Nuka-Cola Corporation 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 standard 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 metal, automotive-style lettering on the left breastplate.
Quantum X-01 power armor is the most powerful armor suit one can find in the game due to it coming pre-upgraded to Mk V. Although X-01 Mk VI parts can be found on the various incomplete power armors scattered around Nuka-World and the Commonwealth, the Quantum armor is the only complete suit that is guaranteed to spawn with such high-level gear.
The Quantum X-01 power armor can only be found in a display case on the first floor of Starport Nuka, in the Galactic Zone, in front of the large mainframe computer that controls many of the park's functions. This display case is unlocked through the main terminal, after obtaining all 35 Star cores, during the quest Star Control.
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 hunt for star cores entirely. Note that the respawning core is not intentional and might eventually get patched out.