My basic theory with research can be put as such: ideas are a type of capital good. That is, broadly defined, they are a good that can be made to make other goods. And research can be roughly described as the action of finding new ideas. At first, it might seem like the state is the best way to fund this, since it has historically directed a lot of resources into research.

However, one of the inputs into undergoing research is existing capital. This can take the form of larger capital stock that allows people to have the productive capacity, and thus wealth, in order to spare the time to do research. It can take the form of new or cheaper technologies that make certain types of research easier. It can take the form of existing ideas, since most new ideas are developments of previous ideas. And so on.

In free and open markets, capital continually accumulates according to compounding production. In a state society, the state necessarily must inhibit the speed of this accumulation in order to fund itself (piles of cash are forms of capital), and in modern progressive societies, states also typically inhibit this accumulation in unnecessary ways (with regards to capital), I.e. regulations, progressive taxes, wars, limiting trade and immigration, etc.

Thus, the question is not one of "is government better at producing research than libertarianism". Instead, the question is. "does government inhibit capital accumulation to the point that research is ultimately inhibited?"

It's actually even worse than that. As long as government funding does not produce a net benefit to accumulation, and as long as total aquirable knowledge does not have a limit, then a libertarian society would actually have more research in the long run, regardless of initial conditions, always. The line y=x+C will always intercept the line y=x/2+C, regardless of C (under standard assumptions).

Now, it might be possible for a theoretical all-knowing technocratic government, that always acts in the public interest, to manage things in such a way as to produce capital accumulation to a point that is the same, or even greater, than a free unhampered market. Too bad those are in short supply.