This is already possible, but it never was a problem. The GPU is just a slave to the CPU.
The biggest problem I could see would be when your computer becomes part of a botnet. Your computer could be used for brute fore encryption cracking. Again, this was also possible with just CUDA or OpenCL.
So in a few years, the govt. will get a report at the end of the year showing that model Swede citizen has bought more than the recommended amount of meat or alcohol or, heavens forbid, has bought tobacco products. And not enough fruit/veggies. So the friendly Swede govt. will "recommend" that model Swede citizen fork over an insane amount of money and "optionally" attend re-education classes in healthy living. Can you spell Big, Big, Big Brother?
It's not really like that in Sweden. A lot more would have to change in Sweden's national government; public policy, especially with regard to privacy issues, are amongst the best in the world. This doesn't mean a super-fascist regime is impossible in the future. But why would a shift away from paper money bring about the changes you propose?
Because with all transactions recorded it would become possible and when something becomes possible, oftentimes govts are tempted. As an exemple, where I live the govt cross-checks your income tax filing with the automobile registration records and if there is, in their opinion, a discrepancy, you have to explain yourself.
Also my comment does not single out Sweden, it is more of a general observation.
And why the fuck should this be on Hacker News??