The objective of this paper concerns at first the motivation and the method of Shor’s algorithm including remarks on quantum computing introducing an algorithmic description of the method.The corner stone of the Shor...The objective of this paper concerns at first the motivation and the method of Shor’s algorithm including remarks on quantum computing introducing an algorithmic description of the method.The corner stone of the Shor’s algorithm is the modular exponentiation that is themost computational component(in time and space).A linear depth unit based on phase estimation is introduced and a description of a generic version of a modular multiplier based on phases is introduced to build block of a gates to efficient modular exponentiation circuit.Our proposal includes numerical experiments achieved on both the IBM simulator using the Qiskit library and on quantum physical optimizers provided by IBM.The shor’s algorithm based on phase estimation succeeds in factoring integer numbers with more than 35 digits using circuits with about 100 qubits.展开更多
The elliptic curve discrete logarithm problem(ECDLP)is a popular choice for cryptosystems due to its high level of security.However,with the advent of the extended Shor’s algorithm,there is concern that ECDLP may soo...The elliptic curve discrete logarithm problem(ECDLP)is a popular choice for cryptosystems due to its high level of security.However,with the advent of the extended Shor’s algorithm,there is concern that ECDLP may soon be vulnerable.While the algorithm does ofer hope in solving ECDLP,it is still uncertain whether it can pose a real threat in practice.From the perspective of the quantum circuits of the algorithm,this paper analyzes the feasibility of cracking ECDLP using an ion trap quantum computer with improved quantum circuits for the extended Shor’s algorithm.We give precise quantum circuits for extended Shor’s algorithm to calculate discrete logarithms on elliptic curves over prime felds,including modular subtraction,three diferent modular multiplication,and modular inverse.Additionally,we incorporate and improve upon windowed arithmetic in the circuits to reduce the CNOTcounts.Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit,we focus on minimizing the number of CNOT gates in the circuit,which greatly afects the running time of the algorithm on an ion trap quantum computer.Specifcally,we begin by presenting implementations of basic arithmetic operations with the lowest known CNOT-counts,along with improved constructions for modular inverse,point addition,and windowed arithmetic.Next,we precisely estimate that,to execute the extended Shor’s algorithm with the improved circuits to factor an n-bit integer,the CNOT-count required is1237n^(3)/log n+2n^(2)+n.Finally,we analyze the running time and feasibility of the extended Shor’s algorithm on an ion trap quantum computer.展开更多
The computational accuracy and efficiency of modeling the stress spectrum derived from bridge monitoring data significantly influence the fatigue life assessment of steel bridges.Therefore,determining the optimal stre...The computational accuracy and efficiency of modeling the stress spectrum derived from bridge monitoring data significantly influence the fatigue life assessment of steel bridges.Therefore,determining the optimal stress spectrum model is crucial for further fatigue reliability analysis.This study investigates the performance of the REBMIX algorithm in modeling both univariate(stress range)and multivariate(stress range and mean stress)distributions of the rain-flowmatrix for a steel arch bridge,usingAkaike’s Information Criterion(AIC)as a performance metric.Four types of finitemixture distributions—Normal,Lognormal,Weibull,and Gamma—are employed tomodel the stress range.Additionally,mixed distributions,including Normal-Normal,Lognormal-Normal,Weibull-Normal,and Gamma-Normal,are utilized to model the joint distribution of stress range and mean stress.The REBMIX algorithm estimates the number of components,component weights,and component parameters for each candidate finite mixture distribution.The results demonstrate that the REBMIX algorithm-based mixture parameter estimation approach effectively identifies the optimal distribution based on AIC values.Furthermore,the algorithm exhibits superior computational efficiency compared to traditional methods,making it highly suitable for practical applications.展开更多
This study is trying to address the critical need for efficient routing in Mobile Ad Hoc Networks(MANETs)from dynamic topologies that pose great challenges because of the mobility of nodes.Themain objective was to del...This study is trying to address the critical need for efficient routing in Mobile Ad Hoc Networks(MANETs)from dynamic topologies that pose great challenges because of the mobility of nodes.Themain objective was to delve into and refine the application of the Dijkstra’s algorithm in this context,a method conventionally esteemed for its efficiency in static networks.Thus,this paper has carried out a comparative theoretical analysis with the Bellman-Ford algorithm,considering adaptation to the dynamic network conditions that are typical for MANETs.This paper has shown through detailed algorithmic analysis that Dijkstra’s algorithm,when adapted for dynamic updates,yields a very workable solution to the problem of real-time routing in MANETs.The results indicate that with these changes,Dijkstra’s algorithm performs much better computationally and 30%better in routing optimization than Bellman-Ford when working with configurations of sparse networks.The theoretical framework adapted,with the adaptation of the Dijkstra’s algorithm for dynamically changing network topologies,is novel in this work and quite different from any traditional application.The adaptation should offer more efficient routing and less computational overhead,most apt in the limited resource environment of MANETs.Thus,from these findings,one may derive a conclusion that the proposed version of Dijkstra’s algorithm is the best and most feasible choice of the routing protocol for MANETs given all pertinent key performance and resource consumption indicators and further that the proposed method offers a marked improvement over traditional methods.This paper,therefore,operationalizes the theoretical model into practical scenarios and also further research with empirical simulations to understand more about its operational effectiveness.展开更多
