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Quantum support vector machine for big feature and big data classification.

Patrick Rebentrost, Masoud Mohseni, Seth Lloyd

Published Jul 1, 2013• 38 citations
OpenAlex

Abstract

Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized linear and non-linear binary classifier, can be implemented on a quantum computer, with exponential speedups in the size of the vectors and the number of training data examples. At the core of the algorithm is a non-sparse matrix simulation technique to efficiently perform a principal component analysis and matrix inversion of the training data kernel matrix. We thus provide an example of a quantum big feature and big data algorithm and pave the way for future developments at the intersection of quantum computing and machine learning.

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Supervised machine learning is the classification of new data based on
already classified training examples. In this work, we show that the support
vector machine, an optimized linear and non-linear binary classifier, can be
implemented on a quantum computer, with exponential speedups in the size of the
vectors and the number of training data examples. At the core of the algorithm
is a non-sparse matrix simulation technique to efficiently perform a principal
component analysis and matrix inversion of the training data kernel matrix. We
thus provide an example of a quantum big feature and big data algorithm and
pave the way for future developments at the intersection of quantum computing
and machine learning.
Supervised machine learning is the classification of new data based on already classified training examples. In this work, we show that the support vector machine, an optimized binary classifier, can be implemented on a quantum computer, with complexity logarithmic in the size of the vectors and the number of training examples. In cases where classical sampling algorithms require polynomial time, an exponential speedup is obtained. At the core of this quantum big data algorithm is a nonsparse matrix exponentiation technique for efficiently performing a matrix inversion of the training data inner-product (kernel) matrix.
Quantum computing is an interdisciplinary field at the intersection of computer science, mathematics, and physics that studies information processing tasks on a quantum computer. A quantum computer is a device whose operations are governed by the laws of quantum mechanics. As building quantum computers is nearing the era of commercialization and quantum supremacy, it is essential to think of potential applications that we might benefit from. Among many applications of quantum computation, one of the emerging fields is quantum machine learning. We focus on predictive models for binary classification and variants of Support Vector Machines that we expect to be especially important when training data becomes so large that a quantum algorithm with a guaranteed speedup becomes useful. We present a quantum machine learning algorithm for training Sparse Support Vector Machine for problems with large datasets that require a sparse solution. We also present the first quantum semi-supervised algorithm, where we still have a large dataset, but only a small fraction is provided with labels. While the availability of data for training machine learning models is steadily increasing, oftentimes it is much easier to collect feature vectors to obtain the corresponding labels. Here, we present a quantum machine learning algorithm for training Semi-Supervised Kernel Support Vector Machines. The algorithm uses recent advances in quantum sample-based Hamiltonian simulation to extend the existing Quantum LS-SVM algorithm to handle the semi-supervised term in the loss while maintaining the same quantum speedup as the Quantum LS-SVM.
The paper presents an influence of machine learning techniques that can be efficiently solved using quantum computer. Machine learning is considered as a core concept of modeling and mining large dimensionality data and visualizes data instances as a vector. Support vector is one of the trending methods that use supervised learning for classifying the data efficiently irrespective of linearity or non-linearity in data. As the quantum computations already proved to be inherently parallel and can become one of the key factors in order to reduce the complexities while dealing with the classical machine learning model. The quantum machine learning takes advantage of parallelism and can explore the number of vectors and their dimensions in logarithmic time, and hence it can achieve the computational speedup. Now, the main contribution of the paper is to discuss quantum fundamental, quantum algorithm and then to review over classical equivalent quantum support vector machine along with their training algorithm and the comparative analysis between algorithmic growths. At last, the paper justifies the quantum support vector as a highly efficient model that gains the computational speedup and also shows the possibility of further improvement.
Support vector machine (SVM) is a particularly powerful and flexible supervised learning model that analyzes data for both classification and regression, whose usual algorithm complexity scales polynomially with the dimension of data space and the number of data points. To tackle the big data challenge, a quantum SVM algorithm was proposed, which is claimed to achieve exponential speedup for least squares SVM (LS-SVM). Here, inspired by the quantum SVM algorithm, we present a quantum-inspired classical algorithm for LS-SVM. In our approach, an improved fast sampling technique, namely indirect sampling, is proposed for sampling the kernel matrix and classifying. We first consider the LS-SVM with a linear kernel, and then discuss the generalization of our method to nonlinear kernels. Theoretical analysis shows our algorithm can make classification with arbitrary success probability in logarithmic runtime of both the dimension of data space and the number of data points for low rank, low condition number, and high dimensional data matrix, matching the runtime of the quantum SVM.
The evolution of quantum computers and quantum machine learning (QML) algorithms have started demonstrating exponential speed-ups. In machine learning problems, the efficient handling and manipulation of linear algebra subroutines defines the complexity of the task to be performed. Quantum computers handle big datasets in the form of vectors and matrix operations very efficiently. In this paper, quantum support vector machine (QSVM) algorithm is used to solve a classification problem using a benchmarking MNIST dataset of handwritten images of digits. Quantum SVM variational and kernel matrix algorithms are implemented to analyze quantum speedup on quantum simulator and physical quantum processor back-ends. The study compared classical and quantum SVM algorithms in terms of execution time and accuracy. The results explicitly prove quantum speed-up achieved by quantum classifiers on quantum back-ends for machine learning applications.
This paper evaluates the performance of two quantum machine learning algorithms: Quantum Support Vector Machine (QSVM) and Quantum Principal Component Analysis (Q-PCA). The scalability of QSVM is analyzed across increasing dataset sizes and feature dimensions, revealing faster-than-linear growth in kernel evaluation and SVM training times. However, this computational overhead is accompanied by a notable decline in classification accuracy, highlighting challenges in applying QSVM to larger or more complex datasets. In contrast, Q-PCA demonstrates near-linear scalability with dataset size and qubit count, leveraging the independence of quantum circuit execution and structured operations to maintain efficiency. A QNN-based implementation of Q-PCA further validates its ability to align data with principal components, maximize variance, and enable effective dimensionality reduction. These findings emphasize the potential of Q-PCA for efficient quantum-enhanced data analysis while underscoring the need for optimization in QSVM and further exploration of Q-PCA’s capabilities for real-world quantum computing applications.