There's a structure called a carry-bypass adder[1] that lets you add two numbers in O(√n) time for only O(n) gates. That or a similar structure is what modern CPUs use and they allow you two add two numbers in a single clock cycle which is all you care about from a software perspective.

There are also tree adders which add in O(log(n)) time but use O(n^2) gates if you really need the speed, but AFAIK nobody actually does need to.

[1]https://en.wikipedia.org/wiki/Carry-skip_adder