand side of the circuit, I've stated what the outcomes need to be. Hopefully, you also recognise some of those outputs (particularly the top qubit), so that rapidly starts to formulate the circuit...