Problem of mentioning the department affiliated with PhD degrees
My question is, what is the best way in our cases to "market" our research focus and experiences to other people? 


Marketing is all about knowing your audience. And thus you may market your PhD to different audiences in different ways.

If you're talking to a person on the street, you could say "PhD in Computer Science". 

If you're talking to a Computer Scientist, you could say "PhD in Quantum Information". 

If you're talking to a specialist in Quantum Information, you could say "PhD in X, Y &amp; Z" where X, Y and Z are detailed topics.

The truth of the matter is that a PhD is a PhD. Officially speaking, there's no such thing as a "PhD in ...". So for the given person in front of you, try to communicate the area of your PhD as best as you can.

And as for the "IT guy" stuff, if that's how they chose to stereotype what you are doing, then that's their problem, not yours/your boyfriends.