A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Quantum Algorithms for Element Distinctness
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Polynomial Degree vs. Quantum Query Complexity
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
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In 1994, Peter Shor discovered a polynomial time quantum algorithm for factoring and discrete logarithm. Two years later, in 1996, Lov Grover discovered a search algorithm which is quadratically better than conventional search. By now, each of the two algorithms has developed into a line of research which goes well beyond the original algorithm. Shor's algorithm has inspired the study of quantum Fourier sampling which has resulted in more quantum algorithms for number-theoretic and group-theoretic problems. Grover's algorithm has developed into the area of quantum query algorithms.I will survey the developments in quantum query algorithms. The topics will include: applications of Grover's algorithm to element distinctness and other problems, lower bounds on quantum algorithms and the use of quantum random walks to design better search algorithm. I will also describe how some of techniques in this area can be used as "quantum black boxes" in an otherwise classical algorithm.