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

TL;DR

This paper introduces a new quantum algorithm for linear combination of unitaries with complexity depending only on the number of unitaries and precision, leading to new insights into Grover's search and efficient quantum state preparation.

Contribution

The paper proposes a novel LCU algorithm with improved complexity dependence and applies it to develop new quantum search algorithms and efficient quantum state preparation methods.

Findings

Complexity depends only on number of unitaries and precision.

New quantum algorithms for search problems inspired by Grover's algorithm.

Efficient preparation of quantum states from classical vectors.

Abstract

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