How would you apply quantum computing to break a symmetric key system?

How would you apply quantum computing to break a symmetric key system? Do you attack the plaintext password by bruteforcing? Or the mathematical process of the encryption algorithm?

Generally, symmetric key cryptography has much less structure than asymmetric-key cryptography, so there usually isn't much benefit to looking for mathematical structure that you can use. If no helpful structure can be found (such as with AES), you treat the encryption as a black box, and try to break it from there. The usual model for this is a known plaintext attack: you know a plaintext message $$m$$, and you know the encrypted value $$c$$, but you don't know the key that encrypted it. In this case, you would use Grover's algorithm to search the keyspace. In theory this offers a square-root speed-up over a classical brute-force search, although this only holds for a single processor doing the search. In practice you would use many parallel machines for a big search like this, and the more parallel machines you use, the smaller the advantage of Grover's algorithm.