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Implementable Quantum Classifier for Nonlinear Data

Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao

Published Sep 17, 2018• 10 citations
OpenAlex

Abstract

In this Letter, we propose a quantum machine learning scheme for the classification of classical nonlinear data. The main ingredients of our method are variational quantum perceptron (VQP) and a quantum generalization of classical ensemble learning. Our VQP employs parameterized quantum circuits to learn a Grover search (or amplitude amplification) operation with classical optimization, and can achieve quadratic speedup in query complexity compared to its classical counterparts. We show how the trained VQP can be used to predict future data with $O(1)$ {query} complexity. Ultimately, a stronger nonlinear classifier can be established, the so-called quantum ensemble learning (QEL), by combining a set of weak VQPs produced using a subsampling method. The subsampling method has two significant advantages. First, all $T$ weak VQPs employed in QEL can be trained in parallel, therefore, the query complexity of QEL is equal to that of each weak VQP multiplied by $T$. Second, it dramatically reduce the {runtime} complexity of encoding circuits that map classical data to a quantum state because this dataset can be significantly smaller than the original dataset given to QEL. This arguably provides a most satisfactory solution to one of the most criticized issues in quantum machine learning proposals. To conclude, we perform two numerical experiments for our VQP and QEL, implemented by Python and pyQuil library. Our experiments show that excellent performance can be achieved using a very small quantum circuit size that is implementable under current quantum hardware development. Specifically, given a nonlinear synthetic dataset with $4$ features for each example, the trained QEL can classify the test examples that are sampled away from the decision boundaries using $146$ single and two qubits quantum gates with $92\%$ accuracy.

