So why you need static linking? Qt is a part of Ubuntu distribution!
To deploy app you should use debian packege system with marked required dependency. Then when package is installed calling:
will install all needed libraries (of Qt).sudo apt-get install -f
In case when package is install from package repository everything should be automatic.
If you don't know how to make debian package just install qt-libraries manually:
will install most of the Qt libraries.sudo apt-get install libqt4-dev
You can try install minimal number of packages (depending what kind of features you app is using):
sudo apt-get install libqt4-gui libqt4-core
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