基于Grover算法的图着色问题求解

Grover量子搜索算法是针对非结构化搜索问题设计的著名量子算法，可用于解决图着色、最短路径排序等问题，也可以有效破译密码系统。图着色问题是最著名的NP-完全问题之一，文中首先将图着色问题转化为数学上的无向图；然后采用布尔表达式将其转换为布尔可满足性问题，介绍了量子线路图解决布尔表达式的步骤原理以及图着色问题向布尔可满足性问题的转换过程；最后在IBMQ云平台上，对三节点的2-着色问题以及4-着色问题进行模拟仿真。实验结果验证了使用Grover算法求解图着色问题的可行性，在搜索空间为8的2-着色问题和搜索空间为64的4-着色问题中，分别以近82%和97%的成功概率搜索到目标项。文中使用Grover算法解决了4-着色问题，拓展了该算法在此问题领域上的实验规模，且改进了现有实验的量子线路，使量子位成本更低，结果的成功率更高，展示了Grover算法在大型搜索问题中显著的加速效果。     

改进量子搜索算法及其在核属性求解上的应用

大部分的量子算法都必须先求解目标分量占比，否则算法的迭代次数无法确定。迭代次数自适应Grover算法有效地避开了目标分量占比求解这个步骤，但其性能相对于Grover算法来说并没有任何改善。致力于提升迭代次数自适应Grover算法的性能，提出了一种改进量子搜索算法，并将其应用于求解粗糙集的核属性。经过仿真实验，改进算法不仅实现了迭代次数自适应，而且整体上提升了获得目标分量的概率，使得获得目标分量的概率恒高于85%。     

逼近全概率Grover算法的搜索次数计算

文章研究了Grover量子搜索算法,该算法进行o(N√)次搜索后只能以大于0.5的概率获得正确结果,并且没有确定最佳的搜索次数。针对这两个问题,提出了一种确定搜索次数的计算方法,使Grover算法逼近全概率地获得搜索目标。仿真结果表明,该计算方法行之有效。     

基于固定目标权重的量子搜索算法

针对现有量子搜索算法均未考虑目标对象重要性的差异,提出了一种对已分配权重的目标对象进行搜索的量子搜索算法。首先对改变叠加态初态幅值会对迭代结果产生的影响进行了分析;在此基础上得出了保证算法有效性前提下,引入权重系数必须满足的条件;基于该条件,构建了含有目标权重信息的量子叠加态,并使算法同时保持了Grover算法的原有性质。仿真结果表明,提出的算法能够以权重值的概率,对成功搜索到的目标态得到满意的结果。     

基于量子计算机的数据库搜索

Grover提出的量子搜索算法,可以用O(N1/2)的时间复杂度完成对规模为N的非结构化数据集的搜索,这在经典计算机上需要O(N)的复杂度。其中量子黑盒(又称为Oracle)依赖于具体问题,根据数据库搜索的要求,设计了量子黑盒的内部结构和相应的量子线路,给出了适合于数据库搜索的量子算法。     

求最优装载的量子算法

随着Grover量子搜索算法的不断发展,它的实际应用价值也在逐渐体现.通过介绍量子并行计算和量子算法的基本思想以及对改进的Grover搜索算法进行研究的基础上,分析给出了一个时间复杂度为O(√N)的求解最优装载问题的量子算法.对于最优装载问题,分别用经典计算机上的贪心算法和量子算法来求解,得出了这两种算法的时间复杂度,从而可以看出量子算法相对于经典算法具有更快的搜索速度.     

Grover量子搜索算法的线路优化

Grover算法是能够高效查找到目标态的量子搜索算法，但随着搜索数据量的增大，它的量子线路面临着复杂的门分解问题。在如今的NISQ时代资源非常有限，因此线路的深度成为一种重要的度量标准。介绍了一种基于分治思想的二阶段量子搜索算法，能够在量子计算机上快速地并行运行。提出一种线路优化方法，应用块级的Oracle线路来减少迭代次数。将该方法与分治思想相结合，提出2P-Grover算法。在量子计算框架Cirq上进行模拟实验，与Grover算法进行对比。实验结果表明，2P-Grover算法能够使线路的深度至少减少60%,并且保持了较高的搜索成功率。     

改进的多目标元素量子搜索算法

Grover量子搜索算法解决了未加排序的数据库搜索问题,在2ⁿ个元素中搜索M个目标元素,其计算复杂度为O（（2ⁿ/M）^-2）,相对于经典算法实现了二次加速,但是,当目标元素个数接近2ⁿ/2时该算法成功率只达到50%。从任意相位的Grover变换从发,给出一种改进的多目标元素量子搜索算法,该算法在目标元素个数M≥2ⁿ/4时,只用一次Grover变换就能以概率1完成搜索。     

