We just have our new carpet but now door has become very hard to open and close. Cutting the door isn’t a option as it’s expensive made door. What other ways are there to make the opening and closing of door easy?
You could replace the door hinges with Rising Butt Hinges, these cause the door to lift up by about 10mm as it is opened. They have the side effect that the door wants to "fall" back to the closed position as if a spring is gently pushing it. If you generally want to leave the door open, you will need to use a door-stop or a wedge to hold it open against the force.
After a bit of time depending on its construction, the carpet will begin to fray with some use. This wear will make the door even more difficult to open and close due to the catching of threads on the door material if it's wood, say.
Try removing the carpet underlay material from the area near the doorway so the surface of the carpet is slightly lower than it was. Unfortunately, this will decrease the lifespan of the carpet due to loss of some shock absorbance.
Check that the bottom of the door is free from any burrs, screws, trim, or splinters that may interfere with opening and closing. Planing a few millimetres from the door bottom leading and trailing edges will help and won't be noticeable.
If the original flooring was wood parquet, there's a chance a couple of the tiles are slightly higher than their neighbours which would very slightly increase the average height.
Finally, ensure that the door is parallel to the floor surface.
If none of these suggestions work for you, the alternative is to choose between modifying the door or a different decorative treatment than a carpeted doorway.