As a hybrid of artificial intelligence and quantum computing, quantum neural networks (QNNs) have gained significant attention as a promising application on near-term, noisy intermediate-scale quantum devices. Conventional QNNs are described by parametrized quantum circuits, which perform unitary operations and measurements on quantum states. In this Letter, we propose a novel approach to enhance the expressivity of QNNs by incorporating randomness into quantum circuits. Specifically, we introduce a random layer, which contains single-qubit gates sampled from a trainable ensemble pooling. The prediction of QNN is then represented by an ensemble average over a classical function of measurement outcomes. We prove that our approach can accurately approximate arbitrary target operators using Uhlmann's theorem for majorization, which enables observable learning. Our proposal is demonstrated with extensive numerical experiments, including observable learning, Rényi entropy measurement, and image recognition. We find the expressivity of QNNs is enhanced by introducing randomness for multiple learning tasks, which could have broad application in quantum machine learning.
As a hybrid of artificial intelligence and quantum computing, quantum neural networks (QNNs) have gained significant attention as a promising application on near-term, noisy intermediate-scale quantum devices. Conventional QNNs are described by parametrized quantum circuits, which perform unitary operations and measurements on quantum states. In this Letter, we propose a novel approach to enhance the expressivity of QNNs by incorporating randomness into quantum circuits. Specifically, we introduce a random layer, which contains single-qubit gates sampled from a trainable ensemble pooling. The prediction of QNN is then represented by an ensemble average over a classical function of measurement outcomes. We prove that our approach can accurately approximate arbitrary target operators using Uhlmann's theorem for majorization, which enables observable learning. Our proposal is demonstrated with extensive numerical experiments, including observable learning, Rényi entropy measurement, and image recognition. We find the expressivity of QNNs is enhanced by introducing randomness for multiple learning tasks, which could have broad application in quantum machine learning.
The expressivity of quantum neural networks (QNNs) is a key issue in quantum machine learning (QML), which determines the capability to learn and represent complex functions. The recently proposed Randomness-Enhanced QNNs [Wu et al., Phys. Rev. Lett. 132, 010602 (2024)] use a mixed-unitary channel to accurately express arbitrary single observables. In this work, we generalize the mixed-unitary channel to a unital channel and prove that the unital channel can accurately express multiple arbitrary observables. We show that by training and implementing a unital quantum channel as a multiple-output QNN, we can accurately measure the expectation values of multiple incompatible observables simultaneously by Pauli-Z measurements on distinct output qubits. We prove this accurate expressivity for multiple observables by computing the complete positivity of the unital quantum channel. An iterative algorithm is proposed to compute the theoretical maximal scaling factor of the QNNs' outcomes. Notably, our theoretical analysis shows that it requires fewer copies of state when measuring some incompatible observables by the multiple-output QNNs, which demonstrates a resource efficiency advantage over separately applying projective measurements.
Quantum neural networks (QNNs), represented by parameterized quantum circuits, can be trained in the paradigm of supervised learning to map input data to predictions. Much work has focused on theoretically analyzing the expressive power of QNNs. However, in almost all literature, QNNs' expressive power is numerically validated using only simple univariate functions. We surprisingly discover that state-of-the-art QNNs with strong expressive power can have poor performance in approximating even just a simple sinusoidal function. To fill the gap, we propose four expressibility-enhancing strategies for QNNs: Sinusoidal-friendly embedding, redundant measurement, post-measurement function, and random training data. We analyze the effectiveness of these strategies via mathematical analysis and/or numerical studies including learning complex sinusoidal-based functions. Our results from comparative experiments validate that the four strategies can significantly increase the QNNs' performance in approximating complex multivariable functions and reduce the quantum circuit depth and qubits required.
Quantum Neural Networks (QNN) are considered a candidate for achieving quantum advantage in the Noisy Intermediate Scale Quantum computer (NISQ) era. Several QNN architectures have been proposed and successfully tested on benchmark datasets for machine learning. However, quantitative studies of the QNN-generated entanglement have been investigated only for up to few qubits. Tensor network methods allow to emulate quantum circuits with a large number of qubits in a wide variety of scenarios. Here, we employ matrix product states to characterize recently studied QNN architectures with random parameters up to fifty qubits showing that their entanglement, measured in terms of entanglement entropy between qubits, tends to that of Haar distributed random states as the depth of the QNN is increased. We certify the randomness of the quantum states also by measuring the expressibility of the circuits, as well as using tools from random matrix theory. We show a universal behavior for the rate at which entanglement is created in any given QNN architecture, and consequently introduce a new measure to characterize the entanglement production in QNNs: the entangling speed. Our results characterise the entanglement properties of quantum neural networks, and provides new evidence of the rate at which these approximate random unitaries.
