Quantum algorithms: A?survey of some recent results

Quantum algorithms are a field of growing interest within the theoretical computer science as well as the physics community. Surprisingly, although the number of researchers working on the subject is ever-increasing, the number of quantum algorithms found so far is quite small. In fact, the task of designing new quantum algorithms has been proven to be extremely difficult. In this paper we give an overview of the known quantum algorithms and briefly describe the underlying ideas. Roughly, the algorithms presented are divided into hidden subgroup type algorithms and in amplitude amplification type algorithms. While the former deal with problems of group-theoretical nature and have the promise to yield strong separations of classical and quantum algorithms, the latter have been proved to be a prolific source of algorithms in which a polynomial speed-up as compared to classical algorithms can be achieved. We also discuss quantum algorithms which do not fall under these two categories and give a survey of techniques of general interest in quantum computing such as adiabatic computing, lower bounds for quantum algorithms, and quantum interactive proofs.