Are hash functions really quantum resistant for commitment schemes?

Question

It is commonly stated that hash functions remain secure in a post quantum world, the justification being that a quantum computer only has the advantage of Grover's search to give it a quadratic speedup.

However, there is active research into, for example, commitments such as https://eprint.iacr.org/2015/361.pdf. These state that in a quantum setting (whatever that means), the old way of doing commitments by requiring collision resistance doesn't work.

My understanding (possibly wrong) is that one cannot implement a hash function with quantum operators and gates because the quantum operators are unitary and therefore the hash function becomes invertible. The classical hash function uses compression to lose information and is hence, non-invertible. Is there then a possibility that if we attempted a quantum implementation of SHA-2, that would end up showing us how to break it?

Consider $x \rightarrow y = H(x)$. If I have an implementation of $H$ with quantum gates, I could take any $y$, put the remaining registers into whatever state I want (corresponding to the lost information) and then invert to obtain a valid $x$. My only assumption here is that I implemented the quantum $H$ but I can't see why this is forbidden, or even difficult.