I found this very interesting article on qt-project.org - "Processing a Qt PDF project."
This page discusses the various options available for working with PDF documents in the Qt application. The page doesn't exactly show how to "open and display an existing PDF document," but it will help you get something useful out of everything that is explained there.
Here on the page it says:
To display pages or elements from existing PDF documents into image files or images in memory (useful, for example, to create thumbnails or embed custom viewers), third-party libraries can be used (for example: poppler-qt4 (freedesktop.org) and muPDF (mupdf .com)).
Alternatively, the task can be delegated to an existing command line (for example, poppler-utils (freedesktop.org) and muPDF (mupdf.com)).
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