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Journal of Physics A Mathematical and Theoretical

Quantum speedup of Bayes’ classifiers

Changpeng Shao

Published Nov 29, 2019• 9 citations
Publisher

Abstract

Abstract Data classification is a fundamental problem in machine learning. We study quantum speedup of the supervised data classification algorithms (quadratic, linear and naïve Bayes classifiers) based on Bayes’ theory. The main technique we use to achieve quantum speedup is block-encoding. However, to apply this technique effectively, we propose a general method to construct the block-encoding. As an application, we show that all the three classifiers achieve exponential speedup at the number of samples over their classical counterparts. As for the dimension of the space, quantum quadratic and linear classifiers achieve varying degrees of polynomial speedup, while quantum naïve Bayes’ classifier achieves an exponential speedup. The only assumption we make is the qRAM to prepare quantum states of the input data.

DOI
10.1088/1751-8121/ab5d77

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Abstract Data classification is a fundamental problem in machine learning. We study quantum speedup of the supervised data classification algorithms (quadratic, linear and naïve Bayes classifiers) based on Bayes’ theory. The main technique we use to achieve quantum speedup is block-encoding. However, to apply this technique effectively, we propose a general method to construct the block-encoding. As an application, we show that all the three classifiers achieve exponential speedup at the number of samples over their classical counterparts. As for the dimension of the space, quantum quadratic and linear classifiers achieve varying degrees of polynomial speedup, while quantum naïve Bayes’ classifier achieves an exponential speedup. The only assumption we make is the qRAM to prepare quantum states of the input data.
Abstract Classification is at the core of data‐driven prediction and decision‐making, representing a fundamental task in supervised machine learning. Recently, several quantum machine learning algorithms that use quantum kernels as a measure of similarities between data have emerged to perform binary classification on datasets encoded as quantum states. The potential advantages of quantum kernels arise from the ability of quantum computers to construct kernels that are more effective than their classical counterparts in capturing patterns in data or computing kernels more efficiently. However, existing quantum kernel‐based classification algorithms do not harness the capability of having data samples in quantum superposition for additional enhancements. This work demonstrates how such capability can be leveraged in quantum kernelized binary classifiers (QKCs) through Quantum Amplitude Estimation (QAE) for quadratic speed‐up. Additionally, new quantum circuits are proposed for the QKCs in which the number of qubits is reduced by one, and the circuit depth is reduced linearly with respect to the number of sample data. The quadratic speed‐up over previous methods is verified through numerical simulations on the Iris dataset.
Bayesian networks are powerful tools for probabilistic analysis and have been widely used in machine learning and data science. Unlike the time-consuming parameter training process of neural networks, Bayes classifiers constructed on Bayesian networks can make decisions based solely on statistical data from samples. In this paper we focus on constructing quantum Bayes classifiers (QBCs). We design both a na\"{\i}ve QBC and three semina\"{\i}ve QBCs (SN-QBCs). These QBCs are then applied to image classification tasks. To reduce computational complexity, we design a local feature sampling method to extract a limited number of feature attributes from an image. These attributes serve as nodes of the Bayesian networks to generate the QBCs. We simulate these QBCs on the MindQuantum platform and evaluate their performance on the MNIST and Fashion-MNIST data sets. Our results demonstrate that these QBCs achieve good classification accuracies even with a limited number of attributes. The classification accuracies of QBCs on the MNIST data set surpass those of classical Bayesian networks and quantum neural networks that utilize all available feature attributes. Additionally, we simulate these QBCs in a quantum noise environment.
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.
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.
A central task in the field of quantum computing is to find applications where quantum computer could provide exponential speedup over any classical computer. Machine learning represents an important field with broad applications where quantum computer may offer significant speedup. Several quantum algorithms for discriminative machine learning have been found based on efficient solving of linear algebraic problems, with potential exponential speedup in runtime under the assumption of effective input from a quantum random access memory. In machine learning, generative models represent another large class which is widely used for both supervised and unsupervised learning. Here, we propose an efficient quantum algorithm for machine learning based on a quantum generative model. We prove that our proposed model is exponentially more powerful to represent probability distributions compared with classical generative models and has exponential speedup in training and inference at least for some instances under a reasonable assumption in computational complexity theory. Our result opens a new direction for quantum machine learning and offers a remarkable example in which a quantum algorithm shows exponential improvement over any classical algorithm in an important application field.
Perceptrons, which perform binary classification, are the fundamental building blocks of neural networks. Given a data set of size NNN and margin $\ensuremath{\gamma}$ (how well the given data are separated), the query complexity of the best-known quantum training algorithm scales as either $(\sqrt{N}/{\ensuremath{\gamma}}^{2})log(1/{\ensuremath{\gamma}}^{2})$ or $N/\sqrt{\ensuremath{\gamma}}$, which is achieved by a hybrid of classical and quantum search. In this paper, we improve the version space quantum training method for perceptrons such that the query complexity of our algorithm scales as $\sqrt{N/\ensuremath{\gamma}}$. This is achieved by constructing an oracle for the perceptrons using quantum counting of the number of data elements that are correctly classified. Once such an oracle is constructed, bounded-error quantum search can be used to search over the hyperplane instances. The optimality of our algorithm is proven by reducing the evaluation of a two-level and-or tree (for which the query complexity lower bound is known) to a multicriterion search. Our quantum training algorithm can be generalized to train more complex machine learning models such as neural networks, which are built on a large number of perceptrons.