A long way at that level. See OpenWorm [1], which is still struggling to simulate C. elegans, a nematode. It's close to the simplest organism with a nervous system. Fully mapped, 302 neurons. They have the wiring diagram, but not the weights.
That project needs more money, and the Human Brain Project[2] needs less money. The Human Brain Project was an effort to understand the human brain in ten years. This is year seven.
Grinding up from the bottom is a thankless task, but necessary. It's bad for your career in some ways. Years ago, Rod Brooks was promoting Cog, an attempt to get to human-level AI in one big jump. (It failed.) I asked him "You did a good robot insect. Why not try for a mouse next." He answered "Because I don't want to go down in history as the builder of the world's best robotic mouse" In the end, he went down in history as the inventor of the first production robotic vacuum cleaner, the Roomba.
This is a classic problem with AI as a field. People keep thinking that they're one big idea away from general artificial intelligence. Hubris. I've seen four cycles of that in my lifetime. There's definite progress, but it's very slow.
Computationally? Think something like "NP hard", except the "verifier" function isn't even plausibly cheap either.
Experimentally from the real thing? I'm not even sure we know how many new pieces of technology we'd need.
It's not even clear that "the weights" are the only variable we're still needing. Even in the pure-computer-science conceptualization of neural networks, things like the activation functions matter; so, it's not unreasonable to suppose there are similar important features to track in biological systems... and whatever those are, we probably aren't getting them captured in a purely geographic connectivity scan.
Doesn't it really depend on how you define the weights? If you can't answer that question, you'll have no idea what to look for. For example, I understand that FIB-SEM microscopy can currently sample the chemical makeup of the voxels where the synapses lie. Those should be the answer to this question, if we agree that the neurotransmitter type and density are the weights. However, if you define the weights as the type and quality of dendrites that lead up to the synapse, you'll get a different answer. Perhaps a better answer to the question is that we are still studying what the weights should be, and having a complete connectome will make it much easier to test the different theories.
Not an original comment, but people occasionally point out that neurons like other cells are descended from self-sufficient life-forms that do all tasks all by themselves, so it would be weird if they were as simple as the mathematical concept we want to assign to them. If a neuron is not itself intelligent in some sense, how does a single cell "decide" what to do?
Currently the process involves experimentally recording activation and trying to replicate that. It’s not as easy as it sounds, though. Organic neurons don’t have floating-point weight values, nor do neural maps propagate in the feed-forward way that makes ANNs efficient on computers.
The best approximations use Spiking Neural Networks [1] and comparatively little research is done there. And again, this is just a different propagation strategy that looks more similar to how organic brains fire. It’s not a fully accurate simulation of the complex chemical ion channels found in the real thing.
That project needs more money, and the Human Brain Project[2] needs less money. The Human Brain Project was an effort to understand the human brain in ten years. This is year seven.
Grinding up from the bottom is a thankless task, but necessary. It's bad for your career in some ways. Years ago, Rod Brooks was promoting Cog, an attempt to get to human-level AI in one big jump. (It failed.) I asked him "You did a good robot insect. Why not try for a mouse next." He answered "Because I don't want to go down in history as the builder of the world's best robotic mouse" In the end, he went down in history as the inventor of the first production robotic vacuum cleaner, the Roomba.
This is a classic problem with AI as a field. People keep thinking that they're one big idea away from general artificial intelligence. Hubris. I've seen four cycles of that in my lifetime. There's definite progress, but it's very slow.
[1] http://openworm.org/ [2] https://www.humanbrainproject.eu/en/