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Quantum algorithm for random-walk-assisted nonlinear dimensionality reduction

Apimuk Sornsaeng, Thiparat Chotibut, Ninnat Dangniam

Published Jan 1, 2022• 0 citations
Publisher

Abstract

Dimensionality reduction is an unsupervised learning technique that uncovers the underlying low-dimensional structure in highdimensional data sets. Random walks on graphs for dimensionality reduction, such as diffusion maps (DM) and the Potential of Heat-diffusion for Affinity-based Trajectory Embedding (PHATE), have emerged as powerful tools for automatically identifying low-dimensional data. These methods have applications in various fields, including many-body physics and bioinformatics. However, the classical implementations of DM and PHATE suffer from a computational bottleneck caused by the eigendecomposition step, which has a cubic time complexity to the size of the data set, rendering them impractical for large data sets. Leveraging the quantum computational speedup, we propose a quantum analog of nonlinear dimensionality reduction, named the quantum diffusion map (qDM), which can significantly reduce the computational cost of constructing the Markov transition matrix. Specifically, our algorithm can construct the Markov transition matrix for N classical data vectors in O(log3 N) time and extract the eigenpairs of the transition matrix classically to construct the DM with an expected runtime of N2polylogN. Moreover, the quantum subroutines utilized in our algorithm have potential applications beyond dimensionality reduction, for example, in any random-walk-based problems requiring Markov transition matrix construction and spectral analysis.

