The left bulb

Start with the left bulb. It represents the left digit in the answer.

Look at the four sums again:

0 + 0 = 00

0 + 1 = 01

1 + 0 = 01

1 + 1 = 10

The left bulb should light in exactly one case: when A and B are both 1.

That is the rule you already know as AND.

The AND gate gives addition its left digit. That solves the left bulb. The right bulb needs a different rule.