Rand's Objectivisms' central tenets are that reality exists independently of consciousness, that human beings have direct contact with reality through sense perception, that one can attain objective knowledge from perception through the process of concept formation and inductive logic.
Rand didn't say much about induction except that she didn't know how it works from the appendix to "Introduction to Objectivist Epistemology":
Prof. M: The question is: when does one stop? When does one decide that enough confirming evidence exists? Is that in the province of the issue of induction?
AR: Yes. That’s the big question of induction. Which I couldn’t begin to discuss—because (a) I haven’t worked on that subject enough to even begin to formulate it, and (b) it would take an accomplished scientist in a given field to illustrate the whole process in that field.
An idea that Rand freely admits she didn't understand can hardly be regarded as central to Objectivism.
Quantum physical experiment after experiment has shown that if we assume that the particles that make up ordinary objects have an objective, observer-independent existence, we get the wrong answers.
This is false. In quantum mechanics, a system in general does not have a single value for each measurable quantity. Rather, the system exists in multiple instances that are partitioned into versions, with each version having some particular measurable value. So there is a version of my coffee mug sitting in its current position on the table next to me, and another version one millimetre away from that, and many other versions of the mug. We can tell that these different versions of a system exist by doing experiments such as interference experiments. See "The Fabric of Reality" by David Deutsch, Chapter 2.
When you measure a system, you evolve so that after the measurement there are multiple versions of you, one for each version of the system you measured. Before the measurement there is no single fact of the matter about what outcome you will see. Some people are confused by this into thinking that the system doesn't exist before you do the measurement or similarly silly ideas. But this is refuted by the fact that you can manipulate the system before measuring it, which you could hardly do if it didn't exist.
Quantum mechanics doesn't refute the existence of an objective world. Rather, it claims that the world is very different from how people imagined it before quantum mechanics.
Does quantum mechanics refute the idea that I have direct contact with reality through sense perception? No. It just says that the reality I have contact with through sense perception is a lot more complex than it appears at first glance.
Nor does quantum mechanics refute the idea that we can understand reality. Arguably it puts in a better position to understand reality, since it allows the construction of computers that can simulate any physical system to any desired degree of accuracy:
Some people claim that quantum mechanics is so strange that the laws of logic don't apply to it and that a new quantum logic if needed. In reality, ordinary logic still applies to quantum mechanics, provided you take the propositions as being about the whole of physical reality, i.e. - a description in terms of all of the versions of each system and how the versions interfere with one another.