Though the question is from digital logic, answer is purely from automata. As per question, we just need to count the difference of the number of 0's and 1's in the first k bit of a number. And we just need to count till this count reaches 2 or -2 (negative when number of 0's is less than number of 1's) . So, the possibilities are -2, -1, 0, 1 and 2 which represents the five states of the state transition diagram.
For state -2, the output of the circuit will be 01, for state 2, output will be 10 (both these states not having any outgoing transitions) and for other 3 states, output will be 00 as per the given description of the circuit.