Can I break a White Hawk Software protection with a quantum computer?

We don’t know; we haven’t seen a quantum computer yet and don’t know what its real limitations will be. Once quantum computers are available a lot of algorithms will need to be modified. Protections (and cryptanalysis) are no exceptions. For the near future we expect quantum computers to be used for “standard” algorithms. We believe it will take somewhat longer until quantum computing can be applied to such unstructured problems as reverse engineering protected code.

April 2016: NIST publshed a paper “Report on Post-Quantum Cryptography” (NISTIR 8105)
That is a summary; it has a substantial list of scientific references.