Parkinson’s disease is a neurodegenerative disorder that inflicts irreversible damage on humans.Some experimental data regarding Parkinson’s patients are redundant and irrelevant,posing significant challenges for di...Parkinson’s disease is a neurodegenerative disorder that inflicts irreversible damage on humans.Some experimental data regarding Parkinson’s patients are redundant and irrelevant,posing significant challenges for disease detection.Therefore,there is a need to devise an effective method for the selective extraction of disease-specific information,ensuring both accuracy and the utilization of fewer features.In this paper,a Binary Hybrid Artificial Hummingbird and Flower Pollination Algorithm(FPA),called BFAHA,is proposed to solve the problem of Parkinson’s disease diagnosis based on speech signals.First,combining FPA with Artificial Hummingbird Algorithm(AHA)can take advantage of the strong global exploration ability possessed by FPA to improve the disadvantages of AHA,such as premature convergence and easy falling into local optimum.Second,the Hemming distance is used to determine the difference between the other individuals in the population and the optimal individual after each iteration,if the difference is too significant,the cross-mutation strategy in the genetic algorithm(GA)is used to induce the population individuals to keep approaching the optimal individual in the random search process to speed up finding the optimal solution.Finally,an S-shaped function converts the improved algorithm into a binary version to suit the characteristics of the feature selection(FS)tasks.In this paper,10 high-dimensional datasets from UCI and the ASU are used to test the performance of BFAHA and apply it to Parkinson’s disease diagnosis.Compared with other state-of-the-art algorithms,BFAHA shows excellent competitiveness in both the test datasets and the classification problem,indicating that the algorithm proposed in this study has apparent advantages in the field of feature selection.展开更多
Shor proposed a polynomial time algorithm for computing the order of one element in a multiplicative group using a quantum computer. Based on Miller’s randomization, he then gave a factorization algorithm. But the al...Shor proposed a polynomial time algorithm for computing the order of one element in a multiplicative group using a quantum computer. Based on Miller’s randomization, he then gave a factorization algorithm. But the algorithm has two shortcomings, the order must be even and the output might be a trivial factor. Actually, these drawbacks can be overcome if the number is an RSA modulus. Applying the special structure of the RSA modulus, an algorithm is presented to overcome the two shortcomings. The new algorithm improves Shor’s algorithm for factoring RSA modulus. The cost of the factorization algorithm almost depends on the calculation of the order of 2 in the multiplication group.展开更多
Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equa...Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.展开更多
It is widely believed that Shor＇s factoring algorithm provides a driving force to boost the quantum computing research.However, a serious obstacle to its binary implementation is the large number of quantum gates. No...It is widely believed that Shor＇s factoring algorithm provides a driving force to boost the quantum computing research.However, a serious obstacle to its binary implementation is the large number of quantum gates. Non-binary quantum computing is an efficient way to reduce the required number of elemental gates. Here, we propose optimization schemes for Shor＇s algorithm implementation and take a ternary version for factorizing 21 as an example. The optimized factorization is achieved by a two-qutrit quantum circuit, which consists of only two single qutrit gates and one ternary controlled-NOT gate. This two-qutrit quantum circuit is then encoded into the nine lower vibrational states of an ion trapped in a weakly anharmonic potential. Optimal control theory（OCT） is employed to derive the manipulation electric field for transferring the encoded states. The ternary Shor＇s algorithm can be implemented in one single step. Numerical simulation results show that the accuracy of the state transformations is about 0.9919.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
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
文摘The objective of this paper concerns at first the motivation and the method of Shor’s algorithm including remarks on quantum computing introducing an algorithmic description of the method.The corner stone of the Shor’s algorithm is the modular exponentiation that is themost computational component(in time and space).A linear depth unit based on phase estimation is introduced and a description of a generic version of a modular multiplier based on phases is introduced to build block of a gates to efficient modular exponentiation circuit.Our proposal includes numerical experiments achieved on both the IBM simulator using the Qiskit library and on quantum physical optimizers provided by IBM.The shor’s algorithm based on phase estimation succeeds in factoring integer numbers with more than 35 digits using circuits with about 100 qubits.
