Grover量子搜索算法的算子相干动力学分析

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"探究Grover量子搜索算法中算子的相干动力学特性" Grover量子搜索算法是量子计算领域的一个重要突破，它利用量子力学的性质，显著地提高了在无结构数据库中查找目标项的效率。这篇研究论文"Operator coherence dynamics in Grover’s quantum search algorithm"由Minghua Pan和Daowen Qiu等人发表在PHYSICAL REVIEW A 100, 012349(2019)上，深入探讨了Grover算法中算子的相干动力学。量子相干是量子信息科学中最基本的概念之一，它代表了量子系统中不同状态之间的无干扰叠加。在Grover算法中，量子相干起着关键作用，因为它允许量子位（qubits）以非经典的方式处理信息，这是经典计算机无法实现的。作者们在论文中对每次基本算子应用后状态的相干动力学进行了分析，以理解其在Grover搜索过程中的核心作用。 Grover算法的核心操作是Grover算子G的重复应用。这个算子由两部分组成：反演关于平均（Inversion About the Mean, IAM）操作和位翻转（Bit Flip）操作。IAM操作使得非目标状态与目标状态之间的相位差异逐渐增大，而位翻转操作则翻转所有状态的相位。这种精心设计的组合使得每次迭代都能更接近目标状态，而相干性在此过程中扮演了至关重要的角色。论文中，作者们可能通过量子相干度量（如相对熵相干度、l1范数等）来追踪每个操作后量子态的相干性变化。他们可能分析了算法的初期阶段，即相干性如何初始化，并在随后的迭代中如何受到破坏（由于退相干效应）。此外，他们可能还讨论了在理想情况下（无噪声环境）和现实情况（有环境交互导致的退相干）下，相干性的动态演变以及这对算法性能的影响。此外，他们可能还研究了如何通过优化操作序列或使用不同的相干保持策略来增强算法的性能。这可能包括对算法的迭代次数进行优化，以在相干性损失到不可忽视的程度之前找到目标状态，或者探索使用量子纠错码来抵抗退相干。这篇论文对Grover量子搜索算法中算子的相干动力学进行了深入研究，揭示了量子相干在搜索过程中的关键作用，这对于理解和改进量子搜索算法以及量子计算的其他应用具有重要意义。

PHYSICAL REVIEW A 100, 012349 (2019)

Operator coherence dynamics in Grover’s quantum search algorithm

Minghua Pan

1,2,3,4,*

and Daowen Qiu

1,3,5,†

Institute of Computer Science Theory, School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China

School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 510006, China

The Guangdong Key Laboratory of Information Security Technology, Sun Yat-sen University, 510006, China

School of Computer Science and Information Security, Guilin University of Electronic Technology, Guilin 541004, China

Instituto de Telecomunicações, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisbon, Portugal

(Received 8 December 2018; published 31 July 2019)

Coherence is one of the most basic concepts and resources in quantum information. To clear the role coherence

plays on the essential operator level in Grover’s search algorithm, here we discuss the coherence dynamics of

the state after each basic operator is applyied. As it is known, Grover’s search algorithm repeats the application

of Grover operator G, which can be decomposed into G = H

⊗n

O,whereH is Hadamard operator, P is

the condition phase-shift operator, and O is the oracle operator. First, we show that O and P are incoherent

operators while H

⊗n

is coherent. Second, we prove that the amount of the operator coherence of the ﬁrst H

⊗n

and the operator coherence produced or depleted by H

⊗n

depends not only on the size of the database and the

success probability, but also on target states. Moreover, the amount of operator coherence is larger when the

superposition state of targets is entangled rather than product. Third, we show that the two H

⊗n

have different

effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a

major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence

is in depletion.

DOI: 10.1103/PhysRevA.100.012349

I. INTRODUCTION

Coherence, as one of the essential quantum properties

which derived from the superposition principle of quantum

states, plays an important role in quantum physics [1,2] and

quantum information [3–5]. Coherence not only exists in bi-

partite and multipartite systems, but also exists in a single sys-

tem. However, a rigorous framework for quantifying coher-

ence was proposed by Baumgratz et al. recently [6]. From the

viewpoint of the resource theory, the authors established the

quantitative theory of coherence following the approach that

has been established for entanglement [7–9]. Following their

footsteps, the characterization and quantiﬁcation of coherence

aroused a great deal of interest [10–15]. It is worth noting that

coherence can be converted to other quantum resources, such

as entanglement and discord, by suitable operations [16–19].

Egloff et al. [20] uniﬁed entanglement, discord, and coherence

as different aspects of a single underlying resource theory.

The role of coherence in quantum algorithms has attracted

people’s attention [21–27]. For the Deutsch-Jozsa algorithm,

Hillery showed that coherence can be viewed as a resource

in the sense that a bigger amount of coherence decreases

the failure of this algorithm [21]. In deterministic quantum

computation with one qubit (DQC1), Matera et al. [22]dis-

played that the precision is directly related to the recoverable

coherence.

panmhwz@qq.com

†

Corresponding author: issqdw@mail.sysu.edu.cn

Grover’s search algorithm (GSA) [28], as one of the fa-

mous quantum algorithms, was introduced to speed up the

search process which is a quadratic improvement over its

classical one [29]. To achieve the speedup, it was proven that

multipartite entanglement is necessary for GSA operating on

pure state [30]. Therefore, a great deal of research works have

been done to investigate the properties of entanglement in

GSA [31–36]. These works showed that entanglement plays

an important role in GSA and relates to the success proba-

bility. As an important quantum resource, coherence in GSA

has been investigated [24–26]. Anand and Pati [24] studied

the relation between coherence and success probability in

the analog GSA, which was based on adiabatic Hamiltonian

evolution. They found that the success probability of the

algorithm is related to the amount of coherence. Shi et al.

[25] investigated the role of quantum coherence dynamics

in GSA. They showed that the behavior about the quantum

coherence depletion enhances the success probability. Refer-

ence [26] introduced a discrete coherence monotone named

the coherence number and showed the similar conclusion that

the enhancement of success probability consumes coherence

as iteration number increases. These works discussed the

coherence of the state after each iteration of G in GSA.

GSA repeats the application of Grover iteration G which

consists of G = H

⊗n

O, where H is Hadamard operator,

P is the condition phase-shift operator, and O is the oracle

operator. To investigate how these operators contribute to

coherence and the relationship among them and the success

probability, in this paper, we would investigate the coherence

dynamics of each basic operator in GSA. We show that O

and P are incoherent operators while two H

⊗n

are coherent

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