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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 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.

In efficient quantum algorithms, 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 a $2^n \times 2^n$ 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.

1Thanks for the extensive response. So the answer is for me sufficiently clear for the points

Hamiltonian simulationandQuantum algorithm for linear systems of equationsbut for the other points the link with a real world problem is missing. For me most of those quantum algorithms are very theoretical and I don't see how they can be used for a real world problem. Linking them to an actual real world problem (even very simple) would already make it much clearer. – JanVdA – 2018-06-20T06:55:23.2901@JanVdA I already mentioned the real world use of Discrete Fourier Transforms. Please read that again. Problems in graph theory are extremely relevant to both computer science as well as statistical physics (QAOA). VQE would be relevant to computational chemistry. If that's not "real world" I don't know what is. – Sanchayan Dutta – 2018-06-20T06:58:18.293

I thought that the first point is not about DFT but about QFT. The links about QFT explain

whatit is not, but doesn't explainhowit can be used for a real world problem. VQE addresses indeed a real world problem, sorry for not mentioning it in my comment (I had classified it under Hamiltonian Simulation). I am aware that several problems in graph theory can be improved by a quantum algorithm but I am still looking for the first real world problem that can be addressed by such an algorithm. – JanVdA – 2018-06-20T07:33:08.157@JanVdA QFT could be used for the same purposes DFT is used. Would be simply more efficient. – Sanchayan Dutta – 2018-06-20T07:34:13.957

@JanVdA Another common use of QFT is in Quantum Phase Estimation which is in particular used for the "System of linear equations" quantum algorithm. I'm a bit busy now, but if you insist on it I'll elaborate a bit more on the answer. – Sanchayan Dutta – 2018-06-20T07:58:51.627

Thanks for a clear an human answer that doesn't involve all the problems and complexities in QC theory. – not2qubit – 2018-10-15T12:24:56.637