The current Grover quantum searching algorithm can not identify the difference in importance of the marked items when it is applied to an unsorted quantum database, and the probability for each marked item is equal. To solve the problem, a Grover searching algorithm based on the weighted marked items is proposed. Firstly, each marked item is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the marked items are represented as a quantum superposition state. Secondly, the novel Grover searching algorithm based on the quantum superposition of the weighting marked items is constructed. Using this algorithm the probability of getting each marked item can be approximated to the corresponding weight coefficient, which the flexibility of this algorithm is shown. Finally, the validity of the algorithm is validated by a simple searching example.
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