Quantum search algorithms
Andris Ambainis
我们回顾了一些搜索问题的量子算法：Grover的搜索算法，它的推广到振幅放大，振幅放大的各种问题的应用和最近的量子算法基于量子行走。
We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
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We analyse the resilience of the quantum search algorithm in the presence of quantum noise modelled as trace preserving completely positive maps. We study the influence of noise on computational complexity of the quantum search algorithm. We show that only for small amounts of noise the quantum search algorithm is still more efficient than any classical algorithm.
In this work, we show that the usage of a quantum gate that gives extra information about the solution searched permits to improve the performance of the search algorithm by switching from quantum to classical search in the appropriated moment. A comparison to the case where only quantum search is used is also realized.
Traditional tree search algorithms supply a blueprint for modeling problem solving behaviour. A diverse spectrum of problems can be formulated in terms of tree search. Quantum computation, in particular Grover's algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. In this work we consider the impact of incorporating classical search concepts alongside Grover's algorithm into a hybrid quantum search system. Some of the crucial points examined include: (1) the reverberations of contemplating the use of non-constant branching…
Search-base algorithms have widespread applications in different scenarios. Grover's quantum search algorithms and its generalization, amplitude amplification, provide a quadratic speedup over classical search algorithms for unstructured search. We consider the problem of searching with prior knowledge. More preciously, search for the solution among N items with a prior probability distribution. This letter proposes a new generalization of Grover's search algorithm which performs better than the standard Grover algorithm in average under this setting. We prove that our new algorithm achieves t…
Detection of symmetry is vital to problem solving. Most of the problems of computer vision and computer graphics and machine intelligence in general, can be reduced to symmetry detection problem. Unstructured search problem can also be looked upon from symmetry detection point of view. Unstructured search can be thought as searching a binary string satisfying some search condition in an unsorted list of binary strings. In this paper unstructured search problem is reduced to function bilateral symmetry detection problem with polynomial overhead in terms of the size of the input. Keywords: Unstr…
In this paper we present a quantum algorithm which increases the amplitude of the states corresponding to the solutions of the search problem by a factor of almost two.
Quantum walk based search algorithms
在这篇综述论文中，我们给出了一个直观的处理离散时间量化的经典马尔可夫链。Grover搜索和Ambainis，Szegedy和Magniez等人的基于量子行走的搜索算法将被描述为经典搜索过程的量子模拟。我们提出了一个相当详细的描述有点简化版本的MNRS算法。最后，在查询复杂性模型中，我们展示了量子行走如何应用于以下搜索问题：元素清晰度，矩阵乘积验证，限制范围关联性，三角形和组交换性。
Quantum Search Algorithm made a big impact by being able to solve the search problem for a set with $N$ elements using only $O(\sqrt{N})$ steps. Unfortunately, it is impossible to reduce the order of the complexity of this problem, however, it is possible to make improvements by a constant factor. In this paper we pursued such improvements for search problem in sets with known probability distributions. We have shown that by using a modified version of quantum search algorithm, it is possible to decrease the expected number of iterations for such sets.
A quantum search algorithm based on the partial adiabatic evolution\cite{Tulsi2009} is provided. We calculate its time complexity by studying the Hamiltonian in a two-dimensional Hilbert space. It is found that the algorithm improves the time complexity, which is $O(\sqrt{N/M})$, of the local adiabatic search algorithm\cite{Roland2002}, to $O(\sqrt{N}/M)$.
