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Recently, the implementation of quantum neural networks is based on noisy intermediate-scale quantum (NISQ) devices. Parameterized quantum circuit (PQC) is such the method, and its current design just can handle linear data classification. However, data in the real world often shows a topological structure. In the machine learning field, the classical graph convolutional layer (GCL)-based graph convolutional network (GCN) can well handle the topological data. Inspired by the architecture of a classical GCN, in this paper, to expand the function of the PQC, we design a novel PQC architecture to realize a quantum GCN (QGCN). More specifically, we first implement an adjacent matrix based on linear combination unitary and a weight matrix in a quantum GCL, and then by stacking multiple GCLs, we obtain the QGCN. In addition, we first achieve gradients decent on quantum circuit following the parameter-shift rule for a GCL and then for the QGCN. We evaluate the performance of the QGCN by conducting a node classification task on Cora dataset with topological data. The numerical simulation result shows that QGCN has the same performance as its classical counterpart, the GCN, in contrast, requires less tunable parameters. Compared to a traditional PQC, we also verify that deploying an extra adjacent matrix can significantly improve the classification performance for quantum topological data.

Quantum machine learning is one of the most promising applications of quantum computing in the Noisy Intermediate-Scale Quantum(NISQ) era. Here we propose a quantum convolutional neural network(QCNN) inspired by convolutional neural networks(CNN), which greatly reduces the computing complexity compared with its classical counterparts, with $O((log_{2}M)^6) $ basic gates and $O(m^2+e)$ variational parameters, where $M$ is the input data size, $m$ is the filter mask size and $e$ is the number of parameters in a Hamiltonian. Our model is robust to certain noise for image recognition tasks and the parameters are independent on the input sizes, making it friendly to near-term quantum devices. We demonstrate QCNN with two explicit examples. First, QCNN is applied to image processing and numerical simulation of three types of spatial filtering, image smoothing, sharpening, and edge detection are performed. Secondly, we demonstrate QCNN in recognizing image, namely, the recognition of handwritten numbers. Compared with previous work, this machine learning model can provide implementable quantum circuits that accurately corresponds to a specific classical convolutional kernel. It provides an efficient avenue to transform CNN to QCNN directly and opens up the prospect of exploiting quantum power to process information in the era of big data., Comment: we need a major revision for our manuscript and add some new contexts

Utilizing quantum computers to deploy artificial neural networks (ANNs) will bring the potential of significant advancements in both speed and scale. In this paper, we propose a kind of quantum spike neural networks (SNNs) as well as comprehensively evaluate and give a detailed mathematical proof for the quantum SNNs, including its successful probability, calculation accuracy, and algorithm complexity. The proof shows the quantum SNNs' computational complexity that is log-polynomial in the data dimension. Furthermore, we provide a method to improve quantum SNNs' minimum successful probability to nearly 100%. Finally, we present the good performance of quantum SNNs for solving pattern recognition from the real-world.

Finding a maximum or minimum is a fundamental building block in many mathematical models. Compared with classical algorithms, Durr, Hoyer's quantum algorithm (DHA) achieves quadratic speed. However, its key step, the quantum exponential searching algorithm (QESA), which is based on Grover algorithm, is not a sure-success algorithm. Meanwhile, quantum circuits encounter the gate decomposition problem due to variation of the scale of data. In this paper, we propose an optimized quantum algorithm for searching maximum and minimum, based on DHA and the optimal quantum exact search algorithm. Furthermore, we provide the corresponding quantum circuits, together with three equivalent simplifications. In circumstances when we can exactly estimate the ratio of the number of solutions M and the searched space N, our method can improve the successful probability close to 100%. Furthermore, compared with DHA, our algorithm shows an advantage in complexity with large databases and in the gate complexity of constructing oracles. Experiments have been executed on an IBM superconducting processor with two qubits, and a practical problem of finding the minimum from Titanic passengers' age was numerically simulated. Both showed that our optimized maximum or minimum performs more efficiently compared with DHA. Our algorithm can serve as an important subroutine in various quantum algorithms which involves searching maximum or minimum.