The field of machine learning has experienced rapid development over the last decades and has found numerous applications across diverse domains ranging from industrial to personal settings. Quantum computing, on the other hand, is an emerging field offering a fundamentally different computational paradigm that can potentially surpass traditional computational approaches in certain domains.In this thesis, we explore the intersection between these two fields, commonly referred to as quantum machine learning (QML). In particular, we focus on Quantum neural networks (QNNs), which have gained attention in the field of QML due to the unparalleled success of their classical counterparts. However, the limitations of the current quantum computers have led to hybrid approaches combining classical and quantum devices. We, therefore, use hybrid quantum neural networks (HQNNs) to explore the potential advantages of QNNs in various practical applications.First, we explore the design space of HQNNs as there is no standardized methodology for designing quantum layers in HQNNs. Various quantum layer designs with different data encoding techniques are investigated, identifying the optimal set of quantum gates for different encoding strategies.We also investigate the impact of cost function globality and locality on the performance of HQNNs for real-world applications, namely: binary and multi-class classification. We exploit a single parameterized quantum circuit (PQC) structure with both single (local) and all-qubit (global) measurements. Our results show that the global cost function significantly outperforms the local cost function on multi-class classification. In contrast, both global and local cost functions have comparable performance on binary classification problems.Moreover, we propose a framework for analyzing the impact of various components of HQNNs with respect to each other on multi-class classification tasks, specifically focusing on the data encoding and quantum layers (expressibility and entanglement between qubits). Two ansatz structures, entangled and unentangled, are used to investigate the existence of the barren plateau (BP) in HQNNs. We further evaluate the impact of entanglement on trainability and provide recommendations for different constraint scenarios. The results show that the overall expressibility of the quantum layers impacts the presence of the BP phenomenon and that entanglement plays a role in the training landscapes of HQNNs, depending on the data encoding technique. Finally, we propose a new architecture for quantum neural networks (QNNs) called residual quantum neural networks (ResQNets), inspired by the classical residual neural networks (ResNets), to address the issue of BP in QNNs. ResQNets split the conventional QNN architecture into multiple quantum nodes (QNs), each with its own PQC, allowing residual connections between QNs. The results show that incorporating residual connections results in improved training performance, making ResQNets a promising solution for overcoming the problem of BP in QNNs.
Quantum neural networks are a prominent model of quantum machine learning. Their training consists in the minimization of a given loss function over a parametrized family of quantum circuits, mathematically described by unitary operators, or, more generally, completely positive linear maps. In this work, we extend the notion of quantum neural network, using random sampling and classical data processing to enlarge the optimization space in a way that includes linear combinations of completely positive maps. Our extended model, called virtual quantum neural networks, leverages its enlarged optimization space to achieve increased expressivity and improved noise robustness. These benefits are illustrated in three representative tasks: quantum error mitigation, binary classification, and estimation of ground-state energies. Overall, virtual quantum neural networks offer a flexible learning paradigm that expands the space of achievable computations and strengthens the applications of near-term quantum hardware.
Quantum machine learning (QML) is promising for potential speedups and improvements in conventional machine learning (ML) tasks. Existing QML models that use deep parametric quantum circuits (PQC) suffer from a large accumulation of gate errors and decoherence. To circumvent this issue, we propose a new QML architecture called QNet. QNet consists of several small quantum neural networks (QNN). Each of these smaller QNN’s can be executed on small quantum computers that dominate the NISQ-era machines. By carefully choosing the size of these QNN’s, QNet can exploit arbitrary size quantum computers to solve supervised ML tasks of any scale. It also enables heterogeneous technology integration in a single QML application. Through empirical studies, we show the trainability and generalization of QNet and the impact of various configurable variables on its performance. We compare QNet performance against existing models and discuss potential issues and design considerations. In our study, we show 43% better accuracy on average over the existing models on noisy quantum hardware emulators. More importantly, QNet provides a blueprint to build noise-resilient QML models with a collection of small quantum neural networks with near-term noisy quantum devices.