A Memory-efficient Simulation Method of Grover’s Search Algorithm

Grover’s search algorithm is one of the most significant quantum algorithms,which can obtain quadratic speedup of the extensive search problems.Since Grover's search algorithm cannot be implemented on a real quantum computer at present,its quantum simulation is regarded as an effective method to study the search performance.When simulating the Grover's algorithm,the storage space required is exponential,which makes it difficult to simulate the high-qubit Grover’s algorithm.To this end,we deeply study the storage problem of probability amplitude,which is the core of the Grover simulation algorithm.We propose a novel memory-efficient method via amplitudes compression,and validate the effectiveness of the method by theoretical analysis and simulation experimentation.The results demonstrate that our compressed simulation search algorithm can help to save nearly 87.5%of the storage space than the uncompressed one.Thus under the same hardware conditions,our method can dramatically reduce the required computing nodes,and at the same time,it can simulate at least 3 qubits more than the uncompressed one.Particularly,our memory-efficient simulation method can also be used to simulate other quantum algorithms to effectively reduce the storage costs required in simulation.

O(logN) Algorithm for Amplitude Amplification and O(logN) Algorithms for Amplitude Transfer in Grover’s Algorithm

Grovers algorithm is a category of quantum algorithms that can be applied to many problems through the exploitation of quantum parallelism. The Amplitude Amplification in Grovers algorithm is T = O(N). This paper introduces two new algorithms for Amplitude Amplification in Grovers algorithm with a time complexity of T = O(logN), aiming to improve efficiency in quantum computing. The difference between Grovers algorithm and our first algorithm is that the Amplitude Amplification ratio in Grovers algorithm is an arithmetic series and ours, a geometric one. Because our Amplitude Amplification ratios converge much faster, the time complexity is improved significantly. In our second algorithm, we introduced a new concept, Amplitude Transfer where the marked state is transferred to a new set of qubits such that the new qubit state is an eigenstate of measurable variables. When the new qubit quantum state is measured, with high probability, the correct solution will be obtained.

Time Complexity of the Oracle Phase in Grover’s Algorithm

Since Grover’s algorithm was first introduced, it has become a category of quantum algorithms that can be applied to many problems through the exploitation of quantum parallelism. The original application was the unstructured search problems with the time complexity of O(). In Grover’s algorithm, the key is Oracle and Amplitude Amplification. In this paper, our purpose is to show through examples that, in general, the time complexity of the Oracle Phase is O(N), not O(1). As a result, the time complexity of Grover’s algorithm is O(N), not O(). As a secondary purpose, we also attempt to restore the time complexity of Grover’s algorithm to its original form, O(), by introducing an O(1) parallel algorithm for unstructured search without repeated items, which will work for most cases. In the worst-case scenarios where the number of repeated items is O(N), the time complexity of the Oracle Phase is still O(N) even after additional preprocessing.

Routing Protocol Based on Grover’s Searching Algorithm for Mobile Ad-hoc Networks

In Mobile Ad-hoc Networks (MANETs), routing protocols directly affect various indices of network Quality of Service (QoS), so they play an important role in network performance. To address the drawbacks associated with traditional routing protocols in MANETs, such as poor anti-fading performance and slow convergence rate, for basic Dynamic Source Routing (DSR), we propose a new routing model based on Grover's searching algorithm. With this new routing model, each node maintains a node vector function, and all the nodes can obtain a node probability vector using Grover's algorithm, and then select an optimal routing according to node probability. Simulation results show that compared with DSR, this new routing protocol can effectively extend the network lifetime, as well as reduce the network delay and the number of routing hops. It can also significantly improve the anti-jamming capability of the network.

A Four-Phase Improvement of Grover＇s Algorithm

When applying Grover＇s algorithm to an unordered database, the probabifity of obtaining correct results usually decreases as the quantity of target increases. A four-phase improvement of Grover＇s algorithm is proposed to fix the deficiency, and the unitary and the phase-matching condition are also proposed. With this improved scheme, when the proportion of target is over 1/3, the probability of obtaining correct results is greater than 97.82% with only one iteration using two phases. When the computational complexity is O（ √M/N）, the algorithm can succeed with a probability no less than 99.63%.

Adaptive Phase Matching in Grover’s Algorithm

When the Grover’s algorithm is applied to search an unordered database, the successful probability usually decreases with the increase of marked items. In order to solve this problem, an adaptive phase matching is proposed. With application of the new phase matching, when the fraction of marked items is greater , the successful probability is equal to 1 with at most two Grover iterations. The validity of the new phase matching is verified by a search example.

Grover’s Algorithm in a 4-Qubit Search Space

This paper provides an introduction to a quantum search algorithm,known as Grover’s Algorithm,for unsorted search purposes.The algorithm is implemented in a search space of 4 qubits using the Python-based Qiskit SDK by IBM.While providing detailed proof,the computational complexity of the algorithm is generalized to n qubits.The implementation results obtained from the IBM QASM Simulator and IBMQ Santiago quantum backend are analyzed and compared.Finally,the paper discusses the challenges faced in implementation and real-life applications of the algorithm hitherto.Overall,the implementation and analysis depict the advantages of this quantum search algorithm over its classical counterparts.

