Though the question is from digital logic, the answer is purely from automata. As per the 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 the number of $0's$ is less than the 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 $2$, the output will be $10$ (both these states not having any outgoing transitions) and for other $3$ states, the output will be $00$ as per the given description of the circuit.
Correct Answer: $A$