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 quan...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.展开更多
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 wit...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.展开更多
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...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.展开更多
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 th...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.展开更多
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 i...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.展开更多
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 ...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.展开更多
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 algorit...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.展开更多
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,...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.展开更多
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...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.展开更多
The advent of Grover’s algorithm presents a significant threat to classical block cipher security,spurring research into post-quantum secure cipher design.This study engineers quantum circuit implementations for thre...The advent of Grover’s algorithm presents a significant threat to classical block cipher security,spurring research into post-quantum secure cipher design.This study engineers quantum circuit implementations for three versions of the Ballet family block ciphers.The Ballet‑p/k includes a modular-addition operation uncommon in lightweight block ciphers.Quantum ripple-carry adder is implemented for both“32+32”and“64+64”scale to support this operation.Subsequently,qubits,quantum gates count,and quantum circuit depth of three versions of Ballet algorithm are systematically evaluated under quantum computing model,and key recovery attack circuits are constructed based on Grover’s algorithm against each version.The comprehensive analysis shows:Ballet-128/128 fails to NIST Level 1 security,while when the resource accounting is restricted to the Clifford gates and T gates set for the Ballet-128/256 and Ballet-256/256 quantum circuits,the design attains Level 3.展开更多
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 propos...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%.展开更多
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 pr...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.展开更多
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 algorith...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.展开更多
Based on Fermat＇s principle and the automatic optimization mechanism in the propagation process of light, an optimal searching algorithm named light ray optimization is presented, where the laws of refraction and ref...Based on Fermat＇s principle and the automatic optimization mechanism in the propagation process of light, an optimal searching algorithm named light ray optimization is presented, where the laws of refraction and reflection of light rays are integrated into searching process of optimization. In this algorithm, coordinate space is assumed to be the space that is full of media with different refractivities, then the space is divided by grids, and finally the searching path is assumed to be the propagation path of light rays. With the law of refraction, the search direction is deflected to the direction that makes the value of objective function decrease. With the law of reflection, the search direction is changed, which makes the search continue when it cannot keep going with refraction. Only the function values of objective problems are used and there is no artificial rule in light ray optimization, so it is simple and easy to realize. Theoretical analysis and the results of numerical experiments show that the algorithm is feasible and effective.展开更多
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 intr...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.展开更多
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 uns...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.展开更多
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 corre...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.展开更多
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...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.展开更多
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）...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.展开更多
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 Electro...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.展开更多
基金This work was supported by Funding of National Natural Science Foundation of China(Grant No.61571226,Grant No.61701229).
文摘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.
基金supported by Zhejiang Provincial Key Laboratory of Communication Networks and Applications and National Natural Science Foundation of China under Grant No.60872020
文摘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.
基金the National Natural Science Foundation of China (60773065).
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975132 and 61772295)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2019YQ01)Shandong Province Higher Educational Science and Technology Program,China(Grant No.J18KZ012).
文摘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.
基金Project supported by Fok Ying Tung Education Foundation （Grant No 81008）, the National Natural Science Foundation of China （Grant Nos 60008003 and 10225421）, and Funds from Fuzhou University, China.
文摘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.
文摘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.
文摘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.
基金supported by Sichuan Normal University,China (Grant No 06lk002)
文摘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.
基金Supported by the National High Technology Research and Development Program(No.2011AA010803)the National Natural Science Foundation of China(No.U1204602)
文摘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.
基金State Key Lab of Processors,Institute of Computing Technology,Chinese Academy of Sciences(CLQ202516)the Fundamental Research Funds for the Central Universities of China(3282025047,3282024051,3282024009)。
文摘The advent of Grover’s algorithm presents a significant threat to classical block cipher security,spurring research into post-quantum secure cipher design.This study engineers quantum circuit implementations for three versions of the Ballet family block ciphers.The Ballet‑p/k includes a modular-addition operation uncommon in lightweight block ciphers.Quantum ripple-carry adder is implemented for both“32+32”and“64+64”scale to support this operation.Subsequently,qubits,quantum gates count,and quantum circuit depth of three versions of Ballet algorithm are systematically evaluated under quantum computing model,and key recovery attack circuits are constructed based on Grover’s algorithm against each version.The comprehensive analysis shows:Ballet-128/128 fails to NIST Level 1 security,while when the resource accounting is restricted to the Clifford gates and T gates set for the Ballet-128/256 and Ballet-256/256 quantum circuits,the design attains Level 3.
基金Supported by the National Basic Research Program of China under Grant No 2013CB338002the National Natural Science Foundation of China under Grant No 11504430
文摘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%.
文摘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.
基金Supported by National Natural Science Foundation of China （ No. 60773065 ）.
文摘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.
基金Supported by Natural Science Foundation of Heilongjiang Province of China (Grant No. F200931)
文摘Based on Fermat＇s principle and the automatic optimization mechanism in the propagation process of light, an optimal searching algorithm named light ray optimization is presented, where the laws of refraction and reflection of light rays are integrated into searching process of optimization. In this algorithm, coordinate space is assumed to be the space that is full of media with different refractivities, then the space is divided by grids, and finally the searching path is assumed to be the propagation path of light rays. With the law of refraction, the search direction is deflected to the direction that makes the value of objective function decrease. With the law of reflection, the search direction is changed, which makes the search continue when it cannot keep going with refraction. Only the function values of objective problems are used and there is no artificial rule in light ray optimization, so it is simple and easy to realize. Theoretical analysis and the results of numerical experiments show that the algorithm is feasible and effective.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
文摘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.
