Quantum Adiabatic Evolution for Pattern Recognition Problem

Quantum pattern recognition algorithm for two-qubit systems has been implemented by quantum adiabatic evolution.We will estimate required running time for this algorithm by means of an analytical solution of timedependent Hamiltonian since the time complexity of adiabatic quantum evolution is a limitation on the quantum computing.These results can be useful for experimental implementation.     

On Models of Nonlinear Evolution Paths in Adiabatic Quantum Algorithms

In this paper,we study two different nonlinear interpolating paths in adiabatic evolution algorithms for solving a particular class of quantum search problems where both the initial and final Hamiltonian are one-dimensional projector Hamiltonians on the corresponding ground state.If the overlap between the initial state and final state of the quantum system is not equal to zero,both of these models can provide a constant time speedup over the usual adiabatic algorithms by increasing some another corresponding "complexity".But when the initial state has a zero overlap with the solution state in the problem,the second model leads to an infinite time complexity of the algorithm for whatever interpolating functions being applied while the first one can still provide a constant running time.However,inspired by a related reference,a variant of the first model can be constructed which also fails for the problem when the overlap is exactly equal to zero if we want to make up the "intrinsic" fault of the second model - an increase in energy.Two concrete theorems are given to serve as explanations why neither of these two models can improve the usual adiabatic evolution algorithms for the phenomenon above.These just tell us what should be noted when using certain nonlinear evolution paths in adiabatic quantum algorithms for some special kind of problems.     

基于相位匹配的量子行走搜索算法及电路实现

量子行走是经典随机行走在量子力学框架下的对应, 理论上可以用来解决一类无序数据库的搜索问题. 因为携带信息的量子态的扩散速度与经典相比有二次方式的增长, 所以量子行走优于经典随机行走, 量子行走的特性值得加以利用. 量子行走作为一种新发现的物理现象的数学描述, 引发了一种新的思维方式, 孕育了一种新的理论计算模型. 最新研究表明, 量子行走本身也是一种通用计算模型, 可被视为设计量子算法的高级工具, 因此受到部分计算机理论科学领域学者的关注和研究. 对于多数问题求解方案的量子算法的设计, 理论上可以只在量子行走模型下进行考虑. 基于Grover算法的相位匹配条件, 本文提出了一个新的基于量子行走的搜索算法. 理论演算表明: 一般情况下本算法的时间复杂度与Grover算法相同, 但是当搜索的目标数目多于总数的1/3时, 本算法搜索成功的概率要大于Grover算法. 本文不但利用Grover算法中相位匹配条件构造了一个新的量子行走搜索算法, 而且在本研究室原有的量子电路设计研究成果的基础上给出了该算法的量子电路表述.     

QCF: Quantum Collaborative Filtering Recommendation Algorithm

With the rapid development of the Internet, e-commerce plays an important role in people’s lives, and the recommendation system is one of the most critical technologies. However, as the number of users and the scale of goods increase sharply, the traditional collaborative filtering recommendation algorithm has a large computational complexity in the part of calculating the user similarity, which leads to a low recommendation efficiency. In response to the above problems, this paper introduces the concept of quantum computing theory. The user score vector is first prepared into a quantum state, the similarity score is calculated in parallel, then the similarity information is saved into the quantum bit, and finally the similar user is searched by the Grover search algorithm. Compared with the traditional collaborative filtering recommendation algorithm, the time complexity of the collaborative filtering recommendation algorithm based on Grover algorithm can be effectively reduced under certain conditions.

     

An Alternative Adiabatic Quantum Algorithm for the Hamiltonian Cycle Problem

We put forward an alternative quantum algorithm for finding Hamiltonian cycles in any N-vertex graph based on adiabatic quantum computing. With a von Neumann measurement on the final state, one may determine whether there is a Hamiltonian cycle in the graph and pick out a cycle if there is any. Although the proposed algorithm provides a quadratic speedup, it gives an alternative algorithm based on adiabatic quantum computation, which is of interest because of its inherent robustness.     

Quantum Algorithm for Hilbert's Tenth Problem

We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain Hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church—Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.     

