A similar issue is found elsewhere but only with 32bit programs.
I have this problem when trying to run clementine - which is 64bit, i checked the binary i have - and ubuntu-one-control-panel-qt
I'm starting to feel like I've tried everything. Is there a way to restore my qt installation to some kind of default state?
I think this might have happened after I uninstalled Rhythmbox with 'sudo apt-get remove rhythmbox'. I've since reinstalled rhythmbox though, and thought any removed dependencies would be reinstalled with apt....
EDIT: this is my apt history log: https://gist.github.com/2646562
Line 215 is when I first installed Clementine. It was obviously working at this point. Removing Rhythmbox at line 305 doesn't look like it did too much damage. Yet that is the point when it stopped working, as most of the rest of the log is me trying to fix Clementine.
Just to repeat: I don't think any Qt apps are working on my installation. I'm about to research how to completely remove and reinstall Qt, see if that's a hopeful avenue.