On the Hidden Subgroup Problem as a Pivot in Quantum Complexity Theory

Quantum computing has opened the way to new algorithms that can efficiently solve problems that have always been deemed intractable. However, since quantum algorithms are hard to design, the necessity to find a generalization of these problems arises. Such necessity is satisfied by the hidden subgroup problem (HSP), an abstract problem of group theory which successfully generalizes a large number of intractable problems. The HSP plays a significant role in quantum complexity theory, as efficient algorithms that solve it can be employed to efficiently solve other valuable problems, such as integer factorization, discrete logarithms, graph isomorphism and many others. In this thesis we give a computational definition of the HSP. We then prove the reducibility of some of the aforementioned problems to the HSP. Lastly, we introduce some essential notions of quantum computing and we present two quantum algorithms that efficiently solve the HSP on Abelian groups.