State of the art of SAT on a quantum computer

Disclaimer: I don't understand quantum computing.

Given a CNF boolean formula $$\phi$$ in $$n$$ variables and quantum computer with $$q$$ qubits, what is the complexity of solving $$\phi$$ as a function of $$n,q$$?

I am mainly interested when $$q$$ is polynomial in $$n$$.

Can we get $$2^{o(n)}$$ (small Oh, Exponential time hypothesis)?

I believe on a traditional computer the complexity is $$C^n$$ for a constant $$C < 2$$.

This is an open question. People who try to prove $$\text{BQP} = \text{NP}$$ usually use the SAT formulation, in my experience. But we don't know of any algorithm that addresses the structure of the SAT problem.
Adding to psitae's answer, in 2 recent papers by Aaronson, Chia, Lin, Wang and Zhang (arXiv:1911.01973) and Buhrman, Patro and Speelman (arXiv:1911.05686) the Quantum Strong Exponential Time Hypothesis (QSETH) was formulated. This hypothesis in fact claims that there is no $$2^{o(n)}$$ quantum algorithm for solving $$k-\mathrm{SAT}$$.