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Linear combination of unitaries (LCU for short) is one of the most importanttechniques in designing quantum algorithms. In this paper, we propose a newquantum algorithm in three different forms to achieve LCU. Different fromprevious algorithms [Childs-linear-system,Clader,Long11], the complexity nowonly depends on the number of the unitaries and the precision. So it will playmore important role in the design of quantum algorithms when the number ofunitaries is small, such as quantum iteration algorithms. %Since in iterationalgorithms, $m$ often refers to the iteration steps, which is small if theiteration algorithms are efficient. Moreover, as an application of the new LCU,three new quantum algorithms to the searching problem will be proposed, whichwill provide us new insights into Grover's searching algorithm. We also showthat the problem of LCU is closely related to the problem of if we canefficiently implement $U^t$ for $0<t<1$ when $U$ is an efficiently implementedunitary operator? This problem is not hard to solve. However, it becomesinefficient when it contains a strict requirement on precision, such as inGrover's algorithm. Finally, as an application of the new LCU technique, wewill show that the quantum state of any real classical vector can be preparedefficiently in quantum computer. So this solves the "input problem" in quantumcomputer efficiently.

From linear combination of quantum states to Grover's searching algorithm