Training classical neural networks generally requires a large number of training samples. Using entangled training samples, Quantum Neural Networks (QNNs) have the potential to significantly reduce the amount of training samples required in the training process. However, to minimize the number of incorrect predictions made by the resulting QNN, it is essential that the structure of the training samples meets certain requirements. On the one hand, the exact degree of entanglement must be fixed for the whole set of training samples. On the other hand, training samples must be linearly independent and non-orthogonal. However, how failing to meet these requirements affects the resulting QNN is not fully studied. To address this, we extend the proof of the Quantum No-Free-Lunch theorem to (i) provide a generalization of the theorem for varying degrees of entanglement. This generalization shows that the average degree of entanglement in the set of training samples can be used to predict the expected quality of the QNN. Furthermore, we (ii) introduce new estimates for the expected accuracy of QNNs for moderately entangled training samples that are linearly dependent or orthogonal. Our analytical results are (iii) experimentally validated by simulating QNN training and analyzing the quality of the QNN after training.
Training classical neural networks generally requires a large number of training samples. Using entangled training samples, Quantum Neural Networks (QNNs) have the potential to significantly reduce the amount of training samples required in the training process. However, to minimize the number of incorrect predictions made by the resulting QNN, it is essential that the structure of the training samples meets certain requirements. On the one hand, the exact degree of entanglement must be fixed for the whole set of training samples. On the other hand, training samples must be linearly independent and non-orthogonal. However, how failing to meet these requirements affects the resulting QNN is not fully studied. To address this, we extend the proof of the Quantum No-Free-Lunch theorem to (i) provide a generalization of the theorem for varying degrees of entanglement. This generalization shows that the average degree of entanglement in the set of training samples can be used to predict the expected quality of the QNN. Furthermore, we (ii) introduce new estimates for the expected accuracy of QNNs for moderately entangled training samples that are linearly dependent or orthogonal. Our analytical results are (iii) experimentally validated by simulating QNN training and analyzing the quality of the QNN after training.
Abstract Training classical neural networks generally requires a large number of training samples. Using entangled training samples, Quantum Neural Networks (QNNs) have the potential to significantly reduce the amount of training samples required in the training process. However, to minimize the number of incorrect predictions made by the resulting QNN, it is essential that the structure of the training samples meets certain requirements. On the one hand, the exact degree of entanglement must be fixed for the whole set of training samples. On the other hand, training samples must be linearly independent and non-orthogonal. However, how failing to meet these requirements affects the resulting QNN is not fully studied. To address this, we extend the proof of the Quantum No-Free-Lunch theorem to (i) provide a generalization of the theorem for varying degrees of entanglement. This generalization shows that the average degree of entanglement in the set of training samples can be used to predict the expected quality of the QNN. Furthermore, we (ii) introduce new estimates for the expected accuracy of QNNs for moderately entangled training samples that are linearly dependent or orthogonal. Our analytical results are (iii) experimentally validated by simulating QNN training and analyzing the quality of the QNN after training.
Training classical neural networks generally requires a large number of training samples. Using entangled training samples, Quantum Neural Networks (QNNs) have the potential to significantly reduce the amount of training samples required in the training process. However, to minimize the number of incorrect predictions made by the resulting QNN, it is essential that the structure of the training samples meets certain requirements. On the one hand, the exact degree of entanglement must be fixed for the whole set of training samples. On the other hand, training samples must be linearly independent and non-orthogonal. However, how failing to meet these requirements affects the resulting QNN is not fully studied. To address this, we extend the proof of the QNFL theorem to (i) provide a generalization of the theorem for varying degrees of entanglement. This generalization shows that the average degree of entanglement in the set of training samples can be used to predict the expected quality of the QNN. Furthermore, we (ii) introduce new estimates for the expected accuracy of QNNs for moderately entangled training samples that are linear dependent or orthogonal. Our analytical results are (iii) experimentally validated by simulating QNN training and analyzing the quality of the QNN after training.