A new quantum lower bound method, with an application to strong direct product theorem for quantum search
我们提出了一种新的方法来证明量子查询算法的下界。通过分析问题的特征空间结构，提出了一种改进的对抗法。利用这种新方法，我们证明了量子搜索的一个强直积定理。这一结果先前由Klauck、Spalek和de Wolf（quant-ph/0402123）使用多项式方法证明。以前没有使用对抗方法的证明。
We present experimental results on running 4-qubit unstructured search on IBM quantum processors. Our best attempt attained probability of success around 24.5%. We try several algorithms and use the most recent developments in quantum search to reduce the number of entangling gates that are currently considered the main source of errors in quantum computations. Comparing theoretical expectations of an algorithm performance with the actual data, we explore the hardware limits, showing sharp, phase-transition-like degradation of performance on quantum processors. We conclude that it is extremely…
Optimization of Partial Search
量子Grover搜索算法可以在数据库中找到一个目标项目比任何经典算法更快。人们可以用准确性换取速度，甚至更快地找到包含目标项的数据库的一部分（一个块），这就是部分搜索。部分搜索算法最近由Grover和Radhakrishnan提出。这里我们优化它。搜索算法的效率是通过对oracle的查询次数来衡量的。作者提出了新版本的Grover-Radhakrishnan算法，该算法使用最少数量的查询。该算法可以在与Grover算法相同的硬件上运行。
A Quantum Random Walk Search Algorithm
图上的量子随机游动已经显示出许多有趣的性质，包括与经典同行相比的指数快速命中时间。然而，目前还不清楚如何使用这些新的属性，以获得一个算法的速度超过经典算法。在本文中，我们提出了一个量子搜索算法的基础上的量子随机游走架构，提供了这样的加速。它将被证明，该算法执行Oracle搜索的数据库上的$N$项目与$O(\sqrt{N})$调用的Oracle，产生类似于其他量子搜索算法的速度。看来量子随机游走公式具有相当大的灵活性，为其他可能新颖的量子算法的发展提供了有趣的机会。
This article highlights some of the key operating principles of Grover algorithm. These principles were used to develop a new oracle function, that illustrates the possibility of using Grover algorithm for solving more realistic and specific search problems, like searching for a solution to a simple mathematical equation.
Quantum search without entanglement
量子变量的纠缠通常被认为是获得信息处理任务（如搜索数据库）量子加速的先决条件。本文介绍了量子搜索的方法，提供了一个速度超过经典的方法，但不需要纠缠。这些方法反而依赖于干扰来提供加速。没有纠缠的搜索是有代价的：尽管它们的性能优于类似的经典设备，但执行搜索的量子设备不是通用的量子计算机，并且需要比使用纠缠操作的量子计算机更大的开销。没有纠缠的量子搜索与使用波的经典搜索进行了比较。
Implementation of a Quantum Search Algorithm on a Nuclear Magnetic Resonance Quantum Computer
我们展示了一个基于胞嘧啶的双量子比特NMR量子计算机上的量子搜索算法的实现。
Ion Trap Proposal for Quantum Search
在这封信中，我们表明，激光哈密顿可以执行量子搜索。我们还证明了量子搜索的过程是初始态和目标态之间的共振，这意味着自然界已经有了一个量子搜索系统来使用能量的跃迁。此外，我们还提出了基于囚禁离子的量子搜索算法的具体实现方案。
A Non-Oracle Quantum Search Algorithm and Its Experimental Implementation
Grover的算法取得了巨大的成功。但是由于Grover的Oracle，量子搜索算法仍然不是完整的算法。我们关注这个问题，并在绝热模型中提出了一种新的量子搜索算法。我们分析了量子搜索算法中的一般困难，并展示了如何在本算法中解决这些困难。该算法可以处理单解和多解的搜索而无需修改。我们还在NMR量子计算机上实现了该算法。这是第一个进行真正的量子数据库搜索而不是标记态搜索的实验。
In this work, a constant time Quantum searching algorithm over a datasets is proposed and subsequently the algorithm is executed in real chip quantum computer developed by IBM Quantum experience (IBMQ). QISKit, the software platform developed by IBM is used for this algorithm implementation. Quantum interference, Quantum superposition and π phase shift of quantum state applied for this constant time search algorithm. The proposed quantum algorithm is executed in QISKit SDK local backend 'local_qasm_simulator', real chip 'ibmq_16_melbourne' and 'ibmqx4' IBMQ. Result also suggest that real chip …
We propose a continuous time quantum search algorithm using a generalization of the Jaynes-Cummings model. In this model the states of the atom are the elements among which the algorithm realizes the search, exciting resonances between the initial and the searched states. This algorithm behaves like Grover's algorithm; the optimal search time is proportional to the square root of the size of the search set and the probability to find the searched state oscillates periodically in time. In this frame, it is possible to reinterpret the usual Jaynes-Cummings model as a trivial case of the quantum …
A Hybrid Quantum Search Engine: A Fast Quantum Algorithm for Multiple Matches
在本文中，我们将提出一个量子算法，该算法在搜索空间内的多个匹配的情况下非常有效，在很少匹配的情况下，该算法执行经典。这使我们能够提出一个混合量子搜索引擎，它集成了Grover算法和这里提出的算法，具有比任何纯经典或量子搜索算法更好的一般性能。
We consider in this paper the possibility of embedding a quantum search algorithm within a classical binary search framework. The result appears promising: taking full advantage of quantum parallelism, we show that it may actually be possible to search an unstructured list in O(lg(N)), provided we are willing to restart the quantum search multiple times with a different sequence of qubits and perform a series of non-unitary measurements at the end of each.