A hybrid quantum encoding algorithm of vector quantization for image compression

Many classical encoding algorithms of vector quantization （VQ） of image compression that can obtain global optimal solution have computational complexity O（N）. A pure quantum VQ encoding algorithm with probability of success near 100% has been proposed, that performs operations 45√N times approximately. In this paper, a hybrid quantum VQ encoding algorithm between the classical method and the quantum algorithm is presented. The number of its operations is less than √N for most images, and it is more efficient than the pure quantum algorithm.

A quantum search algorithm of two entangled registers to realize quantum discrete Fourier transform of signal processing

The discrete Fourier transform （DFT） is the base of modern signal processing. 1-dimensional fast Fourier transform （1D FFT） and 2D FFT have time complexity O（N log N） and O（N^2 log N） respectively. Since 1965, there has been no more essential breakthrough for the design of fast DFT algorithm. DFT has two properties. One property is that DFT is energy conservation transform. The other property is that many DFT coefficients are close to zero. The basic idea of this paper is that the generalized Grover＇s iteration can perform the computation of DFT which acts on the entangled states to search the big DFT coefficients until these big coefficients contain nearly all energy. One-dimensional quantum DFT （1D QDFT） and two-dimensional quantum DFT （2D QDFT） are presented in this paper. The quantum algorithm for convolution estimation is also presented in this paper. Compared with FFT, 1D and 2D QDFT have time complexity O（v/N） and O（N） respectively. QDFT and quantum convolution demonstrate that quantum computation to process classical signal is possible.

Design of quantum VQ iteration and quantum VQ encoding algorithm taking O（√N） steps for data compression

Vector quantization （VQ） is an important data compression method. The key of the encoding of VQ is to find the closest vector among N vectors for a feature vector. Many classical linear search algorithms take O（N） steps of distance computing between two vectors. The quantum VQ iteration and corresponding quantum VQ encoding algorithm that takes O（√N） steps are presented in this paper. The unitary operation of distance computing can be performed on a number of vectors simultaneously because the quantum state exists in a superposition of states. The quantum VQ iteration comprises three oracles, by contrast many quantum algorithms have only one oracle, such as Shor＇s factorization algorithm and Grover＇s algorithm. Entanglement state is generated and used, by contrast the state in Grover＇s algorithm is not an entanglement state. The quantum VQ iteration is a rotation over subspace, by contrast the Grover iteration is a rotation over global space. The quantum VQ iteration extends the Grover iteration to the more complex search that requires more oracles. The method of the quantum VQ iteration is universal.

QUANTUM COLLISION SEARCH ALGORITHM AGAINST NEW FORK-256

In order to improve the attack efficiency of the New FORK-256 function, an algorithm based on Grover's quantum search algorithm and birthday attack is proposed. In this algorithm, finding a collision for arbitrary hash function only needs O(2m/3) expected evaluations, where m is the size of hash space value. It is proved that the algorithm can obviously improve the attack efficiency for only needing O(2 74.7) expected evaluations, and this is more efficient than any known classical algorithm, and the consumed space of the algorithm equals the evaluation.

Novel Quantum Algorithms to Minimize Switching Functions Based on Graph Parti

After Google reported its realization of quantum supremacy,Solving the classical problems with quantum computing is becoming a valuable research topic.Switching function minimization is an important problem in Electronic Design Automation(EDA)and logic synthesis,most of the solutions are based on heuristic algorithms with a classical computer,it is a good practice to solve this problem with a quantum processer.In this paper,we introduce a new hybrid classic quantum algorithm using Grover’s algorithm and symmetric functions to minimize small Disjoint Sum of Product(DSOP)and Sum of Product(SOP)for Boolean switching functions.Our method is based on graph partitions for arbitrary graphs to regular graphs,which can be solved by a Grover-based quantum searching algorithm we proposed.The Oracle for this quantum algorithm is built from Boolean symmetric functions and implemented with Lattice diagrams.It is shown analytically and verified by simulations on a quantum simulator that our methods can find all solutions to these problems.

Behavior of quantum search algorithm with small phase rotations

When the Grover＇ s original algorithm is applied to search an unordered database, the success probability decreases rapidly with the increase of marked items. Aiming at this problem, a general quantum search algorithm with small phase rotations is proposed. Several quantum search algorithms can be derived from this algorithm according to different phase rotations. When the size of phase rotations are fixed at 0. 01π, the success probability of at least 99. 99% can be obtained in 0（√N/M） iterations.

