There are two parts in the security of Bitcoin which could be affected by quantum computers.
One is the security of the elliptic curve cryptography system used when signing transactions. A quantum computer could deduce the private key (and take the funds) if it knows the public key. The private key can be computed solving the discrete logarithm problem, which is efficient in a quantum computer. It would just need a variation of Shor's factoring algorithm:
http://www.math.uwaterloo.ca/~amchilds/teaching/w08/l03.pdfBut, as Etotheipi says, if you only use your address except for spending, the public key is only used once, when you use the funds. The hashing will mask your public key and protect the secret one. There might be a problem if some node see your public key and tries to outrun you and send other transactions and the public keys already in the blockchain would be vulnerable, but I doubt a quantum computer will appear overnight. The system would have time to change to something else.
The second thing is the hash. Grover's algorithm could be used, with some problems, but it is not necessarily the best you can do. There are bounds you can apply directly to collision finding (finding two sequences which hash to the same value). Collision finding (necessary to replace blocks) is not efficient in quantum computers:
http://arxiv.org/abs/quant-ph/0111102You could reduce difficulty, but not so much (at most is like having one fifth of the bits). The impact for a long chain seems small. The model is not completely general, but gives a good taste of what to expect. As far as I know, hash collisions have been studied a lot and this is the best result.
In any case, I think it would be easier to pull a 51% attack amassing a lot of computer power than building a scalable quantum computer, at least in the short/medium term.
Notice that, anyway, the discrete logarithm problem and finding collisions for SHA256 are only SUPPOSED to be hard for classical computers. There is no guarantee that there isn't algorithm that could break elliptic curve cryptography, factor large numbers or find collisions (there could be trapdoor in SHA256). Of course, a lot of people have been trying without success and I think they are probably safe.