Which topics - what skillset in mathematics need the students to possess to be able to proceed with learning quantum mechanics without hitches like need for explaining notation or understanding the underlying calculations? I.e. what should a Mathematics teacher include in the curriculum to make the Physics teacher able to smoothly teach quantum mechanics?
There are no clearly defined mathematical prerequisites that are needed in order to learn quantum mechanics. QM can be taught at a variety of levels. Here are some examples of levels at which it can be taught:
There is a very nice book by Hewitt, Conceptual Physics, that presents a complete survey of physics for a gen ed course with no more mathematics than very basic algebra. Hewitt does a presentation of quantum mechanics that works nicely at that level.
In a typical freshman college calculus-based course, students learn quantum mechanics at a level that is slightly higher than what's in Hewitt, but that still omits a lot of mathematical detail. Students at this level do not know differential equations or linear algebra, so they aren't ready to deal with either the complete Schrodingger picture or the Heisenberg picture.
In an upper-division course for physics majors, one would expect differential equations and linear algebra as prerequisites.
At the lower mathematical levels, the approach I take is that we have two basic quantum mechanical relationships, $E=h\nu$ and $p=h/\lambda$, and everything follows from that. At the lower levels, one has to work around certain difficulties. For example, at the gen ed level, it's not realistic to discuss things in terms of complex wavefunctions. Real wavefunctions work fine for standing waves, but you can't develop a completely accurate description of traveling waves without complex wavefunctions. This means that topics like the WKB approximation have to be omitted, or treated less rigorously.
As happens with many topics in modern physics, the big issue is not mathematical background but physics background. Many mathematicians do not seem to appreciate this, and imagine that they can follow a royal road to understanding advanced topics in physics, simply because they know a lot of math.
A difficulty even at the upper-division undergrad level is that photons are inherently relativistic, and therefore there are foundational issues in trying to present the photon within the same framework as material particles. For example, there is no wavefunction for a photon, if you insist on all the usual axioms of QM. The only really rigorous and detailed way to deal with this is by teaching quantum field theory, but that's unrealistic for undergrads, not because they don't know enough math but because they haven't been exposed to enough physics. As a workaround, one approach is to use the heuristic of treating the electromagnetic wave as the wavefunction of the photon, but this is not rigorously correct.
I'd prefer an answer that contains a more comprehensive list - where first lessons in QM (not just "descriptive"/"popular science" but with actual mathematics behind the phenomena) could start, and what other problems would be solvable with extra additions. Nevertheless, the (stretch) goal is to reach quantum computing and ability to understand some easiest quantum algorithms.
I don't think there's any clear distinction of the type you're imagining between a "descriptive" treatment and one with "actual mathematics." But for the topics you have in mind, you probably don't need any QFT at all, just nonrelativistic quantum mechanics. The mathematical prerequisites are linear algebra and differential equations, and you need the ability to handle those topics over the complex numbers, not just the reals.
As stated above QM can be taught and understood at a variety of mathematical levels.
At a minimum the required concepts are:
1) Linear Algebra
2) Probability Theory
3) Calculus (basic derivatives and integrals, multivariate would be even better)
4) Newtonian physics and vector mathematics
Now all of these topics can be explored at a variety of levels also, but I would say that 1 semester at a college level for each of these is sufficient.
As for almost any course, the best person to ask is whoever is teaching the following course. They are the only ones who really are in position to know what is needed.
Yes, it will happen that they forget to mention some critical topic. And very, very often what they tell you is that "they should know about ..." when they really require the students to be intimately familiar with the application of the technique.
Try to get a sample of exams/homework (with solutions) to work through with your informant(s).
In any case, the smattering of introductory quantum mechanics I got as part of the undergraduate studies required at least three or four full semesters of math, in the lines of what @rushimg states. For an in-depth course it will be quite a bit more (and much physics besides).