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The paper presents and discuses several important problems and challenges of quantum informatics at which algebraic methods have been, or expect to be, very useful, or even of the key importance. Some of these problems came up quite naturally, at a very straightforward quantumization of the classical informatics concepts, models and problems. However, the most attractive/important ones come up when one starts to dig deeper into the quantum world, into its phenomena, processes, laws and limitations. Among them are problems that belong to grand challenges not only of informatics (and physics), but of all current science. To get involved in exploration of such challenges is therefore much desirable for informatics community with algebraic expertise.