The idea that quantum mechanics fundamentally challenges the rules of logic was popular for a while, but has fallen out of favor in recent years.
While intuitively it might seem that quantum superposition (i.e something being in more than one base state at the same time) is what challenges the rules of logic, by invalidating the law of non-contradiction, this is not the case. An electron in a superposition of spin |+> and spin |-> might seem like a contradiction, but it can simply be treated as being in a distinct third state of being "either |+> or |->".
The real challenge to classical logic from Quantum Mechanics comes from the uncertainty principle, which leads to situations where:
[(p and x) or (p and y)] is different from [p and (x or y)].
Birkhoff and Von Neuman proposed in the 1930s that the paradoxes of Quantum Mechanics can be explained if we abandoned classical logic and used some form of Quantum logic instead (Birkhoff, Garrett; von Neumann, John. "The Logic of Quantum Mechanics". Ann. Math. 37 (4): 823–843.). Such a Quantum logic would change or abandon all together some of the rules of classical logic, and would be a perfect case of logical axioms arrived at by observation.
Hilary Putnam discussed this in depth in his paper "Is Logic Empirical?", later republished as "The Logic of Quantum Mechanics." ("The Logic of Quantum Mechanics" in Mathematics, Matter and Method (1975), pp. 174-197). In it, he argued that, just as empirical physical results - relativity - forced us to abandon Euclidean geometry, so it is possible that the results of quantum mechanics will force us to abandon classical logic.
Von Neumann, Birkhoff and Putnam all seemed to have moved away from this position in later years. Quantum logic didn't really solve any physics problems or provide any new insights into the epistemic challenges posed by Quantum Mechanics.
Although Quantum logic is still an active field of study up to the present day, it does not get much attention from most philosophers and has been abandoned completely by physicists. The only people who are paying attention to it are pure mathematicians who study different types of logic as mathematical structures (Quantum Logic's relation to Orthomodular Lattices and its relation to Fuzzy Sets), without paying any attention to the semantic or epistemic value of such non-classical logics. See for example "Quantum Logic, M.L. Dalla Chiara, R. Giuntini, arXiv:quant-ph/0101028".
You will occasionally come across the term "Quantum Logic" used in the quantum computing literature, but by that they do not mean the QL of Von Neumann and Birkhoff. Instead, what is meant by that is classical Boolean logic applied to quantum states and quantum bits.