General Quantum Meet-in-the-Middle Search Algorithm Based on Target Solution of Fixed Weight

Similar to the classical meet-in-the-middle algorithm, the storage and computation complexity are the key factors that decide the efficiency of the quantum meet-in-the-middle algorithm. Aiming at the target vector of fixed weight, based on the quantum meet-in-the-middle algorithm, the algorithm for searching all n-product vectors with the same weight is presented, whose complexity is better than the exhaustive search algorithm. And the algorithm can reduce the storage complexity of the quantum meet-in-the-middle search algorithm. Then based on the algorithm and the knapsack vector of the Chor-Rivest public-key crypto of fixed weight d, we present a general quantum meet-in-the-middle search algorithm based on the target solution of fixed weight, whose computational complexity is ∑_j^d=(0(√C_n-k+1^d-j)+O(C_k^jlogC_k^j)) with ∑_i=0^dC_kⁱ memory cost. And the optimal value of k is given. Compared to the quantum meet-in-the-middle search algorithm for knapsack problem and the quantum algorithm for searching a target solution of fixed weight, the computational complexity of the algorithm is lower. And its storage complexity is smaller than the quantum meet-in-the-middle-algorithm.     

On the Role of Prior Probability in Adiabatic Quantum Algorithms

In this paper, we study the role of prior probability on the efficiency of quantum local adiabatic search algorithm. The following aspects for prior probability are found here: firstly, only the probabilities of marked states affect the running time of the adiabatic evolution; secondly, the prior probability can be used for improving the efficiency of the adiabatic algorithm; thirdly, like the usual quantum adiabatic evolution, the running time for the case of multiple solution states where the number of marked elements are smaller enough than the size of the set assigned that contains them can be significantly bigger than that of the case where the assigned set only contains all the marked states.     

Quantum Algorithm for Approximating Maximum Independent Sets

We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing in the sub-Hilbert space of degenerate ground states,which generates quantum annealing in a secondary Hamiltonian.For both sparse and dense random graphs G,numerical simulation suggests that our algorithm on average finds an independent set of size close to the maximum size α(G) in low polynomial time.The best classical algorithms,by contrast,produce independent sets of size about half of α(G)in polynomial time.     

Improved Handwritten Digit Recognition using Quantum K-Nearest Neighbor Algorithm

Handwritten numeral recognition is a technology for automatic recognition and classification of handwritten numeral input through machine learning model. This is widely used in postal code digital automatic system to sort letters. The classical k-nearest neighbor algorithm is used in the traditional digital recognition training model. The recognized digital image classification is obtained through similarity measure or calculation and K value selection. Nonetheless, as the applied data volume exceeds a certain threshold, the time complexity of the model increases exponentially upon the similarity measure and K value search. This condition makes it hard to apply the model universally. In this paper, we introduce quantum computing, that is where digital image information is stored in the quantum state, and its similarity is calculated in parallel. Also, the most similar K points are obtained through the Grover algorithm. The theoretical analysis of the proposed improved algorithm shows that, handwritten numeral recognition based on quantum k-neighbor algorithm can improved upon time complexity of \( \mathrm{O}\left(\mathrm{R}\sqrt{kM}\right) \) of the existing algorithm.

     

A Parallel Quantum Algorithm for the Satisfiability Problem

In this paper we present a classical parallel quantum algorithm for the satisfiability problem. We have exploited the classical parallelism of quantum algorithms developed in [G.L. Long and L. Xiao, Phys. Rev. A 69 （2004） 052303], so that additional acceleration can be gained by using classical parallelism. The quantum algorithm first estimates the number of solutions using the quantum counting algorithm, and then by using the quantum searching algorithm, the explicit solutions are found.     

Arbitrated Quantum Signature with Hamiltonian Algorithm Based on Blind Quantum Computation

A novel arbitrated quantum signature (AQS) scheme is proposed motivated by the Hamiltonian algorithm (HA) and blind quantum computation (BQC). The generation and verification of signature algorithm is designed based on HA, which enables the scheme to rely less on computational complexity. It is unnecessary to recover original messages when verifying signatures since the blind quantum computation is applied, which can improve the simplicity and operability of our scheme. It is proved that the scheme can be deployed securely, and the extended AQS has some extensive applications in E-payment system, E-government, E-business, etc.     

Quantum Image Encryption Algorithm Based on Quantum Image XOR Operations

A novel encryption algorithm for quantum images based on quantum image XOR operations is designed. The quantum image XOR operations are designed by using the hyper-chaotic sequences generated with the Chen’s hyper-chaotic system to control the control-NOT operation, which is used to encode gray-level information. The initial conditions of the Chen’s hyper-chaotic system are the keys, which guarantee the security of the proposed quantum image encryption algorithm. Numerical simulations and theoretical analyses demonstrate that the proposed quantum image encryption algorithm has larger key space, higher key sensitivity, stronger resistance of statistical analysis and lower computational complexity than its classical counterparts.     

