To work on quantum computing you would need a PhD in physics. It is still mainly an academic subject where nearly all experiments are done at universities or research institutes. The "engineering" part of it is still really experimental physics.

I do know some people who work in the field who are probably more electrical engineers than physicists (they design control electronics etc so spend most of their time using CADENCE and similar software), but they still have PhDs in either physics or microelectronics.