Add only with logical operation.

Add.java

public class Add {
    public static void main(String[] args) {
        int a = 0b01100101;
        int b = 0b00110110;
        System.out.printf("a:%d b:%d a+b:%d%n", a, b, a + b);
        
        int and = a & b;
        System.out.printf("%8s(%d)%n", Integer.toBinaryString(a), a);
        System.out.printf("%8s(%d)%n", Integer.toBinaryString(b), b);
        System.out.printf("%8s(%d) xor%n", Integer.toBinaryString(a ^ b), a ^ b);
        System.out.printf("%8s(%d) and%n", Integer.toBinaryString(and), and);
        System.out.println();
        
        a = a ^ b;
        b = and << 1;
        and = a & b;
        System.out.printf("%8s(%d) and << 1%n", Integer.toBinaryString(b), b);
        System.out.printf("%8s(%d) xor%n", Integer.toBinaryString(a ^ b), a ^ b);
        System.out.printf("%8s(%d) and%n", Integer.toBinaryString(and), and);
        System.out.println();
        
        a = a ^ b;
        b = and << 1;
        and = a & b;
        System.out.printf("%8s(%d) and << 1%n", Integer.toBinaryString(b), b);
        System.out.printf("%8s(%d) xor%n", Integer.toBinaryString(a ^ b), a ^ b);
        System.out.printf("%8s(%d) and%n", Integer.toBinaryString(and), and);
        System.out.println();

    }
}

a:101 b:54 a+b:155
 1100101(101)
  110110(54)
 1010011(83) xor
  100100(36) and

 1001000(72) and << 1
   11011(27) xor
 1000000(64) and

10000000(128) and << 1
10011011(155) xor
       0(0) and