Ok so I'm presuming that an extremely fine grained scan stored with some naive compression is massively more than 10^14 synapse-bits. In order to store all that now in the information theoretic minimum, don't we need some kind of incredibly awesome compression algorithm NOW that we simply don't have?
No, I think the idea is to do coarse-grained scans, which the superintelligence will have to heavily process in order to infer the original brain structure. (Yeah, it's not clear this is possible even with a whole universe worth of computing power and whatever algorithmic breakthroughs a superintelligence might come up with.)
Paul Christiano recently suggested that we can use neuroimaging to form a complete mathematical characterization of a human brain, which a sufficiently powerful superintelligence would be able to reconstruct into a working mind, and the neuroimaging part is already possible today, or close to being possible.
Paul was using this idea as part of an FAI design proposal, but I'm highlighting it here since it seems to have independent value as an alternative or supplement to cryonics. That is, instead of (or in addition to) trying to get your body to be frozen and then preserved in liquid nitrogen after you die, you periodically take neuroimaging scans of your brain and save them to multiple backup locations (1010 bits is only about 1 gigabyte), in the hope that a friendly AI or posthuman will eventually use the scans to reconstruct your mind.
Are there any neuroimaging experts around who can tell us how feasible this really is, and how much such a scan might cost, now or in the near future?
ETA: Given the presence of thermal noise and the fact that a set of neuroimaging data may contain redundant or irrelevant information, 1010 bits ought to be regarded as just a rough lower bound on how much data needs to be collected and stored. Thanks to commenters who pointed this out.