3 Bit Adder Logic Circuit Design
You have C1 correct...A0*B0...but C2 needs to include C1, such that if any two (or 3) of A1, B1, and C1 are true C2 becomes true.  Similarly, C3 is triggered by 2 or more of A2, B2, and C2.  This is a ripple carry architecture, for carry look-ahead each stage n must derive its carry only from the inputs A,B0..n-1.