Cryptanalysis is the study of weaknesses of cryptographic algorithms and it is essential to prove that a cryptographic algorithm is safe. Assuming that in the near or distant future, Quantum Computers will arrive, cryptographic algorithms will need to be safe against standard and quantum cryptanalysis.
Quantum cryptanalysis is the study of weaknesses of cryptographic algorithms which can be exploited only by algorithms running on Quantum Computers. Shore’s algorithm is the most important quantum algorithm because it will break algorithms such as RSA and most of our current IT security.
Post Quantum algorithms are mostly based on hard lattice problems which are safe against the Shor algorithm and thus are safe also in case a full Quantum Computer will be available. But research keeps advancing in the study of quantum cryptanalysis, as shown by this recent paper which, luckily for us, contained an error that invalidated the proof. As commented by Bruce Schneier here, quantum cryptanalysis has still a long way ahead to become a real threat since not only the proof should be formally correct and applicable to the real post-quantum algorithms instead of reduced models, but it should actually be able to run on a Quantum Computer, whenever it will be available.