Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-ve...Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-vector product,matrix-matrix product,the sum of two matrices,and the calculation of determinant and inverse matrix.We encode the matrix entries into the probability amplitudes of the pure initial states of senders.After applying proper unitary transformation to the complete quantum system,the desired result can be found in certain blocks of the receiver’s density matrix.These quantum protocols can be used as subroutines in other quantum schemes.Furthermore,we present an alternative quantum algorithm for solving linear systems of equations.展开更多
Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage,widely employed in the era of noisy intermediate-scale quantum computing.This study presents an advanced va...Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage,widely employed in the era of noisy intermediate-scale quantum computing.This study presents an advanced variational hybrid algorithm(EVQLSE)that leverages both quantum and classical computing paradigms to address the solution of linear equation systems.Initially,an innovative loss function is proposed,drawing inspiration from the similarity measure between two quantum states.This function exhibits a substantial improvement in computational complexity when benchmarked against the variational quantum linear solver.Subsequently,a specialized parameterized quantum circuit structure is presented for small-scale linear systems,which exhibits powerful expressive capabilities.Through rigorous numerical analysis,the expressiveness of this circuit structure is quantitatively assessed using a variational quantum regression algorithm,and it obtained the best score compared to the others.Moreover,the expansion in system size is accompanied by an increase in the number of parameters,placing considerable strain on the training process for the algorithm.To address this challenge,an optimization strategy known as quantum parameter sharing is introduced,which proficiently minimizes parameter volume while adhering to exacting precision standards.Finally,EVQLSE is successfully implemented on a quantum computing platform provided by IBM for the resolution of large-scale problems characterized by a dimensionality of 220.展开更多
In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposi...In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposite-bordered tridiagonal matrix is investigated. Some illustrative examples are given.展开更多
This paper deals with the distributed solving problem of a specific class of linear algebraic equations(LAEs)with block Toeplitz structures.To reduce the communication burden and achieve computation efficiency,a distr...This paper deals with the distributed solving problem of a specific class of linear algebraic equations(LAEs)with block Toeplitz structures.To reduce the communication burden and achieve computation efficiency,a distributed iterative algorithm from the communication-efficient perspective is proposed by incorporating the specific structure of the coefficient matrix tied to any given LAE over a multi-agent network.Each agent possesses a state vector of size smaller than the dimensions of unknown variables related to the LAE and receives information from its neighbors.It is shown that the presented distributed iterative algorithm can solve the specific class of LAEs without requiring any initialization conditions,irrespective of whether it admits a unique solution or multiple solutions.Moreover,an equivalent relation is established between the problem of solving LAEs and the tracking problem of iterative learning control(ILC)systems.The proposed distributed iterative algorithm is leveraged to obtain the distributed control law for ILC systems to realize the tracking objective.Theoretical guarantees are provided for our developed solution results of LAEs,and the effectiveness of them is also verified through simulation examples.展开更多
For current sequential quadratic programming (SQP) type algorithms, there exist two problems; (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and ...For current sequential quadratic programming (SQP) type algorithms, there exist two problems; (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using e-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above.展开更多
This paper observes approaches to algebraic analysis of GOST 28147-89 encryption algorithm (also known as simply GOST), which is the basis of most secure information systems in Russia. The general idea of algebraic an...This paper observes approaches to algebraic analysis of GOST 28147-89 encryption algorithm (also known as simply GOST), which is the basis of most secure information systems in Russia. The general idea of algebraic analysis is based on the representation of initial encryption algorithm as a system of multivariate quadratic equations, which define relations between a secret key and a cipher text. Extended linearization method is evaluated as a method for solving the nonlinear sys- tem of equations.展开更多
Presents information on a study which proposed a superlinearly convergent algorithm of sequential systems of linear equations or nonlinear optimization problems with inequality constraints. Assumptions; Discussion on ...Presents information on a study which proposed a superlinearly convergent algorithm of sequential systems of linear equations or nonlinear optimization problems with inequality constraints. Assumptions; Discussion on lemmas about several matrices related to the common coefficient matrix F; Strengthening of the regularity assumptions on the functions involved; Numerical experiments.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.12031004 and Grant No.12271474,61877054)the Fundamental Research Foundation for the Central Universities(Project No.K20210337)+1 种基金the Zhejiang University Global Partnership Fund,188170+194452119/003partially funded by a state task of Russian Fundamental Investigations(State Registration No.FFSG-2024-0002)。
文摘Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-vector product,matrix-matrix product,the sum of two matrices,and the calculation of determinant and inverse matrix.We encode the matrix entries into the probability amplitudes of the pure initial states of senders.After applying proper unitary transformation to the complete quantum system,the desired result can be found in certain blocks of the receiver’s density matrix.These quantum protocols can be used as subroutines in other quantum schemes.Furthermore,we present an alternative quantum algorithm for solving linear systems of equations.
