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As researchers focus on the field of quantum machine learning, there are more and more quantum algorithms for data fitting. However, the majority of quantum algorithms for data fitting are only applicable to sparse matrices, which bring a large number of limitations to their application. In this paper, to solve this problem, we propose a quantum algorithm for data fitting for general matrices, not only sparse and dense matrices. The quantum algorithm is mainly composed of three subroutines which are generating target state, quantum state tomography and estimating fit quality. The time complexity of the quantum algorithm is O κ 4 ‖ A ‖ F polylog (M N) (M ′ 2 log (M ′) 2 / ɛ 3 + 1 / (ɛ ϵ 2)) , where A ∈ R M × N , κ is the conditional number of matrix A † A ∈ R N × N , | | A | | F is the Frobenius norm of A , M ′ is the maxima number of fit functions in allowing fit, ɛ is the fitting error, and ϵ is the error of fit quality estimation. Moreover, our quantum algorithm is independent of the sparsity of the matrix, so our algorithm has wide application in dense matrices and poses a new state of the art for solving dense data fitting. • Data fitting is a mathematical method to construct the least squares fitting problem. • The proposed quantum algorithm is independent of the sparsity of the matrix. • The algorithm achieves polynomial-scale speedup compared to other algorithms. [ABSTRACT FROM AUTHOR]

Due to the development of quantum algorithms, kinds of quantum generative adversarial networks (QGANs) are proposed to generate suitable samples for different application scenarios. However, the training of QGANs is often not smooth as expected, and the results are unsatisfactory since of the barren plateau. To solve the problems, we propose a QGANs framework based on Rényi divergence (RyQGAN), which avoids the barren plateau phenomenon in the training of QGANs. The quantum circuits to estimate the gradients for the updating parameters are designed, which shows how the gradients can be evaluated by a quantum processor. Finally, based on RyQGAN, we generate 2-qubit, 3-qubit, and 4-qubit Gibbs states. The simulation results show that the barren plateau is eliminated in the RyQGAN, and the average fidelity of RyQGAN with 50 epochs is higher than 99.8%, while that of quantum Wasserstein GANs is no more than 50%. Thus, the RyQGAN can be trained effectively and has further applications in quantum machine learning. [ABSTRACT FROM AUTHOR]

A widely used clustering algorithm, density peak clustering (DPC), assigns different attribute values to data points through the distance between data points, and then determines the number and range of clustering by attribute values. However, DPC is inefficient when dealing with scenes with a large amount of data, and the range of parameters is not easy to determine. To fix these problems, we propose a quantum DPC (QDPC) algorithm based on a quantum D i s t C a l c circuit and a Grover circuit. The time complexity is reduced to O (log (N 2) + 6 N + N) , whereas that of the traditional algorithm is O (N 2) . The space complexity is also decreased from O (N · ⌈ log N ⌉) to O (⌈ log N ⌉) . [ABSTRACT FROM AUTHOR]