@INPROCEEDINGS{Aaronson02quantumlower, author = {Scott Aaronson}, title = {Quantum Lower Bound for the Collision Problem}, booktitle = {}, year = {2002}, pages = {635--642} }

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Abstract

The collision problem is to decide whether a function X : . . . , n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O , but obtaining any lower bound better than# (1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.