Follow up from my last question: $3 \times 3$ Rubik's cube scrambling question
I am talking about $3 \times 3$ Rubik's cubes. Start with a solved cube. Then make some amount of random moves (where moves are defined using the half-turn metric: any twist of the face, i.e. 90 degrees counterclockwise, 90 degrees clockwise, 180 degrees are each one move). After how many moves will each of the 43 quintillion states be equally likely? If the answer is "infinitely many," can someone give some idea of how many moves will be "close enough?"