You might not be able to make a perfect quantum copy, but with iterative refinement you can get arbitrarily close. I think 99.9999999% or so would be good enough.
You might be interested in Greg Egan's [0] story 'The Jewel' [1]. [edit] To clarify, it's actually two stories "Learning to Be Me" and "Closer".
Greg Egan is one of my favourite modern sci-fi writers. He explores interesting and hard ideas about what it means to be human by placing humanity in thought provoking situations.
In 'The Jewel' we have developed an implant that learns how to 'be' the host. At some point your brain is removed and the jewel takes over your functioning. The thing is, what happens when something goes wrong... boom boom booooom!!?!?!
