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Dimensionality reduction (DR) of data is a crucial issue for many machine learning tasks, such as pattern recognition and data classification. In this paper, we present a quantum algorithm and a quantum circuit to efficiently perform linear discriminant analysis (LDA) for dimensionality reduction. Firstly, the presented algorithm improves the existing quantum LDA algorithm to avoid the error caused by the irreversibility of the between-class scatter matrix $S_B$ in the original algorithm. Secondly, a quantum algorithm and quantum circuits are proposed to obtain the target state corresponding to the low-dimensional data. Compared with the best-known classical algorithm, the quantum linear discriminant analysis dimensionality reduction (QLDADR) algorithm has exponential acceleration on the number $M$ of vectors and a quadratic speedup on the dimensionality $D$ of the original data space, when the original dataset is projected onto a polylogarithmic low-dimensional space. Moreover, the target state obtained by our algorithm can be used as a submodule of other quantum machine learning tasks. It has practical application value of make that free from the disaster of dimensionality.

In this paper, we study a three-state quantum walk with a phase defect at a designated position. The coin operator is a parametrization of the eigenvectors of the Grover matrix. We numerically investigate the properties of the proposed model via the position probability distribution, the position standard deviation, and the time-averaged probability at the designated position. It is shown that the localization effect can be governed by the phase defect’s position and strength, coin parameter and initial state.

In this paper, we propose a Newton-based algorithm for nonnegative matrix factorization in image processing. We employ the new algorithm to three real-world databases. Extensive numerical results show the feasibility and validity of the proposed algorithm.

In recent years, nonnegative matrix factorization (NMF) has attracted significant amount of attentions in image processing, text mining, speech processing and related fields. Although NMF has been applied in several application successfully, its simple application on image processing has a few caveats. For example, NMF costs considerable computational resources when performing on large databases. In this paper, we propose two enhanced NMF algorithms for image processing to save the computational costs. One is modified rank-one residue iteration (MRRI) algorithm , the other is element-wisely residue iteration (ERI) algorithm. Here we combine CAPG (a NMF algorithm proposed by Lin), MRRI and ERI with two-dimensional nonnegative matrix factorization (2DNMF) for image processing. The main difference between NMF and 2DNMF is that the former first aligns images into one-dimensional (1D) vectors and then represents them with a set of 1D bases, while the latter regards images as 2D matrices and represents them with a set of 2D bases. The three combined algorithms are named CAPG-2DNMF, MRRI-2DNMF and ERI-2DNMF. The computational complexity and convergence analyses of proposed algorithms are also presented in this paper. Three public databases are used to test the three NMF algorithms and the three combinations, the results of which show the enhancement performance of our proposed algorithms (MRRI and ERI algorithms) over the CAPG algorithm. MRRI and ERI have similar performance. The three combined algorithms have better image reconstruction quality and less running time than their corresponding 1DNMF algorithms under the same compression ratio. We also do some experiments on a real-captured image database and get similar conclusions.

In this paper, a modified conjugate gradient iterative method for solving Sylvester equation is presented. By using this iterative method, the least squares anti-bisymmetric solution and the optimal approximation solution can be obtained. Here we present the derivation and theoretical analysis of our iterative method. Numerical results illustrate the feasibility and effectiveness of the proposed iterative method.

In this paper, we reexamined the security of fair quantum blind signature protocols and found there exists a secure leakage caused by the reuse of signing key. To illustrate it, a new attack strategy is proposed based on unambiguous set discrimination of quantum state. In this attack, the attacker can forge a valid signature for any message when he owns three signatures. Finally, a feasible improved method is proposed to solve this problem.

An arbitrated quantum signature scheme without using entangled states is proposed. In the scheme, by employing a classical hash function and random numbers, the secret keys of signer and receiver can be reused. It is shown that the proposed scheme is secure against several well-known attacks. Specifically, it can stand against the receiver’s disavowal attack. Moreover, compared with previous relevant arbitrated quantum signature schemes, the scheme proposed has the advantage of less transmission complexity.

Decoherence-free subspace (DFS) is a valid solution to realize quantum communication over a collective noise channel, and has been widely studied. Generally speaking, replacing a qubit with a DFS state will cause the reduction of communication efficiency. However, in this letter, it is shown that some kinds of noises may not lower the transmission rate of quantum key distribution. To illustrate it, we propose two quantum key distribution protocols based on Bell states. Here, two nonorthogonal and unbiased sets in a DFS are constructed by linear combination of particles at different positions. Since $n-1$ classical bits are distributed by using $2n$ qubits in our protocols, the transmission rate is close to that of noiseless BB84 protocol. Furthermore, when considering the cost of transmitting classical bits, the efficiencies of these protocols are even higher than that of BB84 protocol.

In this paper, a quantum private comparison protocol with χ-type entangled states is proposed. In the protocol, two users can determine whether their secret inputs are equal, with the help of a semi-trusted third party. Here, the only constraint condition of this third party is that he could not collude with one user. Although the third party takes part in the whole process of the presented protocol, a feature of χ-type entangled state is utilized to design a eavesdropping check method to stand against his attack. Security analysis showed that the presented protocol is secure against some well-known attacks. Meanwhile, all the particles undergo only a one-way trip, which improves the efficiency and security of the presented protocol. Furthermore, only two-particle measurement is required in the implementation of the protocol, which make it more feasible in technique.

In this paper, an efficient quantum anonymous ranking protocol with single particles is proposed. A semitrusted server is introduced to help multiple users achieve this secure task. At the end of the protocol, each user can obtain the rankings of his private data, and keep these data secret. The Chinese remainder theorem is utilized to reduce the level of signal particles and to improve the efficiency of the presented protocol. Meanwhile, a secret transmission order of the signal particles is used to keep the traveling particles secure. Finally, we discuss the security of this protocol and prove it to be secure against certain common attacks under ideal conditions.

A quantum secure direct communication protocol with authentication using only two nonorthogonal states is presented. In the protocol, a one-way hash function is utilized for two-way authentication and two one-particle unitary operations are used for encoding. Moreover, it has been shown that this protocol is secure against some common attacks.