基金supported by the National Natural Science Foundation of China under Grant Nos.62172268 and 62302289the Shanghai Science and Technology Project under Grant Nos.21JC1402800 and 23YF1416200。
文摘Variational quantum algorithms are promising methods with the greatest potential to achieve quantum advantage,widely employed in the era of noisy intermediate-scale quantum computing.This study presents an advanced variational hybrid algorithm(EVQLSE)that leverages both quantum and classical computing paradigms to address the solution of linear equation systems.Initially,an innovative loss function is proposed,drawing inspiration from the similarity measure between two quantum states.This function exhibits a substantial improvement in computational complexity when benchmarked against the variational quantum linear solver.Subsequently,a specialized parameterized quantum circuit structure is presented for small-scale linear systems,which exhibits powerful expressive capabilities.Through rigorous numerical analysis,the expressiveness of this circuit structure is quantitatively assessed using a variational quantum regression algorithm,and it obtained the best score compared to the others.Moreover,the expansion in system size is accompanied by an increase in the number of parameters,placing considerable strain on the training process for the algorithm.To address this challenge,an optimization strategy known as quantum parameter sharing is introduced,which proficiently minimizes parameter volume while adhering to exacting precision standards.Finally,EVQLSE is successfully implemented on a quantum computing platform provided by IBM for the resolution of large-scale problems characterized by a dimensionality of 220.
文摘In the current article we propose a new efficient, reliable and breakdown-free algorithm for solving general opposite-bordered tridiagonal linear systems. An explicit formula for computing the determinant of an opposite-bordered tridiagonal matrix is investigated. Some illustrative examples are given.
基金supported by the National Natural Science Foundation of China under Grant Nos.U2333215,62273018,and U2133210。
文摘This paper deals with the distributed solving problem of a specific class of linear algebraic equations(LAEs)with block Toeplitz structures.To reduce the communication burden and achieve computation efficiency,a distributed iterative algorithm from the communication-efficient perspective is proposed by incorporating the specific structure of the coefficient matrix tied to any given LAE over a multi-agent network.Each agent possesses a state vector of size smaller than the dimensions of unknown variables related to the LAE and receives information from its neighbors.It is shown that the presented distributed iterative algorithm can solve the specific class of LAEs without requiring any initialization conditions,irrespective of whether it admits a unique solution or multiple solutions.Moreover,an equivalent relation is established between the problem of solving LAEs and the tracking problem of iterative learning control(ILC)systems.The proposed distributed iterative algorithm is leveraged to obtain the distributed control law for ILC systems to realize the tracking objective.Theoretical guarantees are provided for our developed solution results of LAEs,and the effectiveness of them is also verified through simulation examples.
基金Project partly supported by the National Natural Science Foundation of China and Tianyuan Foundation of China.
文摘For current sequential quadratic programming (SQP) type algorithms, there exist two problems; (i) in order to obtain a search direction, one must solve one or more quadratic programming subproblems per iteration, and the computation amount of this algorithm is very large. So they are not suitable for the large-scale problems; (ii) the SQP algorithms require that the related quadratic programming subproblems be solvable per iteration, but it is difficult to be satisfied. By using e-active set procedure with a special penalty function as the merit function, a new algorithm of sequential systems of linear equations for general nonlinear optimization problems with arbitrary initial point is presented This new algorithm only needs to solve three systems of linear equations having the same coefficient matrix per iteration, and has global convergence and local superlinear convergence. To some extent, the new algorithm can overcome the shortcomings of the SQP algorithms mentioned above.
文摘This paper observes approaches to algebraic analysis of GOST 28147-89 encryption algorithm (also known as simply GOST), which is the basis of most secure information systems in Russia. The general idea of algebraic analysis is based on the representation of initial encryption algorithm as a system of multivariate quadratic equations, which define relations between a secret key and a cipher text. Extended linearization method is evaluated as a method for solving the nonlinear sys- tem of equations.
基金This research was supported by the National Natural Science Foundation of China(19571001,19971002,79970014)Cross-century Excellent Personnel Award and Teaching and Research Award Program for Outstanding Young Teachers in High EducationMinistry o
文摘Presents information on a study which proposed a superlinearly convergent algorithm of sequential systems of linear equations or nonlinear optimization problems with inequality constraints. Assumptions; Discussion on lemmas about several matrices related to the common coefficient matrix F; Strengthening of the regularity assumptions on the functions involved; Numerical experiments.
