How will Big O be with quantum computers?
Big-O has nothing to do with quantum computers. This should be obvious by the fact that Big-O notation was introduced in 1894, long before we had any knowledge of quantum theory … or computers for that matter.
Big-O is simply a convenient notation for classifying growth rates of functions. Big-O doesn't care how you interpret those functions.
So, the definition of Big-O will not change for quantum computers for the simple reason that its definition has nothing to do with computers.