In this work I describe a classical analog of Grover's quantum searching algorithm, explaining why a quantum algorithm should be able to perform search in O(sqrtN) steps and also acting as a useful pedagogic demonstration.
Quantum searching's underlying SU(2) structure and its quantum decoherence effects
用Grover的量子搜索算法对标记态的搜索操作是SU（2）群中的一个元素，它作用在由标记态和非标记集体态构成的二维空间上.基于这个基本结构，我们以一种相当简洁的方式得到了各种量子搜索算法中步骤的精确界。量子搜索算法的这种重新表述也使得能够详细分析其与环境耦合所引起的退相干效应。结果表明，在完全退相干的情况下，环境不仅会使量子搜索失效，即发现标记态的概率不变，而且还可能使量子搜索算法比预期的要差：当环境表现出量子特性时，会降低这种概率。
Quantum walks have been very successful in the development of search algorithms in quantum information, in particular in the development of spatial search algorithms. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. Here, we demonstrate that continuous-time quantum walk search is possible in two-dimensions by changing the search topology to a graphene lattice, utilising the Dirac point in the energy spectrum. This is made possible by making a change to standard methods of marking a …
Quantum mechanical search induces polynomial speed up in an unsorted database search process. In case of classical linear search the computational time increases with the dimensionality of the query. However, quantum parallelism, inherent to quantum systems, does not let multi-dimensional query processing affect the computational time of the quantum search algorithm. In this letter, a two-dimensional search process has been proposed. It has been shown that a two-dimensional search process can be accomplished without increasing the computational time due to implicit quantum parallelism
We present several families of graphs that allow both efficient quantum walk implementations and efficient quantum walk based search algorithms. For these graphs, we construct quantum circuits that explicitly implement the full quantum walk search algorithm, without reference to a `black box' oracle. These circuits provide a practically implementable method to explore quantum walk based search algorithms with the aim of eventual real-world applications. We also provide a numerical analysis of a quantum walk based search along a twisted toroid family of graphs, which requires O($\sqrt{n}$ log($…
Subspace-Search Variational Quantum Eigen solver (SSVQE) is a searching method of multiple states and relies on the unitarity of transformations to ensure the orthogonality of output states for multiple states. Therefore, this method is thought to be a promising method for quantum chemistry because ordinary Variational Quantum Eigen solver (VQE) can only calculate the excited states step by step from the ground state based on Variational Quantum deflation (VQD). We compare the advantage of VQE, SSVQE with/without the constraint term and/or Tabu search term, which is added by the Lagrange multi…
Quantum search algorithm (also known as Grover's algorithm) lays the foundation for many other quantum algorithms. Although it is very simple, its implementation is limited on noisy intermediate-scale quantum (NISQ) processors. Grover's algorithm was designed without considering the physical resources, such as depth, in the real implementations. Therefore, Grover's algorithm can be improved for NISQ devices. In this paper, we demonstrate how to implement quantum search algorithms better on NISQ devices. We present detailed benchmarks of the five-qubit quantum search algorithm on different quan…
Efficient State Preparation via Ion Trap Quantum Computing and Quantum Searching Algorithm
利用离子阱量子计算和Grover的量子搜索算法，在离子阱中制备一个几率接近1的量子态.
