Variational Quantum Isometric Feature Mapping

Currently, big data often encounters "curse of dimensionality" problems. Using classic nonlinear dimensionality reduction algorithm, the complexity increases sharply with the increase of data dimension and quantity. While, quantum dimensionality reduction algorithm can achieve exponential acceleration for large-scale data processing, it requires a large number of n -qubit operations. To solve the problems above, this paper proposed a variational quantum isometric feature mapping algorithm based on classical nonlinear dimensionality reduction algorithm, quantum machine learning algorithm and variational quantum algorithm. In the data processing stage, the quantum k-nearest neighbor algorithm is used to obtain the K nearest neighbors of the original data, construct the adjacency graph, and calculate the geodesic distance between each point to get the distance matrix. The inner product matrix is obtained by redesigning the quantum circuit. Then the characteristic problem of the inner product matrix is solved by using the variational quantum eigensolver. Finally, the representation of low-dimensional data after dimensionality reduction is obtained, and the nonlinear dimensionality reduction of large-scale high-dimensional data is realized.