John, may I ask what you mean by standard house lighting? What sort of fixture do you have on the tank?
The stock lighting on a 24" flourescent strip fixture ( for a 20 gallon high) is usually 15 watts. A double strip 24" flourescent will give you 30 watts.
If you're using incandescent light, you can expect difficulties as the light spectrum is different and not as favorable to plants like with flourescent.
30 watts of flourescent lighting on that size tank and low level CO2 addition is good despite the fact that it's 1.5wpg. I use those small yeast reactors made by Nutra something or other ( will try to find name, lol ).
The problem with our wpg calculations is that it is very rough. 1.5 wpg is considered low light, however, the height of the tank, the distance of the plants from the light is not taken into account in that calculation and it is a very meaningful factor. So 1.5 wpg on a 20 gallon high is very different from 1.5wpg on a 55 gallon, or a 20 gallon long. In my experience with the 20 high I mentioned, at 30wpg
(of flourescent light) it can be considered a medium light tank. I still wouldn't inject high amounts of CO2, get the little reactors that are meant for that size of tank.
You will get fast growth and it will out compete the algae, be prepared to have to prune hard though.

It will turn into a jungle.
The swords are usually too large for a 20 high. They require deep substrate for their large root systems. You will probably end up having to remove them.