This question is very similar as Is there any general statement about what kinds of problems can be solved more efficiently using a quantum computer?
But the answers provided to that questions mainly looked at it from a theoretical / mathematical point of view.
For this question I am more interested in the practical / engineering point of view. So I would like to understand what kind of a problems can be more efficiently solved by a quantum algorithm than you would currently be able to do with a classical algorithm. So I am really assuming that you do not have all knowledge about all possible classical algorithms that could optimally solve the same problem !
I am aware that the quantum zoo expresses a whole collection of problems for which there exists a quantum algorithm that runs more efficiently than a classical algorithm but I fail to link these algorithms to real world problems.
I understand that Shor's factoring algorithm is very important in the world of cryptography but I have deliberately excluded cryptography from the scope of this question as the world of cryptography is a very specific world which deserves his own questions.
With efficient quantum algorithm, I mean that there must at least be one step in the algorithm that must be translated to a quantum circuit on a n-qubit quantum computer. So basically this quantum circuit is creating a $2^n$ x $2^n$ matrix and its execution will give one of the $2^n$ possibilities with a certain possibility (so different runs might give different results - where the likely hood of each of the $2^n$ possibilities is determined by the constructed $2^n$ x $2^n$ Hermitian matrix.)
So I think to answer my question there must be some aspect / characteristic of the real world problem that can be mapped to an $2^n$ x $2^n$ Hermitian matrix. So what kind of aspects / characteristics of a real world problem can be mapped to such a matrix ?
With real world problem I mean an actual problem that might be solved by a quantum algorithm, I don't mean a domain where there might be a potential use of the quantum algorithm.