Asymptotic Quantum Search and a Quantum Algorithm for Calculation of a Lower Bound of the Probability of Finding a Diophantine Equation That Accepts Integer Solutions
几个数学问题可以建模为在数据库中的搜索。一个例子是寻找函数的最小值的问题。已经提出了解决这个问题的量子算法，它们都使用量子搜索算法作为子程序，并实现了几个中间测量。在这项工作中，提出了一种新的量子算法，用于寻找一个函数的最小值，其中量子搜索不被用作子程序，只需要一个测量。这也被称为渐近量子搜索。作为一个例子，我们提出了一个量子算法的基础上的渐近量子搜索和量子计数能够计算的概率下限找到一个丢番图方程的整数解。
Faster Database search in Quantum Computing
论文已经撤回
Could Grover's quantum algorithm help in searching an actual database?
我调查它是否会在技术上和经济上有意义，建立数据库搜索引擎的基础上格罗弗的量子搜索算法。答案并不完全确定，但我认为是否定的。
Grover's quantum searching algorithm is optimal
本文将Boyer等人（quant-ph/9605034）关于量子搜索的紧界改进为一个匹配界，从而证明了对于任何成功概率Grovers量子搜索算法都是最优的。例如，为了接近确定的成功，我们必须查询oracle pi/4 sqrt{N}次，其中N是搜索空间的大小。我还表明，不幸的是，量子搜索不能更好地并行化，而不是将搜索空间的不同部分分配给独立的量子计算机。早期的结果为更有效的并行化提供了可能性。
An Exponentially Fast Quantum Search Algorithm
这篇论文已被作者撤回。
Solving the quantum search problem in polynomial time on an NMR quantum computer
量子搜索问题是一个重要的问题，因为一般的NP问题可以有效地解决一个unsorted量子搜索算法。这里已经表明，量子搜索问题可以在NMR量子计算机上在多项式时间内解决。NMR系综量子计算是基于量子力学幺正动力学，即封闭量子系统和它的系综都服从同一个量子力学幺正动力学，而不是基于经典NMR量子计算的伪纯态或有效纯态。基于这一新原理，我们发展了常规的NMR多量子谱，从实验上解决了搜索问题。首先将搜索问题的解信息加载到由预言机幺正操作和预言机无关幺正操作构造的幺正演化传播子上，用于激发自旋系综中的多重量子相干性.然后利用多量子光谱技术从实验上提取溶液信息.讨论了如何增强量子搜索核磁共振多量子脉冲序列的输出信号，并提出了一些增强核磁共振信号的方法。本文的工作将有助于常规高场核磁共振机有效地解决量子搜索问题。
Robustness of the adiabatic quantum search
研究了局域绝热量子搜索对搜索哈密顿量的瞬时本征基中退相干的鲁棒性。我们证明了理想的封闭情况下的渐近时间复杂性被保存，只要存在的哈密顿动力学。在纯退相干的特殊情况下，环境监测的搜索哈密顿量，它表明，局部绝热量子搜索执行的经典搜索。
A new adiabatic quantum search algorithm
本文提出了一种新的绝热量子搜索算法。使用简化的复杂度分析优化了新算法。
Quantum search algorithm as an open system
我们将量子搜索算法作为一个开放系统来研究。更具体地说，我们分析了该算法的外部单色场的响应，以及由测量过程引入的退相干。我们发现，搜索算法是强大的许多外部相互作用。
Grover's quantum search algorithm provides a quadratic speedup over the classical one. The computational complexity is based on the number of queries to the oracle. However, depth is a more modern metric for noisy intermediate-scale quantum computers. We propose a new depth optimization method for quantum search algorithms. We show that Grover's algorithm is not optimal in depth. We propose a quantum search algorithm, which can be divided into several stages. Each stage has a new initialization, which is a rescaling of the database. This decreases errors. The multistage design is natural for p…
Algorithms for searching and sorting data sets on quantum annealing systems are presented. Search algorithms for unordered data sets are developed. A sorting algorithm for data sets is provided, with a consideration of sort stability. Scalability of the algorithms, considering both the number of qubits required and the qubit connectivity, is characterized as a function of problem size.