基金supported by National Natural Science Foundation of China(Grant No.61672517)National Natural Science Foundation of China(Key Program,Grant No.61732021).
文摘The elliptic curve discrete logarithm problem(ECDLP)is a popular choice for cryptosystems due to its high level of security.However,with the advent of the extended Shor’s algorithm,there is concern that ECDLP may soon be vulnerable.While the algorithm does ofer hope in solving ECDLP,it is still uncertain whether it can pose a real threat in practice.From the perspective of the quantum circuits of the algorithm,this paper analyzes the feasibility of cracking ECDLP using an ion trap quantum computer with improved quantum circuits for the extended Shor’s algorithm.We give precise quantum circuits for extended Shor’s algorithm to calculate discrete logarithms on elliptic curves over prime felds,including modular subtraction,three diferent modular multiplication,and modular inverse.Additionally,we incorporate and improve upon windowed arithmetic in the circuits to reduce the CNOTcounts.Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit,we focus on minimizing the number of CNOT gates in the circuit,which greatly afects the running time of the algorithm on an ion trap quantum computer.Specifcally,we begin by presenting implementations of basic arithmetic operations with the lowest known CNOT-counts,along with improved constructions for modular inverse,point addition,and windowed arithmetic.Next,we precisely estimate that,to execute the extended Shor’s algorithm with the improved circuits to factor an n-bit integer,the CNOT-count required is1237n^(3)/log n+2n^(2)+n.Finally,we analyze the running time and feasibility of the extended Shor’s algorithm on an ion trap quantum computer.
基金jointly supported by the Fundamental Research Funds for the Central Universities(Grant No.xzy012023075)the Zhejiang Engineering Research Center of Intelligent Urban Infrastructure(Grant No.IUI2023-YB-12).
文摘The computational accuracy and efficiency of modeling the stress spectrum derived from bridge monitoring data significantly influence the fatigue life assessment of steel bridges.Therefore,determining the optimal stress spectrum model is crucial for further fatigue reliability analysis.This study investigates the performance of the REBMIX algorithm in modeling both univariate(stress range)and multivariate(stress range and mean stress)distributions of the rain-flowmatrix for a steel arch bridge,usingAkaike’s Information Criterion(AIC)as a performance metric.Four types of finitemixture distributions—Normal,Lognormal,Weibull,and Gamma—are employed tomodel the stress range.Additionally,mixed distributions,including Normal-Normal,Lognormal-Normal,Weibull-Normal,and Gamma-Normal,are utilized to model the joint distribution of stress range and mean stress.The REBMIX algorithm estimates the number of components,component weights,and component parameters for each candidate finite mixture distribution.The results demonstrate that the REBMIX algorithm-based mixture parameter estimation approach effectively identifies the optimal distribution based on AIC values.Furthermore,the algorithm exhibits superior computational efficiency compared to traditional methods,making it highly suitable for practical applications.
基金supported by Northern Border University,Arar,Kingdom of Saudi Arabia,through the Project Number“NBU-FFR-2024-2248-03”.
