PhotoQt for Ubuntu 15.04 or newer can be easily installed by adding the following PPA:

Open a terminal and type the following one line at a time:
sudo apt-add-repository ppa:lumas/photoqt
sudo apt update
sudo apt install photoqt

If you prefer to simply install PhotoQt from a deb file (i.e., without adding my PPA), you can do so here for a 32-Bit system and a 64-Bit system.

For Ubuntu 14.04 or earlier, some manual work is required to install the latest version of PhotoQt. This is due to the fact, that Ubuntu 14.04 still ships a very old version of Qt, too old for PhotoQt to work with. Packages of older versions that can be installed without having to tweak your system can be found at the end.

First, we need to make sure that your system is up-to-date. So lets open a terminal (and keep it open, we will use it a couple more times) and run:

sudo apt update
sudo apt upgrade

For all the commands on this page, every line is one command and needs to be executed on its own.

Important: Before we start, if you have followed an older set of these instructions and thus had installed Qt 5.5, you first should remove it:
sudo apt remove qt55-meta-full
Also, you should remove the added line from the file ~/.profile, e.g., by using gedit:
gedit ~/.profile

Now, we need to manually update two things: We need a newer version of Qt and a newer version of GraphicsMagick. Fortunately, both of them can be installed from two different PPAs.
Note, that I do not own either of these PPAs. Use them at your own risk.

In the terminal, run the following two commands. They will add both of the PPAs to your set-up:

sudo apt-add-repository ppa:beineri/opt-qt57-trusty
sudo apt-add-repository ppa:rwky/graphicsmagick

After refreshing the database, we can then proceed to install Qt 5.7, GraphicsMagick 1.3.23, Exiv2 0.23, and LibRaw 0.15.4:

sudo apt update
sudo apt install qt57-meta-full freeglut3 libqt5svg5
sudo apt install graphicsmagick libgraphicsmagick++1-dev libgraphicsmagick++3
sudo apt install exiv2 libraw9

We are almost there now. There's only one more thing we need to do: The Qt packages that you install from the above PPA doesn't install into the system paths. What that means, is that Ubuntu doesn't know about them until we explicitly tell it where to look. For this we need to edit two files.
  1. First, in your terminal run

    gedit ~/.profile

    This should open the file '.profile' in your home directory for editing. Go to the very end of the file and add the following line:

    . "/opt/qt57/bin/qt57-env.sh"

    [Note: The 'dot' followed by the space is on purpose, it all needs to be entered!]
    This will tell Ubuntu where all the new binaries are located.

  2. One very important environment variable, that the line in step 1 adds, will be ignored by Ubuntu. However, without it PhotoQt (and any other Qt app) wont work. So we need to fix this behaviour, so that it is not ignored. Doing that is very simple. In your terminal, execute the following line:

    echo STARTUP=\"/usr/bin/env LD_LIBRARY_PATH=\${LD_LIBRARY_PATH} \${STARTUP}\" | sudo tee /etc/X11/Xsession.d/90preserve_ld_library_path

    Now it is important to restart your computer, otherwise this change wont take any effect. It is not enough to simply log out and back in again!
That's all the preparations necessary, and we're now ready to install PhotoQt. As we had to do a lot of customisations, it's not possible to provide in in a PPA, so you have to manually install either of the two packages below, depending on your system architecture:

And that's everything. Now you should have a working up-to-date version of PhotoQt on your system. I created and tested these instructions on a fresh setup of Ubuntu, however, I might very well have missed some small detail. If you run into any troubles, please let me know!