There are several answers to your problem. But, first, why did it arise in the first place? You chose a carpet you felt will look good, feel good, and which will last for a reasonable amount of time in the area you chose (there are different grades of carpet for different areas of the home). Here, your responsibility, apart from funding the purchase, ends. From there on, responsibility for ensuring your purchase is suitable for fitting in the area in which you choose, and that it is fitted securely without causing problems with the existing infrastructure of you home rests with the supplier/fitter of the carpet. End of story.
Now, to resolve the matter, you must contact the supplier or fitter. They have seen the problem first-hand, will already know about it and must correct it without delay. They should have used their professional experience and/or knowledge to ensure the carpet didn't cause problems and that it wasn't trapping any doors opening onto the carpeted area. Because the fitter will, presumably, have had to close the door to fit the carpet, he/she (I'm not sexist) will have had to force the door over the section of carpet which is trapping the door in order to open it, so, already, they will have caused some damage to this carpet.
The most logical solution is to trim the bottom of the door so that it opens easily. I don't feel there is need to worry about draughts, because the amount removed is very small and will not be noticeable. The supplier should arrange this remedy at no expense to yourself, but you should ensure they use a qualified carpenter and that they know what they are doing.
The suggestion of using rising butt hinges is completely impractical. Certainly, they will do the job of clearing the carpet, but they also will not allow the door to remain open unless you use a doorstop.