Telling PhD supervisor I published a paper about my thesis without telling them or listing them as authors? I am in my 3rd year of PhD. In my program, it is a requirement for PhD students to graduate with a paper that is published in a Scopus Index journal.

Before I entered, and during my PhD, I had already published a few papers to a Scopus index peer reviewed journal (for topics not based on my thesis). At the moment, I am trying to publish a paper as a part of my thesis (with my supervisors' names), but my supervisors keep criticizing the paper and making me rewrite it again and again. This has been going on for two years.

However, I did submit another paper (a part of my thesis) under my own name and without my supervisors' knowledge, and it was accepted. This paper was done without any intellectual contributions from my supervisors - no writing or reviewing. When I told my supervisors about the paper (but not its acceptance,) they were angry that I hadn't included their names, and they asked me to include their names in it. Unfortunately, the editor said I couldn't as I had submitted it initially as one author only.

I am aware that it is my fault that I didn't ask them beforehand, but I know in my heart that if I had included them, it would have taken months or years to even get their approval. In all honesty, they would have ended up demotivating me. I am now really scared to tell them that it has already been published and I cannot add their names.

Can you give me any advice how to handle this situation?
The answers here are generally explaining the fact that supervisors should not be omitted from the authorship as they are supposed to direct and contribute to the research.

The critical problem is, you are saying that they did not contribute at all, and they, naturally, did not write or review the published article. Still they try to add their names as authors when they learned about the article submitted. You can not be an author if you did not even review the article, just this information is sufficient to see that the ethical misconduct is mostly on their side.

Your problem is, you can not both submit an article without making "such" people authors and at the same time maintain a good relationship with them. A one-author paper is, for me, much better than giving credit to the people who disregard the functioning of the academy, by both hardening the situations unnecessarily and also making academic misconducts. Even if you somehow get your advisors' names added to the paper, they will never ever forget this. At the same time, you shake the ethical ground of your publication by this act of "correction", and it may, in the end, cause the rejection to your paper.  All the while, you will still be considered untrustworthy by your advisors, quite rightly in their perspectives.

Your only choice is to forget about updating the author names, and find some other ways to correct your relationship with the advisors. 

And please keep in mind that in academy, at least in Turkey, almost everyone faced in a situation that more than the bulk of work done by someone and just because their advisors don't like them they don't give authorship to the corresponding researcher. And these advisors made their Ph.D. in top-tier universities of USA. I think you will most likely face such problems from now on.