Statistical comparison of ensemble implementations of Grover's search algorithm to classical sequential searches
我们比较伪纯状态集成实现，量化的初始极化和合奏大小，格罗弗的搜索算法的概率经典顺序搜索算法的成功和失败概率。我们提出了一个标准，用于量化的集成实现所使用的资源，通过在整个集成的Oracle调用的总数，并使用此作为与经典的搜索算法进行比较的基础。我们确定一个临界极化的界限，使合奏算法成功的概率比概率经典的顺序搜索。我们的结果表明，临界极化标度为N^（-1/4），其中N是数据库大小，对于典型的室温溶液态NMR，极化使得Grover算法的系综实现在N > 10 ^（22）时是有利的
Practical Quantum Search by Variational Quantum Eigensolver on Noisy Intermediate-scale Quantum Hardware
针对Grover搜索在NISQ硬件上因迭代深致噪声累积的问题，提出用经典优化器更新代替量子迭代的混合架构，通过最小化oracle哈密顿量期望值优化浅层参数化电路，在大于5比特时仍保持高成功概率。
Fixed Phase Quantum Search Algorithm
使用固定算子构建量子器件是简化硬件结构的必要条件。量子搜索引擎也不例外。在本文中，一个固定相位的量子搜索算法，搜索M个匹配的大小为N的非结构化搜索空间。在标准幅度放大中选择1.91684π的相移将使该技术表现得更好，从而在O（sqrt（N/M））中获得至少99.58%的成功概率，优于任何已知的固定算子量子搜索算法。该算法将能够处理搜索空间中的单个匹配或多个匹配。该算法将在O（sqrt（N/M））内找到匹配，无论匹配的数量是否事先已知。
We examine the effect of network heterogeneity on the performance of quantum search algorithms. To this end, we study quantum search on a tree for the oracle Hamiltonian formulation employed by continuous-time quantum walks. We use analytical and numerical arguments to show that the exponent of the asymptotic running time $\sim N^β$ changes uniformly from $β=0.5$ to $β=1$ as the searched-for site is moved from the root of the tree towards the leaves. These results imply that the time complexity of the quantum search algorithm on a balanced tree is closely correlated with certain path-based cen…
This extended abstract reports on on-going research on quantum algorithmic approaches to the problem of generalised tree search that may exhibit effective quantum speedup, even in the presence of non-constant branching factors. Two strategies are briefly summarised and current work outlined.
Grover's search algorithm and the quantum measurement problem
有人建议，测量过程中的个人成果可以理解在标准的量子力学的测量仪器，作为一个量子计算机，执行格罗弗的搜索算法。
In this paper, we show reduction methods for search algorithms on graphs using quantum walks. By using a graph partitioning method called equitable partition for the the given graph, we determine "effective subspace" for the search algorithm to reduce the size of the problem. We introduce the equitable partition for quantum walk based search algorithms and show how to determine "effective subspace" and reduced operator.
Non adiabatic quantum search algorithms
我们提出了两个新的连续时间量子搜索算法类似的绝热搜索算法，但现在没有绝热演化。我们发现，这两种算法的工作范围很广的参数的哈密顿量的值，其中之一，作为一个额外的功能，对于大于一个特征的时间值，它会收敛到一个状态，可以接近搜索状态。
A simple circuit implementation of the oracle for Grover's quantum search of a real unstructured classical database is proposed. The oracle contains a kind of quantumly accessible classical memory, which stores the database.