多目标元素的量子搜索算法

Grover量子搜索算法解决了未加整理的数据库搜索问题,在2ⁿ个元素中搜索M个目标元素时,计算复杂度为O（√2ⁿ/M）,相对于经典算法实现了二次加速,但Grover算法在目标元素个数接近2ⁿ/2时成功率较低。提出了一种针对多目标元素的量子搜索算法,当目标元素个数大于2ⁿ/3时,能以不低于97.36%的概率找到目标元素。     

量子搜索算法

结合Grover和Tad Hogg的算法框架,叙述了量子算法中非结构化和结构化的两类搜索算法的设计思想.在Grover算法中,结合复杂性、临界点、非单调性、完备性和鲁棒性分析总结了一些性质,分析了Grover算法的优缺点.在Tad Hogg算法中对独立于问题的映射和相位调整分别作了介绍.重点分析了一种相位调整策略,解释该策略有效的原因和适用的场合,讨论了影响算法效率的因素.在上述论述的基础上对量子搜索算法与传统搜索算法进行了比较和分析,总结了隐藏在不同量子搜索算法背后的深刻思想.     

Dynamic Grover search: applications in recommendation systems and optimization problems

In the recent years, we have seen that Grover search algorithm (Proceedings, 28th annual ACM symposium on the theory of computing, pp. 212–219, 1996) by using quantum parallelism has revolutionized the field of solving huge class of NP problems in comparisons to classical systems. In this work, we explore the idea of extending Grover search algorithm to approximate algorithms. Here we try to analyze the applicability of Grover search to process an unstructured database with a dynamic selection function in contrast to the static selection function used in the original work (Grover in Proceedings, 28th annual ACM symposium on the theory of computing, pp. 212–219, 1996). We show that this alteration facilitates us to extend the application of Grover search to the field of randomized search algorithms. Further, we use the dynamic Grover search algorithm to define the goals for a recommendation system based on which we propose a recommendation algorithm which uses binomial similarity distribution space giving us a quadratic speedup over traditional classical unstructured recommendation systems. Finally, we see how dynamic Grover search can be used to tackle a wide range of optimization problems where we improve complexity over existing optimization algorithms.     

背包问题的量子计算算法

背包问题属于NP完全问题,经典算法对规模为n的背包问题求解的时间复杂度为O（n²）。给出了基于固定相位的背包问题量子计算算法,证明了该算法在多解的情况下,能够以不低于98%的成功率在O（√N/M）步完成对规模为n的背包问题求解（M是解的数目）,而基于原始Grover算法的背包问题量子计算算法计算复杂度为O（√N/M）,成功率是50%～100%。     

Search an unsorted database with quantum mechanics

In this article, we review quantum search algorithms for unsorted database search problem. Unsorted database search is a very important problem in science and technology. In a quantum computer, a marked state can be found with very high probability using the Grover’s algorithm, or exactly with the Long algorithm. We review the Grover algorithm and related generalizations. In particular, we review the phase matching conditions in quantum search algorithm. Several issues that may cause confusion about the quantum search algorithm are also clarified.     

Quantum Partial Search of a Database with Several Target Items

We consider unstructured database separated into blocks of equal size. Blocks containing target items are called target blocks. Blocks without target items are called non-target blocks. We present a fast quantum algorithm, which finds one of the target blocks. The algorithm uses the same oracle, which the main Grover algorithm does. We study the simplest case, when each target block has the same number of target items. Our algorithm is based on Boyer, Brassard, Hoyer, and Tapp algorithm of searching database with several target items and on Grover–Radhakrishnan algorithm of partial search. We minimize the number of queries to the oracle. We analyze the algorithm for blocks of large size. In next publications we shall consider more general case when the number of target items is different in different target blocks.      

Using the primal-dual interior point algorithm within the branch-price-and-cut method

Branch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm.     

Quest for Fast Partial Search Algorithm

The Grover quantum algorithm can find a target item in a database faster than any classical algorithm. In partial search, one trades accuracy for speed, and a part of the database (a block) containing the target item can be found even faster. We consider different partial search algorithms and argue that the algorithm originally suggested by Grover and Radhakrishnan and modified by Korepin is the optimal one. The efficiency of an algorithm is measured by the number of queries to the oracle.