Abstract
An important and usual sort of search problemsis to find all marked states from an unsorted database with a largenumber of states. Grover's original quantum search algorithm is forfinding single marked state with uncertainty, and it has beengeneralized to the case of multiple marked states, as well as beenmodified to find single marked state with certainty. However, thequery complexity for finding all multiple marked states has not beenaddressed. We use a generalized Long's algorithm with high precisionto solve such a problem. We calculate the approximate querycomplexity, which increases with the number of marked states andwith the precision that we demand. In the end we introduce analgorithm for the problem on a ``duality computer'' and show itsadvantage over other algorithms.
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