It depends upon your employer, and who there you wish to inform.
I am not an alcoholic, but I was dating one. That, too, affected my work performance.
My manager only knew I was going through a hard time. I don't know if he has experience with addiction in his family, and so I didn't tell him, because I did not think he would understand.
We do have a psychologist at work. This psychologist knows everything, and even coordinated to guarantee that my (now ex-)boyfriend was admitted into a good rehab program and that the insurance would cover it. (My ex and I work at the same place.) Our company has very extensive addiction assistance.
The psychologist is still assisting, on both sides (rehab, unfortunately, was not enough, and now my ex has started stalking me as well).
The managers (his and mine) still don't know. (I did tell my manager I may be going to HR regarding someone outside the department - I did not give details or names or even mention the stalking as the reason, and he did not ask, merely offered to help me in any way he could if I needed him).
Again, I'm from the other side of things, but I did want to comment. Do what feels most comfortable for you, depending upon your circumstances. Check what assistance your work provides, as well, as larger companies especially tend to have very good addiction programs.
Pick your answer based on what you feel is best, once you have enough information.
Good luck!