Element Distinctness Revisited
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The element distinctness problem is the problem of determining whether the elements of a list are distinct. Classically, it requires N queries, where N is the number of elements. In the quantum case, it is possible to solve the problem in O(N^2/3) queries. The problem can be extended by asking whether there are k colliding elements, known as element k-distinctness. This work obtains optimal values of two critical parameters of Ambainis' seminal algorithm [SIAM Journal on Computing, 37(1):210-239, 2007]. The first critical parameter is the number of repetitions of the algorithm's main block, which inverts the phase of the marked elements and calls a subroutine. The second parameter is the number of quantum walk steps interlaced by oracle queries. We show that, when the optimal values of the parameters are used, the algorithm's success probability is 1-O(N^1/(k+1)), quickly approaching to 1.
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