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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 $C < 2$.

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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.

The best you can do is a naive implementation of a Grover-like amplitude dispersion, which will give you a square-root speedup. (You can read that in the linked discussion on CS stack exchange.)

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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 $k-\mathrm{SAT}$.

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