Nuka-Cola had strong ties with the U.S. military which allowed them access to their technology, including suits of power armor. The suits were given Nuka-Cola themed paint jobs and used to promote the soft drink. Vim! Pop Incorporated had a similar idea by using a suit of their own and giving it a Vim! inspired paint scheme to promote their drink.
The Quantum X-01 power armor is identical in stats and model to regular X-01 power armor but bears a unique custom paint job in a variety of shades of blue in theme of the then-new Nuka Cola Quantum flavor. It also displays the Nuka-Cola Quantum logo embossed in gold on the left chest.
Quantum X-01 power armor is the most powerful armor suit one can find in the game due to coming pre-upgraded to Mk V. Although X-01 Mk VI parts can be found on the various leveled power armors scattered around Nuka-World, the Quantum armor is the only complete suit that is guaranteed to spawn with such high-level gear. All others are capped at Mk III just like those in the Commonwealth.
As of the current patch, the star core found in the Nuka-World junkyard barn respawns after one in-game week. The armor can therefore be acquired before any of the aforementioned quests have been completed by picking up this core multiple times. With enough patience and by waiting a very long time, one could even skip the tedious hunt for star cores entirely. Note that the respawning core is probably not intentional and might eventually get patched out.