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A simple approach to design quantum neural networks and its applications to kernel-learning methods

Changpeng Shao

Published Oct 19, 2019• 2 citations
OpenAlex

Abstract

We give an explicit simple method to build quantum neural networks (QNNs) to solve classification problems. Besides the input (state preparation) and output (amplitude estimation), it has one hidden layer which uses a tensor product of $\log M$ two-dimensional rotations to introduce $\log M$ weights. Here $M$ is the number of training samples. We also have an efficient method to prepare the quantum states of the training samples. By the quantum-classical hybrid method or the variational method, the training algorithm of this QNN is easy to accomplish in a quantum computer. The idea is inspired by the kernel methods and the radial basis function (RBF) networks. In turn, the construction of QNN provides new findings in the design of RBF networks. As an application, we introduce a quantum-inspired RBF network, in which the number of weight parameters is $\log M$. Numerical tests indicate that the performance of this neural network in solving classification problems improves when $M$ increases. Since using exponentially fewer parameters, more advanced optimization methods (e.g. Newton's method) can be used to train this network. Finally, about the convex optimization problem to train support vector machines, we use a similar idea to reduce the number of variables, which equals $M$, to $\log M$.

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We give an explicit simple method to build quantum neural networks (QNNs) to solve classification problems. Besides the input (state preparation) and output (amplitude estimation), it has one hidden layer which uses a tensor product of log⁡M\log MlogM two-dimensional rotations to introduce log⁡M\log MlogM weights. Here MMM is the number of training samples. We also have an efficient method to prepare the quantum states of the training samples. By the quantum-classical hybrid method or the variational method, the training algorithm of this QNN is easy to accomplish in a quantum computer. The idea is inspired by the kernel methods and the radial basis function (RBF) networks. In turn, the construction of QNN provides new findings in the design of RBF networks. As an application, we introduce a quantum-inspired RBF network, in which the number of weight parameters is log⁡M\log MlogM. Numerical tests indicate that the performance of this neural network in solving classification problems improves when MMM increases. Since using exponentially fewer parameters, more advanced optimization methods (e.g. Newton's method) can be used to train this network. Finally, about the convex optimization problem to train support vector machines, we use a similar idea to reduce the number of variables, which equals MMM, to log⁡M\log MlogM.
Radial-basis-function (RBF) networks are third-layered neural networks that are widely used in function approximation and data classification. Here we propose a quantum model for RBF networks. As in the classical case, we use radial basis functions as activation functions; quantum linear algebraic techniques and coherent states can be applied to implement these functions. We define the state of the weight as a tensor product of single-qubit states. This gives a simple approach to implement the quantum RBF network in quantum circuits. In this model we prove that training is almost quadratically faster than in the classical case. We demonstrate numerically that the quantum RBF network can solve binary classification problems as effectively as the classical RBF network, while the time and number of iterations used for training are much smaller.
Quantum machine learning has established as an interdisciplinary field to overcome limitations of classical machine learning and neural networks. This is a field of research which can prove that quantum computers are able to solve problems with complex correlations between inputs that can be hard for classical computers. This suggests that learning models made on quantum computers may be more powerful for applications, potentially faster computation and better generalization on less data. The objective of this paper is to investigate how training of quantum neural network (QNNs) can be done using quantum optimization algorithms for improving the performance and time complexity of QNNs. A classical neural network can be partially quantized to create a hybrid quantum-classical neural network which is used mainly in classification and image recognition. In this paper, a QNN structure is made where a variational parameterized circuit is incorporated as an input layer named as Variational Quantum Neural Network (VQNNs). We encode the cost function of QNNs onto relative phases of a superposition state in the Hilbert space of the network parameters. The parameters are tuned with an iterative quantum approximate optimisation (QAOA) mixer and problem hamiltonians. VQNNs is experimented with MNIST digit recognition (less complex) and crack image classification datasets (more complex) which converges the computation in lesser time than QNN with decent training accuracy.
Radial basis function (RBF) networks are expanded to incorporate quantum kernel functions enabling a new type of hybrid quantum-classical machine learning algorithm. Using this approach, synthetic examples are introduced which allow for proof of concept on interpolation and classification applications. Quantum kernels have primarily been applied to support vector machines (SVMs), however the quantum kernel RBF network offers potential benefit over quantum kernel based SVMs due to the RBF networks ability to perform multi-class classification natively compared to the standard implementation of the SVM.
Quantum and classical machine learning have been naturally connected through kernel methods, which have also served as proof-of-concept for quantum advantage. Quantum embeddings encode classical data into quantum feature states, enabling the construction of embedding quantum kernels (EQKs) by measuring vector similarities and projected quantum kernels (PQKs) through projections of these states. However, in both approaches, the model is influenced by the choice of the embedding. In this work, we propose using the training of a quantum neural network (QNN) to construct neural quantum kernels, specifically neural EQKs and neural PQKs—problem-inspired kernel functions. Unlike previous approaches, our method requires the kernel matrix to be constructed only once, significantly reducing computational overhead. To achieve this, we introduce a scalable training method for an n -qubit data reuploading QNN. Furthermore, we demonstrate neural quantum kernels can alleviate exponential concentration and enhance generalization capabilities compared to problem-agnostic kernels, positioning them as a scalable and robust solution for quantum machine learning applications.
As quantum computers become increasingly practical, so does the prospect of using quantum computation to improve upon traditional algorithms. Kernel methods in machine learning is one area where such improvements could be realized in the near future. Paired with kernel methods like support-vector machines, small and noisy quantum computers can evaluate classically-hard quantum kernels that capture unique notions of similarity in data. Taking inspiration from techniques in classical machine learning, this work investigates simulated quantum kernels in the context of multiple kernel learning (MKL). We consider pairwise combinations of several classical-classical, quantum-quantum, and quantum-classical kernels in an empirical investigation of their classification performance with support-vector machines. We also introduce a novel approach, which we call QCC-net (quantum-classical-convex neural network), for optimizing the weights of base kernels together with any kernel parameters. We show this approach to be effective for enhancing various performance metrics in an MKL setting. Looking at data with an increasing number of features (up to 13 dimensions), we find parameter training to be important for successfully weighting kernels in some combinations. Using the optimal kernel weights as indicators of relative utility, we find growing contributions from trainable quantum kernels in quantum-classical kernel combinations as the number of features increases. We observe the opposite trend for combinations containing simpler, non-parametric quantum kernels.
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.