Quantum Algorithm for Mining Frequent Patterns for Association Rule Mining

Maximum frequent pattern generation from a large database of transactions and items for association rule mining is an important research topic in data mining. Association rule mining aims to discover interesting correlations, frequent patterns, associations, or causal structures between items hidden in a large database. By exploiting quantum computing, we propose an efficient quantum search algorithm design to discover the maximum frequent patterns. We modified Grover’s search algorithm so that a subspace of arbitrary symmetric states is used instead of the whole search space. We presented a novel quantum oracle design that employs a quantum counter to count the maximum frequent items and a quantum comparator to check with a minimum support threshold. The proposed derived algorithm increases the rate of the correct solutions since the search is only in a subspace. Furthermore, our algorithm significantly scales and optimizes the required number of qubits in design, which directly reflected positively on the performance. Our proposed design can accommodate more transactions and items and still have a good performance with a small number of qubits.

基于B/S结构的异构数据库加密算法仿真

在处理用户与权限中心的密钥时,传统加密方法存在对应层次不均和异构数据库加密时间长等问题。对此,文章提出了一种基于B/S结构的异构数据库加密算法,首先对比了C/S与B/S结构的优缺点;其次构建了基于B/S结构的异构数据库加密体系框架。该框架包括1套全面的密钥管理机制,用于生成各级权限管理中心的密钥,以及用户在对应权限中心的密钥与用户主密钥。通过使用访问策略树对服务器端的明文数据进行加密,该算法可将需要保护的数据转换为密文形式。仿真结果显示,该算法不仅能有效完成异构数据库加密,达到保护数据库安全的目标,且响应时间相比传统方法有显著提升。

基于0.1π旋转相位Grover算法的ECC电压毛刺攻击算法

将Grover算法应用到对公钥密码的故障攻击中,提出一种基于固定相位旋转Grover量子算法,当旋转相位为0.1π时,仿真实验搜索成功率提高到99.23%。进一步与故障攻击结合,提出基于0.1π旋转相位Grover算法的椭圆曲线密码电压毛刺攻击算法,仿真实验以100%的概率攻击了NIST公布的Koblitz安全曲线K-163,其计算复杂度呈指数级降低。这是除Shor算法之外量子计算对公钥密码的一种新的有效攻击途径,有助于拓展量子计算对其他公钥密码体制的攻击。

基于BDD的Grover算法仿真

为了解决仿真量子计算过程中复杂性随量子比特数的增加呈指数级递增的问题,采用二项决策图(BDD)表示矩阵算子仿真Grover提出的量子搜索算法.BDD利用矩阵算子在量子计算过程中呈现出的结构化特性,可以高效地压缩存储空间并实现在压缩数据结构上直接进行矩阵的各种运算.利用改进的BDD实现了仿真过程需要的各种矩阵运算,用C++编写的程序对Grover算法的实例进行仿真,最后从多个角度对违反直观的实验结果进行了分析,阐述了量子算法的内在并行性.

基于混合遗传模拟退火算法的SaaS构件优化放置

目前,对于SaaS优化放置问题的研究都是假定云环境中的虚拟机的种类和数量都是确定的,即,在限定的资源范围内进行优化.然而,在公有云环境下,SaaS提供者所需要的云资源数量是不确定的,其需要根据Iaa S提供者所提供的虚拟机种类以及被部署的SaaS构件的资源需求来确定.为此,站在SaaS提供者角度,提出一种新的SaaS构件优化放置问题模型,并采用混合遗传模拟退火算法(hybrid genetic and simulated annealing algorithm,简称HGSA)对该问题进行求解.HGSA结合了遗传算法和模拟退火算法的优点,克服了遗传算法收敛速度慢和模拟退火算法容易陷入局部最优的缺点,与单独使用遗传算法和模拟退火算法相比,实验结果表明,HGSA在求解SaaS构件优化放置问题方面具有更高的求解质量.所提出的方法为SaaS服务模式的大规模应用提供了理论与方法的支撑.

基于遗传算法的随机(s，S)库存系统仿真优化

提出了一种将系统仿真常用于优化随机库存系统的方案,它将遗传算法与离散事件系统仿真相结合,用遗传算法指导控制变量的选择.在遗传算法中,针对库存系统的随机特性提出了一种新的选择算子,并设计了一个候选策略收集器.基于常用(s,S)库存控制的仿真实例表明,本方案是可行且有效的.

Grover量子搜索算法的仿真实现

利用核磁共振(NMR)实验技术来实现量子计算,是当前各种验证量子算法最为有效的方法之一,但这个方法首先必须把量子算法编译成在现代超导核磁共振谱仪上能够直接执行的NMR脉冲序列,即NMR量子计算程序。在NMR技术中通常只要施加合适的射频脉冲,便可以达到使核自旋翻转以实现某种逻辑功能的目的,该文讨论了如何设计多量子位核磁共振(NMR)脉冲序列来实现Grover量子搜索算法,并在量子仿真器(QCE)上进行了实验验证。