Quantum neural networks (QNNs) use parameterized quantum circuits with data-dependent inputs and generate outputs through the evaluation of expectation values. Calculating these expectation values necessitates repeated circuit evaluations, thus introducing fundamental finite-sampling noise even on error-free quantum computers. We reduce this noise by introducing the variance regularization, a technique for reducing the variance of the expectation value during the quantum model training. This technique requires no additional circuit evaluations if the QNN is properly constructed. Our empirical findings demonstrate the reduced variance speeds up the training and lowers the output noise as well as decreases the number of necessary evaluations of gradient circuits. This regularization method is benchmarked on the regression of multiple functions and the potential energy surface of water. We show that in our examples, it lowers the variance by an order of magnitude on average and leads to a significantly reduced noise level of the QNN. We finally demonstrate QNN training on a real quantum device and evaluate the impact of error mitigation. Here, the optimization is feasible only due to the reduced number of necessary shots in the gradient evaluation resulting from the reduced variance.
Quantum computing is one of the most exciting research areas of the last decades. At the same time, methods of machine learning have started to dominate science, industry and our everyday life. In this thesis we combine these two essential research topics of the 21st century and introduce dissipative quantum neural networks (DQNNs), which are designed for fully quantum learning tasks, are capable of universal quantum computation and have low memory requirements while training. We start the discussion of this interdisciplinary topic by introducing artificial neural networks, which are a very common tool in classical machine learning. Next, we give an overview on quantum information. Here we focus on quantum algorithms and circuits, which are used to implement quantum neural networks. Moreover, we explain the opportunities and challenges arising with today's quantum computers. The discussion of the architecture and training algorithm of the DQNNs forms the core of this work. These networks are optimised with training data pairs in form of input and desired output states and therefore can be used for characterising unknown or untrusted quantum devices. We not only demonstrate the generalisation behaviour of these quantum neural networks using classical simulations, but also implement them successfully on actual quantum computers. To understand the ultimate limits for such quantum machine learning methods, we discuss the quantum no free lunch theorem, which describes a bound on the probability that a quantum device, which can be modelled as a unitary process and is optimised with quantum examples, gives an incorrect output for a random input. This gives us a tool to review the learning behaviour of quantum neural networks in general and the DQNNs in particular. Moreover we expand the area of applications of DQNNs in two directions. In the first case, we include additional information beyond just the training data pairs: since quantum devices are always structured, the resulting data is always structured as well. We modify the DQNN's training algorithm such that knowledge about the graph-structure of the training data pairs is included in the training process and show that this can lead to better generalisation behaviour. Both the original DQNN and the DQNN including graph structure are trained with data pairs in order to characterise an underlying relation. However, in the second extension of the algorithm we aim to learn characteristics of a set of quantum states in order to extend it to quantum states which have similar properties. Therefore we build a generative adversarial model where two DQNNs, called the generator and discriminator, are trained in a competitive way. Overall, we observe that DQNNs can not only be trained efficiently but also, similar to their classical counterparts, modified to suit different applications.
Quantum Neural Networks (QNNs) represent a paradigm shift in neural network design, leveraging the unique properties of quantum mechanics to potentially surpass classical neural networks in terms of efficiency and accuracy. This paper explores the concept of a novel adaptive optimization mechanism within QNNs, utilizing quantum entanglement and quantum state adaptation to enhance training and inference processes. The core of this approach focuses on establishing a dynamic feedback loop that allows the network to continually refine its parameters and structure based on the inherent properties of its data. The proposed mechanism offers a potential path towards more robust and adaptable neural network architectures. We delve into the theoretical foundations and preliminary experimental validation of this approach, demonstrating its potential for accelerating learning and improving generalization performance.
Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical understanding of how the entanglement degree in data affects model performance remains elusive. In this study, we address this knowledge gap by establishing a quantum no-free-lunch (NFL) theorem for learning quantum dynamics using entangled data. Contrary to previous findings, we prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements. With a sufficient number of measurements, increasing the entanglement of training data consistently reduces the prediction error or decreases the required size of the training data to achieve the same prediction error. Conversely, when few measurements are allowed, employing highly entangled data could lead to an increased prediction error. The achieved results provide critical guidance for designing advanced QML protocols, especially for those tailored for execution on early-stage quantum computers with limited access to quantum resources.