Abnormal Quantum State Search Based on Parallel Phase Comparison

We discuss the problem of filtering out abnormal states from a larger number of quantum states. For this type of problem with [Formula: see text] items to be searched, both the traditional search by enumeration and classical Grover search algorithm have the complexity about [Formula: see text]. In this letter a novel quantum search scheme with exponential speed up is proposed for abnormal states. First, a new comprehensive quantum operator is well-designed to extract the superposition state containing all abnormal states with unknown number [Formula: see text] with complexity [Formula: see text] in probability 1 via well-designed parallel phase comparison. Then, every abnormal state is achieved respectively from [Formula: see text] abnormal states via [Formula: see text] times’ measurement. Finally, a numerical example is given to show the efficiency of the proposed scheme.