You can use AMDGPU-Pro Driver "Beta" for Ubuntu 16.04.
From this link:
Installing the AMD GPU-PRO Driver
There are four simple steps involved in the installation of the AMD
GPU-Pro Driver: Download, Extract, Install and Configure. The
instructions to perform the installation have been simplified and
should take less than 10 minutes to complete. Before installing the
driver, a quick note on how to check if your system already has AMD
GPU-PRO installed.
System Check
The easiest way to find out if you have AMD GPU-Pro already installed
on your Ubuntu System is to query the Debian package manager.
Using the following command at a terminal will provide you with the
version of the AMD GPU-Pro stack on your system, or inform you that
there are no packages found:
dpkg -l amdgpu-pro
Download
A direct link to download the Linux AMDGPU-PRO driver is given below
and it is also available on the Driver Download Page. This file
has a tar.xz extension which reflects a more-effective compression
algorithm that (in most cases) creates a smaller archive than the more
common gzip format.
AMD Radeon™ Software AMD GPU-PRO Beta Driver – Linux® for Vulkan™
Version 16.15.1013 for Ubuntu 14.04.4
NOTE: This file can also be located via the Driver Download Page
by locating your card and selecting the Linux Driver link.
Extract
After the archive is downloaded, extract the contents to a temporary
location from which you can install it. The example below assumes you
have downloaded the archive to /tmp and will extract to the same
location. If your file was downloaded into the ~/Downloads/ folder
by default, you can also extract and install from there, and
afterwards you can remove the install files.
cd /tmp
tar -Jxvf amdgpu-pro_16.15.2-277429.tar.xz
Install
Once the archive is expanded on the local machine, run the included
script (amdgpu-pro-install) to install the graphics stack. During
the installation you will be required to provide sudo access, and also
to provide two confirmations to:
- Install the packages, and
Allow installation of "unverified" packages from the AMDGPU-PRO archive. The script will use the package manager to install the
components of the graphics stack, with a short delay during the DKMS
(Dynamic Kernel Module Support) installation. From the directory where
you extracted the archive, issue the following command:
amdgpu-pro-driver/amdgpu-pro-install
After this you can restart your machine to launch using the new
graphics stack.