As it currently stands, bitcoin does depend on the discrete log problem in an elliptic curve group. This is part of the ECDSA signature algorithm. If quantum computing comes to fruition, it would be unwise not to replace this module.
There are plans to change in the near future from ECDSA to a Schnorr signature – Wikipedia. However this scheme is also based on the discrete log problem — it just happens to use less space. As things stand, storing the signature data is the most expensive part of a transaction, and people are eager to reduce the storage cost.
You can read the rest of the answer at: https://www.quora.com/Will-quantum-computing-destroy-Bitcoin