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In this Letter, we propose a quantum machine learning scheme for the classification of classical nonlinear data. The main ingredients of our method are variational quantum perceptron (VQP) and a quantum generalization of classical ensemble learning. Our VQP employs parameterized quantum circuits to learn a Grover search (or amplitude amplification) operation with classical optimization, and can achieve quadratic speedup in query complexity compared to its classical counterparts. We show how the trained VQP can be used to predict future data with O(1)O(1)O(1) {query} complexity. Ultimately, a stronger nonlinear classifier can be established, the so-called quantum ensemble learning (QEL), by combining a set of weak VQPs produced using a subsampling method. The subsampling method has two significant advantages. First, all TTT weak VQPs employed in QEL can be trained in parallel, therefore, the query complexity of QEL is equal to that of each weak VQP multiplied by TTT. Second, it dramatically reduce the {runtime} complexity of encoding circuits that map classical data to a quantum state because this dataset can be significantly smaller than the original dataset given to QEL. This arguably provides a most satisfactory solution to one of the most criticized issues in quantum machine learning proposals. To conclude, we perform two numerical experiments for our VQP and QEL, implemented by Python and pyQuil library. Our experiments show that excellent performance can be achieved using a very small quantum circuit size that is implementable under current quantum hardware development. Specifically, given a nonlinear synthetic dataset with 444 features for each example, the trained QEL can classify the test examples that are sampled away from the decision boundaries using 146146146 single and two qubits quantum gates with 92%92\%92% accuracy.
ABSTRACT Ensemble learning plays a central role in improving the generalization of machine learning models, yet its full potential remains underexplored in the quantum domain. In this work, we propose a Quantum Ensemble Learning (QEL) algorithm that leverages the parallelism of quantum computing to construct diverse classifiers in a shallow‐circuit ensemble. The method uses a control‐qubit‐based sampling strategy to create superpositions of training subsamples, enabling simultaneous evaluation of multiple weak learners. The algorithm proceeds through four stages: quantum state preparation, superposition‐based sampling, learning via quantum interference, and final measurement. Our framework is flexible and allows different quantum classifiers to be used as base learners. We demonstrate this by implementing a robust QSVM‐based ensemble with quantum feature extraction using PauliFeatureMaps. Unlike classical methods, this technique adds classifiers to time complexity additively rather than multiplicatively due to the parallelism of quantum computing. To demonstrate the workflow of the proposed algorithm, we include a step‐by‐step example using the XOR problem. Simulation results further validate the effectiveness of the QEL framework—enhanced by the integration of a Quantum Support Vector Machine (QSVM) classifier with quantum feature extraction—showing improved classification accuracy compared to existing algorithms across several UCI benchmark datasets using IBM Qiskit.
Quantum computing is rapidly advancing and offering new ways to improve artificial intelligence (AI). The most recent advancement is Quantum Machine Learning (QML), an exciting new field that involves the exploration of combining quantum computing principles with machine learning methods to solve complex problems that have high-dimensional problem-solving models that are impossible to address with classical models. The objective of this research was a descriptive-analytical examination of combining classical machine learning with quantum computing principles, specifically using Variational Quantum Classifiers (VQC) to train on a nonlinear XOR dataset, to assess the proposition of QML. The VQC was able to amongst quantum features, and learning how to classify using parameterized variational circuits to model nonlinear decision boundaries as models not achievable using classical linear classifiers. Training convergence was demonstrated with decreasing training and test losses, with the final model achieving test accuracy of nearly 0.85. The training curves and the decision boundary plots illustrated the model’s ability to learn the more complex decision boundaries. This study demonstrated QML's ability to coexist with classical machine learning studies with the ability to exploit quantum entanglement and quantum parallelism. This paper concluded that QML has great potential to challenge the concerns of next generation applications in finance, healthcare, and secure communications and to optimize computational power and scalability.
Quantum machine learning has the potential to provide powerful algorithms for artificial intelligence. The pursuit of quantum advantage in quantum machine learning is an active area of research. For current noisy intermediate-scale quantum computers, various quantum-classical hybrid algorithms have been proposed. One such previously proposed hybrid algorithm is a gate-based variational embedding classifier, which is composed of a classical neural network and a parameterized gate-based quantum circuit. We propose a quantum variational embedding classifier based on an analog quantum computer, where control signals vary continuously in time: our particular focus is an implementation using quantum annealers. In our algorithm, the classical data are transformed into the parameters of the time-varying Hamiltonian of the analog quantum computer by a linear transformation. The nonlinearity needed for a nonlinear classification problem is purely provided by the analog quantum computer through the nonlinear dependence of the final quantum state on the control parameters of the Hamiltonian. We perform numerical simulations that demonstrate the effectiveness of our algorithm for performing binary and multiclass classification on linearly inseparable datasets such as concentric circles and MNIST digits. Our classifier can reach accuracy comparable with that of the best classical classifiers. We find that the performance of our classifier can be increased by increasing the number of qubits, until the performance saturates and fluctuates. Moreover, the number of optimization parameters of our classifier scales linearly with the number of qubits. The increase of the number of training parameters when the size of our model increases is therefore not as fast as that of a neural network. Our algorithm presents the possibility of using current quantum annealers for solving practical machine-learning problems, and it could also be useful to explore quantum advantage in quantum machine learning.
A powerful way to improve performance in machine learning is to construct an ensemble that combines the predictions of multiple models. Ensemble methods are often much more accurate and lower variance than the individual classifiers that make them up but have high requirements in terms of memory and computational time. In fact, a large number of alternative algorithms is usually adopted, each requiring to query all available data. We propose a new quantum algorithm that exploits quantum superposition, entanglement and interference to build an ensemble of classification models. Thanks to the generation of the several quantum trajectories in superposition, we obtain BBB transformations of the quantum state which encodes the training set in only log(B)log\left(B\right)log(B) operations. This implies exponential growth of the ensemble size while increasing linearly the depth of the correspondent circuit. Furthermore, when considering the overall cost of the algorithm, we show that the training of a single weak classifier impacts additively the overall time complexity rather than multiplicatively, as it usually happens in classical ensemble methods. We also present small-scale experiments on real-world datasets, defining a quantum version of the cosine classifier and using the IBM qiskit environment to show how the algorithms work.
Abstract A kernel-based quantum classifier is the most interesting and powerful quantum machine learning technique for hyperlinear classification of complex data, which can be easily realized in shallow-depth quantum circuits such as a SWAP test classifier. A variational quantum approximate support vector machine (VQASVM) can be realized inherently and explicitly on these circuits by introduction of a variational scheme to map the quadratic optimization problem of the support vector machine theory to a quantum-classical variational optimization problem. Probability weight modulation in index qubits of a classifier can designate support vectors among training vectors, which can be achieved with a parameterized quantum circuit (PQC). The classical parameters of PQC is then transferred to many copies of other decision inference circuits. Our VQASVM algorithm is experimented with toy example data sets on cloud-based quantum machines for feasibility evaluation, and numerically investigated to evaluate its performance on a standard iris flower and MNIST data set. The empirical run-time complexity of VQASVM is estimated to be sub-quadratic on the training data set size, while that of the classical solver is quadratic.
Abstract A powerful way to improve performance in machine learning is to construct an ensemble that combines the predictions of multiple models. Ensemble methods are often much more accurate and have a lower variance than the individual classifiers that make them up, but have high requirements in terms of memory and computational time. In fact, a large number of alternative algorithms is usually adopted, each requiring to query all available data. We propose a new quantum algorithm that exploits quantum superposition, entanglement and interference to build an ensemble of classification models. Thanks to the generation of the several quantum trajectories in superposition, we obtain B transformations of the quantum state which encodes the training set in only log ( B ) operations. This implies an exponential growth of the ensemble size while increasing linearly the depth of the correspondent circuit. Furthermore, when considering the overall cost of the algorithm, we show that the training of a single weak classifier impacts additively to the overall time complexity rather than multiplicatively, as it usually happens in classical ensemble methods. We also present small-scale experiments on real-world datasets, defining a quantum version of the cosine classifier and using the IBM qiskit environment to show how the algorithms work.