Great as a show, but more fiction than science I liked this show, I liked that they have gone far with the idea, although I must admint, I might have not liked the show as much, would I have not seen it in one day.
The idea that you can scan a small object and take a computer to calculate all past and even some of the future is interesting. But even me being a "simple" engineer I can see many problems of this even in smaller scales. I am not saying we cannot attempt to predict the future to a useful degree with large computers and samples, that is how weather forecast and engineering simulations work.
However even if the world is determinictic and there are no random things, I do not think we can do what is being done here. For one, while I could reconstruct my home from memory with CAD software to a degree, I could use cameras and laser sensors to make it accurate, none the less it is impossible to make it perfect. Given the imperfections the future or past cannot be determined perfectly, since the initial conditions are not defined with enough precision.
Second, following ideas of the series we could take a bottle cap, and we could predict not just the bottle's shape, but the history down back to dinosaurs. The idea is interesting, I mean a lot of things would affect the bottle cap, but for example one guy being hit by a car on the other side of the planet probably would not. The bottle might be made of oil, made of dinosaurs, but no it would just not contain the info of their skeleton. Again some prediction we could make, but imperfect.
Third, we could take the scan and use our knowledge of the universe to simulate it. Again this would require us to know all the rules of the universe to an infinite degree of precision, even small deviations would cause massive differences over longer time periods, again imperfection.
Fourth: we cannot expect that we can build a machine which simulates the whole of reality and fit it into a small box, that is just not how it works. For one the state of the universe, or even just the region affecting humanity cannot be stored in a small space, unless we use lossy compression, causing imperfections.
Fifth: even if we could store the state one quantum particle could not calculate the state of many more very quickly, that is we would need a machine the size of the simulated area or more and going to the future or past would take a lot of time. Again simplification and imperfection is unavoidable.
Based on this even if the world is deterministic, it would not be possibly to predict the actual behavior for larger stretches of space and time. We could still make useful and realistic predictions. It just would not be an exact match.