Abstract In noisy intermediate-scale quantum era, the research on the combination of artificial intelligence and quantum computing has been greatly developed. Here we propose a quantum circuit-based algorithm to implement quantum residual neural networks, where the residual connection channels are constructed by introducing auxiliary qubits to data-encoding and trainable blocks in quantum neural networks. We prove that when this particular network architecture is applied to a l -layer data-encoding, the number of frequency generation forms extends from one, namely the difference of the sum of generator eigenvalues, to $${{{{{{{\mathcal{O}}}}}}}}({l}^{2})$$ O ( l 2 ) , and the flexibility in adjusting Fourier coefficients can also be improved. It indicates that residual encoding can achieve better spectral richness and enhance the expressivity of various parameterized quantum circuits. Extensive numerical demonstrations in regression tasks and image classification are offered. Our work lays foundation for the complete quantum implementation of classical residual neural networks and offers a quantum feature map strategy for quantum machine learning.
Abstract In noisy intermediate-scale quantum era, the research on the combination of artificial intelligence and quantum computing has been greatly developed. Here we propose a quantum circuit-based algorithm to implement quantum residual neural networks, where the residual connection channels are constructed by introducing auxiliary qubits to data-encoding and trainable blocks in quantum neural networks. We prove that when this particular network architecture is applied to a l -layer data-encoding, the number of frequency generation forms extends from one, namely the difference of the sum of generator eigenvalues, to O(l2) O ( l 2 ) , and the flexibility in adjusting Fourier coefficients can also be improved. It indicates that residual encoding can achieve better spectral richness and enhance the expressivity of various parameterized quantum circuits. Extensive numerical demonstrations in regression tasks and image classification are offered. Our work lays foundation for the complete quantum implementation of classical residual neural networks and offers a quantum feature map strategy for quantum machine learning.
The field of quantum neural networks, an emerging interdisciplinary frontier bridging quantum computing and machine learning, has recently garnered substantial attention in the scientific community. In this work, we propose a complex-valued quantum neural network in the quantum circuit framework through expanding the number domains of conventional quantum machine learning architecture from the real-valued domain to the complex-valued plane. Such an expansion can alleviate the unitary constraints in quantum feature maps and optimization blocks, and introduce enhanced nonlinear characteristics. It is proven theoretically that the expressivity can be significantly enhanced through the frequency-dependent adaptability and the exponential or hyperbolic functional distribution extension for the Fourier coefficients of quantum models' outputs. Moreover, analysis based on covering numbers indicates that our algorithm can achieve an adaptive balance between expressivity and generalization via the trainable scaling factor, while exhibiting superior parameter efficiency relative to unitary quantum learning models. The performance of the proposed algorithms is demonstrated through extensive numerical simulations in regression and classification tasks, including an experimental realization on the superconducting quantum processor. Our work advances the harnessing of imaginarity resources of quantum theory for learning tasks, achieving a theoretically grounded performance enhancement, and underscores the potential of improving quantum neural networks through structured design for future applications.
Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.
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 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.
A possible approach to tackling computationally demanding issues beyond the capabilities of conventional algorithms is the use of quantum circuits, which provide a fundamental foundation for utilizing the power of quantum computation. This abstract examines how quantum circuits are implemented, emphasizing important topics such as circuit optimization, gate operations, and qubit manipulation. We explore different methods of encoding quantum information into topological qubits, trapped ions, and superconducting circuits, among other physical qubit formats. Furthermore, we go over the function of gate operations in quantum circuits, emphasizing crucial gates like phase, Hadamard, and CNOT gates, which serve as the foundation for quantum algorithms. Additionally, we look at methods for maximizing the efficiency and dependability of quantum processing by designing quantum circuits to minimize gate counts, depth, and error rates. To enhance the capabilities of quantum computing, this abstract presents a novel synthesis of quantum neural networks (QNNs) and quantum parameterized circuits. Through the use of parameterized quantum circuits' inherent flexibility, our approach makes it easier to design and optimize quantum circuits with flexible parameters, providing a flexible foundation for a variety of quantum algorithms. Using this amalgamation, we unleash the capacity of quantum computers to address intricate optimization and machine learning assignments with unparalleled effectiveness and expandability. Our solution bridges the gap between traditional machine learning approaches and quantum computing paradigms by enabling the smooth integration of classical data processing techniques with quantum computation. We use quantum states as data representations in our implementation of quantum neural networks, and we optimize trainable parameters to meet predetermined goals. This new paradigm opens up the investigation of the inherent features of quantum states and provides new perspectives on their processing power compared to their classical counterparts. We show, by thorough testing and analysis, how effective quantum parameterized circuits and QNNs are at solving practical problems in a variety of fields. Our work sets the ground for revolutionary breakthroughs in quantum information processing by pushing the boundaries of quantum computation and machine learning integration, offering hitherto unseen insights and solutions to challenging challenges in science, engineering, and beyond.
Quantum neural networks (QNNs), typically implemented via parameterized quantum circuits (PQCs), offer a potential route to quantum advantage in learning. We review recent progress through the Fourier structure of PQCs: the encoding Hamiltonian fixes the accessible frequency set, while training estimates the corresponding Fourier coefficients. This perspective explains how QNNs can mitigate the spectral bias that limits classical deep neural networks and motivates task-specialized designs. In particular, quantum ordinary differential equation (QODE) solver, quantum convolutional neural networks (QCNNs) and quantum reinforcement learning (QRL) exploit frequency pinching to better capture high-frequency image features and structured value landscapes. We further discuss trainability on noisy intermediate-scale quantum (NISQ) hardware, emphasizing Lie-algebraic and Fourier views of barren plateaus and practical mitigation strategies. Together, the Fourier and Lie perspectives yield concrete design principles and motivate quantum-native architectures built around spectral bias avoidance rather than classical imitation. Quantum neural networks (QNNs), implemented via parameterized quantum circuits (PQCs), promise a path to quantum advantage in learning by addressing the limitations of classical deep neural networks. Here, the authors explore the Fourier structure of PQCs to mitigate spectral bias, enhancing applications like quantum ODE solvers and quantum reinforcement learning, with implications for designing quantum-native architectures.