Quantum Image Encryption Algorithm Based on NASS

This paper proposes a quantum image encryption algorithm based on n-qubit normal arbitrary superposition state (NASS) by using the basic scheme of quantum transformation and random phase transformation. According to theoretical analysis and experimental simulation on MATLAB system, we find that key space is an important factor of encryption and decryption algorithm. When the secret key space is large, it is difficult for the attacker to crack the encrypted information. Based on this finding, we perform 2ⁿ +?4 times phase transformation in the encryption process. And each transformation is random, which increases the difficulty of decryption. So there are a total of 2ⁿ +?4 randomly transformed keys. In this paper, we design the implementation circuit of random phase transformation, and because the real quantum computer is not in our grasp, now we use MATLAB software to simulate grayscale image and color image encryption algorithm in classic computer, respectively. And the histogram, complexity and correlation are analyzed. Study shows that the proposed encryption algorithm is valid.     

A Quantum Algorithm for Finding a Hamilton Circuit

A quantum algorithm for solving the classical NP-complete problem-the hamilton circuit is presented.The algorithm employs the quantum SAT and the quantum search algorithms.The algorithm is square-root faster than classical algorithm,and becomes exponentially faster than classical algoriothm if nonlinear quantum mechanical computer is used.     

A Spatial Domain Quantum Watermarking Scheme

This paper presents a spatial domain quantum watermarking scheme. For a quantum watermarking scheme, a feasible quantum circuit is a key to achieve it. This paper gives a feasible quantum circuit for the presented scheme. In order to give the quantum circuit, a new quantum multi-control rotation gate, which can be achieved with quantum basic gates, is designed. With this quantum circuit, our scheme can arbitrarily control the embedding position of watermark images on carrier images with the aid of auxiliary qubits. Besides reversely acting the given quantum circuit, the paper gives another watermark extracting algorithm based on quantum measurements. Moreover, this paper also gives a new quantum image scrambling method and its quantum circuit. Differ from other quantum watermarking schemes, all given quantum circuits can be implemented with basic quantum gates. Moreover, the scheme is a spatial domain watermarking scheme, and is not based on any transform algorithm on quantum images. Meanwhile, it can make sure the watermark be secure even though the watermark has been found. With the given quantum circuit, this paper implements simulation experiments for the presented scheme. The experimental result shows that the scheme does well in the visual quality and the embedding capacity.     

A Necessary Condition for Quantum Adiabaticity Applied to the Adiabatic Grover Search

Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. Recently a practicable necessary condition well suited for many-body systems has been proved. Here we tailor this condition for estimating run times of adiabatic quantum algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that the thus obtained lower bound on the run time of this algorithm reproduces \( \sqrt{N} \) scaling (with N being the number of database entries) of the explicitly known optimum run time. This is in contrast to the poor performance of the known sufficient adiabatic conditions, which guarantee adiabaticity only for a run time on the order of O(N), which does not constitute any speedup over the classical database search. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.     

Quantum Binary Search Algorithm

International Journal of Theoretical Physics - Problems which can be solved on classical computer in polynomial time may not be able to be solved on quantum computer. And a quantum algorithm will...     

Quantum time of flight and the adiabatic limit in scattering processes

The author investigates the adiabatic limit of the exact quantum scattering problem. Various types of long-lived states are considered. The author demonstrates that the characteristic scattering time obtained from the adiabatic condition is not directly related to the quantum time of flight and that the difference between these two parameters can be quite large.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 52–58, March, 1988.The author is grateful to L. M. Baskin for helpful discussions and D. V. Glazanov for computer calculations.     

Quantum information transfer with Cooper-pair box qubits in circuit QED

We propose a feasible scheme to transfer quantum information with Cooper-pair box qubits arrayed in a circuit QED. Qubits interact with a quantum data bus generated by a one-dimensional transmission line resonator. Based on the Raman adiabatic passage, the cavity bus-assisted quantum population transfer between any selected pair of qubits can be controlled by addressing the applied gate pulses. Therefore, the scheme provides the possibility for effectively implementing scalable quantum information transfer with Josephson devices.