The number of elements in a subset: A Grover-kronecker quantum algorithm

Abstract
In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can …nd a single marked object in a database of size N by using only O(pN ) queries of the oracle that identi…es the object. His result was generalized to the case of …nding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either M or K, M < K, our task is to determine which is the case. We show how to solve this problem with a high probability of success using only iterations of Grover’s basic step (and no other algorithm). Let m be the required number of iterations; we prove that under certain restrictions on the sizes of M and K the estimation..
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