DOI
10.58837/chula.the.2022.1377

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Dimensionality reduction is an unsupervised learning technique that uncovers the underlying low-dimensional structure in highdimensional data sets. Random walks on graphs for dimensionality reduction, such as diffusion maps (DM) and the Potential of Heat-diffusion for Affinity-based Trajectory Embedding (PHATE), have emerged as powerful tools for automatically identifying low-dimensional data. These methods have applications in various fields, including many-body physics and bioinformatics. However, the classical implementations of DM and PHATE suffer from a computational bottleneck caused by the eigendecomposition step, which has a cubic time complexity to the size of the data set, rendering them impractical for large data sets. Leveraging the quantum computational speedup, we propose a quantum analog of nonlinear dimensionality reduction, named the quantum diffusion map (qDM), which can significantly reduce the computational cost of constructing the Markov transition matrix. Specifically, our algorithm can construct the Markov transition matrix for N classical data vectors in O(log3 N) time and extract the eigenpairs of the transition matrix classically to construct the DM with an expected runtime of N2polylogN. Moreover, the quantum subroutines utilized in our algorithm have potential applications beyond dimensionality reduction, for example, in any random-walk-based problems requiring Markov transition matrix construction and spectral analysis.
Inspired by random walks on graphs, the diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional data set. In recent years, among its many applications, the DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, a classical DM algorithm is computationally prohibitive for a large data set, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for the DM, termed the quantum diffusion map (qDM). Our qDM takes as an input NNN classical data vectors, performs an eigendecomposition of the Markov transition matrix in time O(log3N)O({log}^{3}N)O(log3N), and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to N2polylogN{N}^{2}\text{polylog}\phantom{\rule{0.16em}{0ex}}NN2polylogN. Finally, quantum subroutines in the qDM for constructing a Markov transition matrix and for analyzing its spectral properties can also be useful for other random-walk-based algorithms.
In the age of big data, unsupervised machine learning plays crucial roles in detecting statistical patterns hidden in gigantic dataset. Taking root in statistical physics of random walks and heat diffusion on networks, Diffusion Maps are one of the most efficient modern classical unsupervised algorithms for clustering high-dimensional dataset. Not only it can automatically discover hidden statistical structure in a high-dimensional dataset, but it can also projects the data into a lower dimensional embedding where the majority of the data structure reside. Such projections are termed nonlinear dimensionality reduction or manifold learning in machine leaning literature. In the first part of this thesis, we begin by reviewing the physics of classical random walks on a graph which motivates the construction of Diffusion Maps. We will discuss how Diffusion Maps can perform clustering as well as nonlinear dimensionality reduction based on the properties of Markov transition matrix defined on a dataset-associated graph. We then showcase the usefulness of Diffusion Maps to learn low dimensional embedding in some real data samples. In the second part of this thesis, we bring diffusion maps into the realm of quantum algorithms. Motivated by advances in modern near-term quantum devices, we explore a construction of Quantum Diffusion Maps. By exploiting coherent state encoding scheme into Quantum RAM, we outline how to achieve both quantum computational speedup as well as quantum storage capacity reduction for quantum computations of Diffusion Maps on a quantum device. Lastly, it’s known that quantum walks can spread faster than its classical counterparts; we construct quantum walk protocols that perhaps can provide an alternative way to perform unsupervised data clustering, given that one can create quantum walks on quantum devices or quantum simulators.
Experimental quantum simulators have become large and complex enough that discovering new physics from the huge amount of measurement data can be quite challenging, especially when little theoretical understanding of the simulated model is available. Unsupervised machine learning methods are particularly promising in overcoming this challenge. For the specific task of learning quantum phase transitions, unsupervised machine learning methods have primarily been developed for phase transitions characterized by simple order parameters, typically linear in the measured observables. However, such methods often fail for more complicated phase transitions, such as those involving incommensurate phases, valence-bond solids, topological order, and many-body localization. We show that the diffusion map method, which performs nonlinear dimensionality reduction and spectral clustering of the measurement data, has significant potential for learning such complex phase transitions unsupervised. This method may work for measurements of local observables in a single basis and is thus readily applicable to many experimental quantum simulators as a versatile tool for learning various quantum phases and phase transitions.
This paper explores the application of quantum computing to the critical tasks of feature selection and dimensionality reduction within machine learning. Traditional methods often struggle with high-dimensional data, leading to computational bottlenecks and potentially reduced model accuracy. We propose a novel approach leveraging the principles of quantum error correction and quantum entanglement to construct an algorithm capable of efficiently identifying and reducing the most relevant features. The core mechanism utilizes quantum machine learning techniques, specifically quantum feature mapping and feature extraction, to transform data into a quantum state that inherently captures feature relationships. This allows for a more effective selection process based on quantum interference and the exploration of complex correlations. We demonstrate, theoretically, the potential advantages of this approach over classical methods, highlighting improvements in both efficiency and predictive accuracy. The algorithm's design incorporates error correction to mitigate the effects of noise inherent in quantum systems, further enhancing its robustness. The research contributes to the growing field of quantum machine learning, offering a new paradigm for tackling complex data challenges.
This paper explores the application of quantum random walks (QRWs) to feature extraction from high-dimensional data. Classical random walks often struggle with the curse of dimensionality, leading to inefficient exploration of the data space. QRWs, leveraging the inherent randomness of quantum mechanics, offer a potential solution by enabling more efficient exploration and potentially uncovering hidden correlations. This work details the theoretical framework for implementing QRWs on graph-based representations of data, presenting algorithms for both discrete and continuous graphs. We investigate the key differences between classical and quantum random walks, focusing on the role of superposition and entanglement in expanding the search space. The core claim is that QRWs can be utilized to extract meaningful features from high-dimensional data. We provide a mathematical formulation for analyzing the convergence and efficiency of QRWs, demonstrating the potential advantages over traditional methods. The paper concludes with a discussion on the challenges and future directions for this emerging field.
Abstract Dimensionality reduction (DR) techniques play an extremely critical role in the data mining and pattern recognition field. However, most DR approaches involve large-scale matrix computations, which cause too high running complexity to implement in the big data scenario efficiently. The recent developments in quantum information processing provide a novel path to alleviate this problem, where a potential quantum acceleration can be obtained comparing with the classical counterpart. Nevertheless, existing proposals for quantum DR methods faced the common dilemma of the nonlinear generalization owing to the intrinsic linear limitation of quantum computation. In this paper, an architecture to simulate the arbitrary nonlinear kernels on a universal quantum computer is illustrated and further propose the quantum kernel principal component analysis (QKPCA) algorithm. The key idea is employing the truncated Taylor expansion to approximate the arbitrary nonlinear kernel within the fixed error and then constructing the corresponding Hamiltonian simulation for the quantum phase estimation algorithm. It is demonstrated theoretically that the QKPCA is qualified for the nonlinear DR task while the exponential speedup is also maintained. In addition, this research has the potential ability to develop other quantum DR approaches and existing linear quantum machine learning models.