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.

Grover quantum searching algorithm based on weighted targets

The current Grover quantum searching algorithm cannot identify the difference in importance of the search targets when it is applied to an unsorted quantum database, and the probability for each search target is equal. To solve this problem, a Grover searching algorithm based on weighted targets is proposed. First, each target is endowed a weight coefficient according to its importance. Applying these different weight coefficients, the targets are represented as quantum superposition states. Second, the novel Grover searching algorithm based on the quantum superposition of the weighted targets is constructed. Using this algorithm, the probability of getting each target can be approximated to the corresponding weight coefficient, which shows the flexibility of this algorithm. Finally, the validity of the algorithm is proved by a simple searching example.

Effects of initial states on the quantum correlations in the generalized Grover search algorithm

We investigate the correlations between two qubits in the Grover search algorithm with arbitrary initial states by numerical simulation.Using a set of suitable bases,we construct the reduced density matrix and give the numerical expression of correlations relating to the iterations.For different initial states,we obtain the concurrence and quantum discord compared with the success probability in the algorithm.The results show that the initial states affect the correlations and the limit point of the correlations in the searching process.However,the initial states do not influence the whole cyclical trend.

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.

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 Fixed-Phase Quantum Search Algorithm with More Flexible Behavior

When the Grover’s algorithm is applied to search an unordered database, the probability of success usually decreases with the increase of marked items. To address this phenomenon, a fixed-phase quantum search algorithm with more flexible behavior is proposed. In proposed algorithm, the phase shifts can be fixed at the different values to meet the needs of different practical problems. If research requires a relatively rapid speed, the value of the phase shifts should be appropriately increased, if search requires a higher success probability, the value of the phase shifts should be appropriately decreased. When the phase shifts are fixed at , the success probability of at least 99.38% can be obtained in iterations.

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.

Grover search algorithm in an ion trap systen

Two schemes for the implementation of the two-qubit Grover search algorithm in the ion trap system are proposed. These schemes might be experimentally realizable with presently available techniques. The experimental implementation of the schemes would be an important step toward more complex quantum computation in the ion trap system.

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.

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.

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.

基于量子衍生涡流算法和T⁃S模糊推理模型的储层岩性识别

鉴于梯度下降法易陷入局部极值、普通群智能优化算法易早熟收敛,提出一种基于量子衍生涡流算法(Quantum Vortex Search Algorithm,QVSA)和T⁃S模糊推理模型的岩性识别方法,QVSA具有操作简单、收敛速度快、寻优能力强等优点,有助于T⁃S模糊推理模型获得最优参数配置,从而实现储层岩性的准确识别。首先利用具有全局搜索能力的QVSA优化T⁃S模糊推理模型的各种参数;然后利用主成分分析方法降低获取的地震属性维度;再利用优化的T⁃S模糊推理模型识别储层岩性。实验结果表明,利用反映储层特征的8个地震属性识别储层岩性时,所提方法的识别正确率达到92%,比普通BP网络方法高5.1%,同时查准率、查全率、F1分数等指标也较BP网络方法提升明显。

基于Grover量子搜索算法的MD5碰撞攻击模型

量子计算天然的并行性使其在密码学领域具有巨大潜力,而在信息安全领域,Hash函数的安全性至关重要。因此,后量子密码学概念的提出使得Hash函数在后量子时代的研究价值凸显。文章提出了一种基于Grover量子搜索算法的MD5碰撞攻击模型,运用模差分分析法,通过对输入的量子叠加态进行约束搜索以找到满足碰撞条件的目标态,再根据差分构造出与之相碰撞的消息。此外,文章探讨了量子搜索算法中的迭代过程及其关键操作,设计了相应的Oracle黑盒的量子线路,并对其进行性能分析,结果表明,与经典算法相比,该模型显著降低了攻击的计算复杂度,为后量子密码时期Hash函数的研究提供了新的思路和方法,也为防御此类攻击提供了有益参考。

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.

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.

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量子搜索算法在“嵩山”超级计算机系统中的模拟

量子计算凭借其叠加性和纠缠性,具有强大的并行计算能力。然而,目前的量子计算机不能在保证大规模量子比特处于稳定叠加态的同时,进行干涉、纠缠等量子操作。因此,当前研究和推动量子计算的有效途径是使用经典计算机模拟量子计算。Grover量子搜索算法针对无序数据库搜索问题设计,将搜索的时间复杂度加速至开平方级,能加速机器学习中的主成分分析。因此,研究和模拟Grover算法,可以促进量子计算与机器学习结合领域的发展,为Grover量子搜索算法的应用以及量子机器学习在“嵩山”超级计算机系统中的模拟奠定基础。通过研究Grover量子搜索算法,模拟出了算法的量子线路。使用Toffoli量子门优化该量子线路,在减少了两个辅助量子比特的同时,提出了Grover算法的通用量子线路。实验基于“嵩山”超级计算机系统的CPU+DCU异构体系,使用了MPI多进程+HIP多线程的两级并行策略。通过调整辅助比特在量子线路中的位置,减少了MPI进程间的通信;使用分片的方式传输数据依赖的量子态。对比串行版本,并行化的模拟算法取得了最高560.33倍的加速,首次实现了31qubits规模的Grover量子搜索算法。

Quantum search via superconducting quantum interference devices in a cavity

We propose a scheme for implementing the Grover search algorithm with two superconducting quantum interference devices （SQUIDs） in a cavity. Our scheme only requires single resonant interaction of the SQUID-cavity system and the required interaction time is very short. The simplicity of the process and the reduction of the interaction time are important for restraining decoherence.

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.