Electrical – Serial 2’s Complementer with One Input and One Output

I know this question has been asked and there already are answers to this question on the Internet. I just want to ask you where I am wrong in my way of thinking.

When I first saw this question, the shortcut -keep copying until you see a 1 and then complement the rest- hadn't occurred to me. So I thought I would just complement the bit, add 1 to it and if I get a carry then I would keep it in the flip flop to add it to the next bit coming from the input.

Meaning that,

In State-0

if the input is 0 then I should output "0" (after complementing) and keep that carry "1" in the flip-flop so flip-flop goes from state-0 to state-1.

If the input is 1 then I should output "1" but the flip-flop remains in the same state which is state-0.

In State-1

if the input is 0 then I should output "1" -since 2's complement would be 10 and I have a carry "1" in the flip-flop- I would get another carry so the state remains the same.

If the input is 1 then I should output "0" and the flip-flop remains in the same state as well.

Here is my state diagram :

+---------+-------+-------+--------+
| Present |       | Next  |        |
| State   | Input | State | Output |
|    A    |   x   |   A   |   y    |
|    0    |   0   |   1   |   0    |
|    0    |   1   |   0   |   1    |
|    1    |   0   |   1   |   1    |
|    1    |   1   |   1   |   0    |
+---------+-------+-------+--------+

After getting the equations from the table above. I get a circuit like so:

^{simulate this circuit – Schematic created using CircuitLab}

Can you please help me to figure out what's wrong with my way of thinking?

Thank you in advance. 🙂