Indeed the installation is a bit tricky for the Intel Gen 12th GPU...

Also, please make sure you follow all setups step here (and ensure you completed the permission settings!)

https://dgpu-docs.intel.com/installation-guides/ubuntu/ubuntu-focal.html

Here is a copied out of that page...

Ubuntu* 20.04 (focal)
To install the latest general purpose GPU (GPGPU) software packages on Ubuntu 20.04 (focal), add the Intel® software package repository and install the appropriate packages.

Add package repository
To install the repositories.intel.com/graphics package repository, add the following to your Ubuntu installation. Prior to copy/pasting to your console, you may want to run sudo ls and enter your password to prevent the commands from being swallowed by the sudo password prompt:

sudo apt-get install -y gpg-agent wget
wget -qO - https://repositories.intel.com/graphics/intel-graphics.key |
  sudo apt-key add -
sudo apt-add-repository \
  'deb [arch=amd64] https://repositories.intel.com/graphics/ubuntu focal main'

The above will make sure your system has gpg-agent and wget installed. It will then download and install the public key used to verify the integrity of the package repository. Finally, it adds the repositories.intel.com/graphics repository to the system.

Install run-time packages
After adding the repository, you can install the GPU software packages by running the following:

sudo apt-get update
sudo apt-get install \
  intel-opencl-icd \
  intel-level-zero-gpu level-zero \
  intel-media-va-driver-non-free libmfx1

OPTIONAL: Install developer packages
If you will be doing development, you may want to install optional development packages. oneAPI users will need to install these packages for some of the oneAPI tools to function correctly:

sudo apt-get install \
  libigc-dev \
  intel-igc-cm \
  libigdfcl-dev \
  libigfxcmrt-dev \
  level-zero-dev

Configuring permissions
In order to access GPU capabilities, a user needs to have the correct permissions on system. The following will list the group assigned ownership of the render nodes, and list the groups the active user is a member of:

stat -c "%G" /dev/dri/render*
groups ${USER}

If a group is listed for the render node which isn’t listed for the user, you will need to add the user to the group using gpasswd. In the following, the active user will be added to the ‘render’ group and a new shell spawned with that group active:

sudo gpasswd -a ${USER} render
newgrp render

Then, lastly follow the step in the wiki and ensure the runtime is installed.

https://github.com/openvinotoolkit/openvino_notebooks/wiki/Ubuntu

sudo apt-get install intel-opencl-icd

Then, magic! It took me a good 2 hours to figure all these out. Hopefully, this will save others some time...

P.S., on Windows you don't have to install anything and will work just right out-of-the-box...