Quantum computing has noteworthy speedup over classical computing by taking advantage of quantum parallelism, i.e., the superposition of states. In particular, quantum search is widely used in various computationally hard problems. Grover's search algorithm finds the target element in an unsorted database with quadratic speedup than classical search and has been proved to be optimal in terms of the number of queries to the database. The challenge, however, is that Grover's search algorithm leads to high numbers of quantum gates, which make it infeasible for the Noise-Intermediate-Scale-Quantum…
Space Searches with a Quantum Robot
量子机器人被描述为移动量子计算机和辅助系统，可以在任意环境中移动并与之交互。它们的动力学被给定为由交替计算和动作阶段序列组成的任务。一个任务的例子被认为是一个量子机器人搜索空间区域，找到一个系统的位置。通过考虑使用Grover算法来处理搜索结果，研究了搜索可以比经典搜索更有效的可能性。对于可逆搜索，这是有问题的，原因有二。一是消除搜索过程中产生的纠缠。另一个是即使可以避免纠缠，在2-D空间区域的搜索过程并不比经典搜索更有效。然而，3维或更多维空间区域的量子搜索比经典搜索更有效。简要总结了量子机器人独立于这些结果而有趣的原因。
Grover's quantum search algorithm drives a quantum computer from a prepared initial state to a desired final state by using selective transformations of these states. Here, we analyze a framework when one of the selective trasformations is replaced by a more general unitary transformation. Our framework encapsulates several previous generalizations of the Grover's algorithm. We show that the general quantum search algorithm can be improved by controlling the transformations through an ancilla qubit. As a special case of this improvement, we get a faster quantum algorithm for the two-dimensiona…
Adiabatic quantum search algorithm for structured problems
量子计算的研究一直是由希望找到有效的量子算法来解决经典难题。在这种情况下，量子算法的局部绝热演化已被证明可以解决一个非结构化的搜索问题的二次加速超过经典的搜索，就像格罗弗的算法。在本文中，我们研究如何结构的搜索问题可能被利用，以进一步提高这些量子绝热算法的效率。我们表明，通过嵌套一个部分搜索到一个全局搜索的变量减少，它是可能的设计量子绝热算法的复杂性，虽然仍然是指数级的，增长与减少的顺序在问题的大小。
Quantum searching with continuous variables
提出了一种连续变量的快速量子搜索算法。其结果是量子连续变量模拟Grover算法最初提出的量子比特。阿达玛的连续变量模拟（即，傅立叶变换）操作与关于量子态的平均值的反演结合使用，以允许以给出使用经典连续变量的搜索算法的平方根加速的方式近似识别未知的量子态。此外，我们表明，这种量子搜索算法是强大的广义傅立叶变换连续变量。
A Framework for Structured Quantum Search
本文提出了一种用于一般组合搜索的量子算法，它利用搜索空间的底层结构来增加找到解的概率。该算法展示了相干量子系统如何与抽象搜索空间的底层结构相匹配，并且在分析上比以前的结构化搜索方法更简单。该算法的经验评估与各种搜索问题，并证明是特别有效的搜索与许多限制。此外，该算法提供了一个简单的框架，利用搜索算法。它还表现出相同的相变的搜索难度，发现复杂的经典搜索方法，这表明它是有效地使用的问题结构。
An Algorithm to Speed Up Quantum Search
由于研究工作仍在进行中，该论文已被撤回。
An Alternative Algorithm for the Database Search Problem on a Quantum Computer
论文已经撤回
Similarity between Grover's quantum search algorithm and classical two-body collisions
通过研究量子搜索算法中平均运算的反演性质，发现量子搜索与两个刚体碰撞过程的相似性。从这种相似性出发，讨论了一些相关问题。
Local Search Methods for Quantum Computers
局部搜索算法利用了搜索状态之间的邻域关系，通常能很好地解决各种NP难的组合搜索问题。本文展示了量子计算机如何也可以使用这些邻域关系。这样的局部量子搜索的一个例子是经验评估的可满足性（SAT）问题，并示出为高度约束的情况下是特别有效的。对于具有中等数量约束的问题，尽管局部方法使用的搜索空间小得多，但它在利用问题结构方面比增量量子方法的效率要低。
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