Consider the second digit of the binary number. First the inputs A and B are added. Furthermore the transfered information from the first digit, the Carry Bit, has to be added. Hence the circuit gets a third input, the CarryIN CI. Such a logical unit is called a full adder.

There are many ways to build a full adder using logic gates. The circuit chosen here consists of two half adders with an additional OR gate.
Now back to the problem of two 2bit numbers to be added.