文摘This study is trying to address the critical need for efficient routing in Mobile Ad Hoc Networks(MANETs)from dynamic topologies that pose great challenges because of the mobility of nodes.Themain objective was to delve into and refine the application of the Dijkstra’s algorithm in this context,a method conventionally esteemed for its efficiency in static networks.Thus,this paper has carried out a comparative theoretical analysis with the Bellman-Ford algorithm,considering adaptation to the dynamic network conditions that are typical for MANETs.This paper has shown through detailed algorithmic analysis that Dijkstra’s algorithm,when adapted for dynamic updates,yields a very workable solution to the problem of real-time routing in MANETs.The results indicate that with these changes,Dijkstra’s algorithm performs much better computationally and 30%better in routing optimization than Bellman-Ford when working with configurations of sparse networks.The theoretical framework adapted,with the adaptation of the Dijkstra’s algorithm for dynamically changing network topologies,is novel in this work and quite different from any traditional application.The adaptation should offer more efficient routing and less computational overhead,most apt in the limited resource environment of MANETs.Thus,from these findings,one may derive a conclusion that the proposed version of Dijkstra’s algorithm is the best and most feasible choice of the routing protocol for MANETs given all pertinent key performance and resource consumption indicators and further that the proposed method offers a marked improvement over traditional methods.This paper,therefore,operationalizes the theoretical model into practical scenarios and also further research with empirical simulations to understand more about its operational effectiveness.
基金supported by the National Natural Science Foundation of China under Grant Nos.U21A20464,62066005the Innovation Project of Guangxi Graduate Education under Grant No.YCSW2023259.
文摘Parkinson’s disease is a neurodegenerative disorder that inflicts irreversible damage on humans.Some experimental data regarding Parkinson’s patients are redundant and irrelevant,posing significant challenges for disease detection.Therefore,there is a need to devise an effective method for the selective extraction of disease-specific information,ensuring both accuracy and the utilization of fewer features.In this paper,a Binary Hybrid Artificial Hummingbird and Flower Pollination Algorithm(FPA),called BFAHA,is proposed to solve the problem of Parkinson’s disease diagnosis based on speech signals.First,combining FPA with Artificial Hummingbird Algorithm(AHA)can take advantage of the strong global exploration ability possessed by FPA to improve the disadvantages of AHA,such as premature convergence and easy falling into local optimum.Second,the Hemming distance is used to determine the difference between the other individuals in the population and the optimal individual after each iteration,if the difference is too significant,the cross-mutation strategy in the genetic algorithm(GA)is used to induce the population individuals to keep approaching the optimal individual in the random search process to speed up finding the optimal solution.Finally,an S-shaped function converts the improved algorithm into a binary version to suit the characteristics of the feature selection(FS)tasks.In this paper,10 high-dimensional datasets from UCI and the ASU are used to test the performance of BFAHA and apply it to Parkinson’s disease diagnosis.Compared with other state-of-the-art algorithms,BFAHA shows excellent competitiveness in both the test datasets and the classification problem,indicating that the algorithm proposed in this study has apparent advantages in the field of feature selection.
文摘Shor proposed a polynomial time algorithm for computing the order of one element in a multiplicative group using a quantum computer. Based on Miller’s randomization, he then gave a factorization algorithm. But the algorithm has two shortcomings, the order must be even and the output might be a trivial factor. Actually, these drawbacks can be overcome if the number is an RSA modulus. Applying the special structure of the RSA modulus, an algorithm is presented to overcome the two shortcomings. The new algorithm improves Shor’s algorithm for factoring RSA modulus. The cost of the factorization algorithm almost depends on the calculation of the order of 2 in the multiplication group.
文摘Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.
基金supported by the National Natural Science Foundation of China(Grant No.61205108)the High Performance Computing(HPC)Foundation of National University of Defense Technology,China
文摘It is widely believed that Shor＇s factoring algorithm provides a driving force to boost the quantum computing research.However, a serious obstacle to its binary implementation is the large number of quantum gates. Non-binary quantum computing is an efficient way to reduce the required number of elemental gates. Here, we propose optimization schemes for Shor＇s algorithm implementation and take a ternary version for factorizing 21 as an example. The optimized factorization is achieved by a two-qutrit quantum circuit, which consists of only two single qutrit gates and one ternary controlled-NOT gate. This two-qutrit quantum circuit is then encoded into the nine lower vibrational states of an ion trapped in a weakly anharmonic potential. Optimal control theory（OCT） is employed to derive the manipulation electric field for transferring the encoded states. The ternary Shor＇s algorithm can be implemented in one single step. Numerical simulation results show that the accuracy of the state transformations is about 0.9919.
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.
