First off please bear with me that I am entering the realm of learning about both of these topics. However from what I have gathered objectivism says that reality is the only reality and that there is nothing but, thus being no point in finding something else (most religions in regards to their "afterlife"), you know, A is A. So my understanding of this is that unless something can be proven then it does not exist. As stated on the Ayn Rand Lexicon, "Reality exists as an objective absolute—facts are facts, independent of man’s feelings, wishes, hopes or fears." This would be saying that something such as the Quantum Immortality theory (basically says that if a person dies in a "schrodinger's box" then they will simply wake up in a tangent created by a superposition dividing) which can not be proven, at least to my knowledge, would not be considered reality and thus should not be sought after to prove and explain. SOMEWHAT SPOILERS AHEAD TO ATLAS SHRUGGED I'm at the part now where Dagny is talking to Galt and all of her old friends and his new friends on why they left it all behind. So if stuff like this is further explained and answered later in this book or in The Fountainhead please answer starting off saying that, then continue to answer if it dives deeper than what the book explains.
From your post
"Reality exists as an objective absolute—facts are facts, independent of man’s feelings, wishes, hopes or fears."
And the wikipedia article on objectivism:
Objectivism's central tenets are that reality exists independently of consciousness, that human beings have direct contact with reality through sense perception,[...]
So Realism is a central component of Objectivism.
There are more than one interpretation of Quantum Mechanics (henceforth QM), so answering the question how do objectivism and QM mix is contingent on which interpretation you subscribe to. At least two interpretations (Copenhagen, Wigner) assign an active role to the observer in the collapse of the wave-function, meaning that they don't go well with Objectivism's hard realist stance.
Moreover, one of QM's central results, Heisenberg's uncertainty principle, dictates a Kantian ontology, where the human subject can never access the things in themselves. Again this flies directly in the face of objectivism's absolute realism, and in particular is in direct conflict with "[..]human beings have direct contact with reality through sense perception,[..]".
In so far as objectivism is a philosophical theory of anything at all, it is mainly a values and ethics theory that Ayn Rand used to build her political worldview on. The ontological and epistemic positions about realism and materialism that are associated with objectivism are merely cosmetic presuppositions designed to inflate it into a complete system, and to try to elevate it above being just a bunch of folk maxims.
Further relevant research on Objectivism, QM and relativity can be found in Sokal's groundbreaking work on Quantum Hermeneutics.