The subset sum problem is a combinatorial optimization problem,and its complexity belongs to the nondeterministic polynomial time complete(NP-Complete)class.This problem is widely used in encryption,planning or schedu...The subset sum problem is a combinatorial optimization problem,and its complexity belongs to the nondeterministic polynomial time complete(NP-Complete)class.This problem is widely used in encryption,planning or scheduling,and integer partitions.An accurate search algorithm with polynomial time complexity has not been found,which makes it challenging to be solved on classical computers.To effectively solve this problem,we translate it into the quantum Ising model and solve it with a variational quantum optimization method based on conditional values at risk.The proposed model needs only n qubits to encode 2ndimensional search space,which can effectively save the encoding quantum resources.The model inherits the advantages of variational quantum algorithms and can obtain good performance at shallow circuit depths while being robust to noise,and it is convenient to be deployed in the Noisy Intermediate Scale Quantum era.We investigate the effects of the scalability,the variational ansatz type,the variational depth,and noise on the model.Moreover,we also discuss the performance of the model under different conditional values at risk.Through computer simulation,the scale can reach more than nine qubits.By selecting the noise type,we construct simulators with different QVs and study the performance of the model with them.In addition,we deploy the model on a superconducting quantum computer of the Origin Quantum Technology Company and successfully solve the subset sum problem.This model provides a new perspective for solving the subset sum problem.展开更多
It is well known that almost all subset sum problems with density less than 0.9408… can be solved in polynomial time with an SVP oracle that can find a shortest vector in a special lattice.In this paper,the authors s...It is well known that almost all subset sum problems with density less than 0.9408… can be solved in polynomial time with an SVP oracle that can find a shortest vector in a special lattice.In this paper,the authors show that a similar result holds for the k-multiple subset sum problem which has k subset sum problems with exactly the same solution.Specially,for the single subset sum problem(k=1),a modified lattice is introduced to make the proposed analysis much simpler and the bound for the success probability tighter than before.Moreover,some extended versions of the multiple subset sum problem are also considered.展开更多
This paper indicates the possible difficulties for applying the interior point method to NPcomplete problems,transforms an NP-complete problem into a nonconvex quadratic program and then develops some convexity theori...This paper indicates the possible difficulties for applying the interior point method to NPcomplete problems,transforms an NP-complete problem into a nonconvex quadratic program and then develops some convexity theories for it. Lastly it proposes an algorithm which uses Karmarkar's algorithm as a subroutine. The finite convergence of this algorithm is also proved.展开更多
Let ε ： y^2 = x3 ＋ Ax ＋ B be an elliptic curve defined over the finite field Zp（p 〉 3） and G be a rational point of prime order N on ε. Define a subset of ZN, the residue class ring modulo N, asS：={n：n∈ZN,n...Let ε ： y^2 = x3 ＋ Ax ＋ B be an elliptic curve defined over the finite field Zp（p 〉 3） and G be a rational point of prime order N on ε. Define a subset of ZN, the residue class ring modulo N, asS：={n：n∈ZN,n≠0,（X（nG）/p）=1} where X（nG） denotes the x-axis of the rational points nC and （＊/P） is the Legendre symbol. Some explicit results on quasi-randomness of S are investigated. The construction depends on the intrinsic group structures of elliptic curves and character sums along elliptic curves play an important role in the proofs.展开更多
基金supported by the National Key R&D Program of China(Grant No.2019YFA0308700)the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0301500)。
文摘The subset sum problem is a combinatorial optimization problem,and its complexity belongs to the nondeterministic polynomial time complete(NP-Complete)class.This problem is widely used in encryption,planning or scheduling,and integer partitions.An accurate search algorithm with polynomial time complexity has not been found,which makes it challenging to be solved on classical computers.To effectively solve this problem,we translate it into the quantum Ising model and solve it with a variational quantum optimization method based on conditional values at risk.The proposed model needs only n qubits to encode 2ndimensional search space,which can effectively save the encoding quantum resources.The model inherits the advantages of variational quantum algorithms and can obtain good performance at shallow circuit depths while being robust to noise,and it is convenient to be deployed in the Noisy Intermediate Scale Quantum era.We investigate the effects of the scalability,the variational ansatz type,the variational depth,and noise on the model.Moreover,we also discuss the performance of the model under different conditional values at risk.Through computer simulation,the scale can reach more than nine qubits.By selecting the noise type,we construct simulators with different QVs and study the performance of the model with them.In addition,we deploy the model on a superconducting quantum computer of the Origin Quantum Technology Company and successfully solve the subset sum problem.This model provides a new perspective for solving the subset sum problem.
基金supported by the National Natural Science Foundation of China under Grant Nos.11201458,11471314in part by 973 Project under Grant No.2011CB302401in part by the National Center for Mathematics and Interdisciplinary Sciences,Chinese Academy of Sciences
文摘It is well known that almost all subset sum problems with density less than 0.9408… can be solved in polynomial time with an SVP oracle that can find a shortest vector in a special lattice.In this paper,the authors show that a similar result holds for the k-multiple subset sum problem which has k subset sum problems with exactly the same solution.Specially,for the single subset sum problem(k=1),a modified lattice is introduced to make the proposed analysis much simpler and the bound for the success probability tighter than before.Moreover,some extended versions of the multiple subset sum problem are also considered.
文摘This paper indicates the possible difficulties for applying the interior point method to NPcomplete problems,transforms an NP-complete problem into a nonconvex quadratic program and then develops some convexity theories for it. Lastly it proposes an algorithm which uses Karmarkar's algorithm as a subroutine. The finite convergence of this algorithm is also proved.
基金Supported by the National Natural Science Foundation of China(No.61170246)the Program for New Century Excellent Talents in Fujian Province University of China(No.JK2010047)the Open Funds of State Key Laboratory of Information Security (Chinese Academy of Sciences)(No.01-01-1)
文摘Let ε ： y^2 = x3 ＋ Ax ＋ B be an elliptic curve defined over the finite field Zp（p 〉 3） and G be a rational point of prime order N on ε. Define a subset of ZN, the residue class ring modulo N, asS：={n：n∈ZN,n≠0,（X（nG）/p）=1} where X（nG） denotes the x-axis of the rational points nC and （＊/P） is the Legendre symbol. Some explicit results on quasi-randomness of S are investigated. The construction depends on the intrinsic group structures of elliptic curves and character sums along elliptic curves play an important role in the proofs.
