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arXiv:1910.12164 · Physical Review A

Variational quantum algorithms for dimensionality reduction and classification

Jin‐Min Liang, Shu‐Qian Shen, Ming Li, Lei Li

Published Mar 16, 2020• 41 citations
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Abstract

In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil $(\mathcal{G},\mathcal{S})$. As a proof of principle, we implement our algorithm to solve ${2}^{5}\ifmmode\times\else\texttimes\fi{}{2}^{5}$ generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.

DOI
10.1103/physreva.101.032323

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In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil (G,S)(\mathcal{G},\mathcal{S})(G,S). As a proof of principle, we implement our algorithm to solve 25\ifmmode×\else\texttimes\fi25{2}^{5}\ifmmode\times\else\texttimes\fi{}{2}^{5}25\ifmmode×\else\texttimes\fi25 generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.
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Variational hybrid quantum classical algorithms to optimizations are important applications for near-term quantum computing. This paper proposes two quantum algorithms (the second one is variational) for training neural networks. Both of them obtain exponential speedup at the number of samples and polynomial speedup at the dimension of the samples over classical training algorithms. Moreover, the proposed quantum algorithms return the classical information of the training weight so that the outputs can be used directly to solve other problems. For practicality, we draw the quantum circuits to implement the two algorithms. Finally, as an inspiration, we show how to apply the variational algorithm to achieve speedup at the number of constraints in solving convex optimization problems.
With the development of quantum algorithms, high-cost computations are being scrutinized in the hope of a quantum advantage. While graphs offer a convenient framework for multiple real-world problems, their analytics still comes with high computation and space. By mapping the graph data into a low dimensional space, in which graph structural information is preserved, the eigenvectors of the Laplacian matrix constitute a powerful node embedding, called Laplacian Eigenmaps. Computing these embeddings is on its own an expensive task knowing that using specific sparse methods, the eigendecomposition of a Laplacian matrix has a cost of O(rn2rn^2rn2), rrr being the ratio of nonzero elements. We propose a method to compute a Laplacian Eigenmap using a quantum variational circuit. The idea of our algorithm is to reach the eigenstates of the laplacian matrix, which can be considered as a hamiltonian operator, by adapting the variational quantum eigensolver algorithm. By estimating the ddd first eigenvectors of the Laplacian at the same time, our algorithm directly generates a ddd dimension quantum embedding of the graph. We demonstrate that it is possible to use the embedding for graph machine learning tasks by implementing a quantum classifier on the top of it. The overall circuit consists in a full quantum node classification algorithm. Tests on 32 nodes graph with a quantum simulator shows that we can achieve similar performances as the classical laplacian eigenmap algorithm. Although mathematical properties of this approximate approach are not fully understood, this algorithm opens perspectives for graph pre-processing using noisy quantum computers.
As a general framework, matrix exponential dimensionality reduction (MEDR) deals with the small-sample-size problem that appears in linear dimensionality-reduction (DR) algorithms. High complexity is the bottleneck in this type of DR algorithm because one has to solve a large-scale matrix exponential generalized eigenproblem. To address it, here we design a general quantum algorithm framework for MEDR based on the block-encoding technique. This framework is configurable, that is, by selecting suitable methods to design the block encodings of the data matrices, a series of efficient quantum algorithms can be derived from this framework. Specifically, by constructing the block encodings of the data matrix exponentials, we solve the generalized eigenproblem and then obtain the digital-encoded quantum state corresponding to the compressed low-dimensional data set, which can be directly utilized as the input state for other quantum machine learning tasks to overcome the curse of dimensionality. As applications, we apply this framework to four linear DR algorithms and design their quantum algorithms, which all achieve a polynomial speedup in the dimension of the sample over their classical counterparts.
This paper explores the application of quantum algorithms for feature selection and dimensionality reduction within the field of quantum machine learning. Traditional methods for these tasks often suffer from computational bottlenecks, particularly when dealing with high-dimensional datasets. We propose a framework that leverages the principles of quantum computing to accelerate these processes. Specifically, we investigate the transformation of feature selection and dimensionality reduction problems into quantum optimization problems. We then examine the utilization of quantum annealing and variational quantum sampling algorithms for solving these problems. The core claim is that by employing quantum algorithms, we can significantly improve the efficiency and accuracy of machine learning models. The theoretical framework presented outlines a novel approach to tackling these challenges, offering a potential pathway towards more powerful and scalable quantum machine learning systems. This work provides a foundational exploration into this emerging area, focusing on the algorithmic strategies and potential benefits. ---