Hey I'm not familiar with this way of comparing classical and quantum computation. Can you point me to some more details? I have Nielson's book but don't remember seeing this analogy before!
I presume its explained in Scott Asronsons book, its implicitly there in Nielsen and Chuang. But the best way to understand it is by example - try to write out how you would describe classical probabilistic computation on two classical bits to mimic the quantum circuit type of picture, and if you succeed the generalization will be obvious.