Variational quantum circuit algorithm is suggested as a machine learning paradigm to utilize the power of Noisy Intermediate-Scale Quantum hardware. Recent efforts lead to better theoretical understanding of the expressibility and generalization ability of quantum circuits. In this work, we compare our previous results on the quantum circuit learnability based on VC theory to some preliminary results based on PAC-Bayesian method. Some recent developments based on other generalization bounds due to other research groups are also reviewed and discussed.
Variational quantum circuit algorithm is suggested as a machine learning paradigm to utilize the power of Noisy Intermediate-Scale Quantum hardware. Recent efforts lead to better theoretical understanding of the expressibility and generalization ability of quantum circuits. In this work, we compare our previous results on the quantum circuit learnability based on VC theory to some preliminary results based on PAC-Bayesian method. Some recent developments based on other generalization bounds due to other research groups are also reviewed and discussed.
Applying quantum processors to model a high-dimensional function approximator is a typical method in quantum machine learning with potential advantage. It is conjectured that the unitarity of quantum circuits provides possible regularization to avoid overfitting. However, it is not clear how the regularization interplays with the expressibility under the limitation of current Noisy-Intermediate Scale Quantum devices. In this article, we perform simulations and theoretical analysis of the quantum circuit learning problem with hardware-efficient ansatz. Thorough numerical simulations show that the expressibility and generalization error scaling of the ansatz saturate when the circuit depth increases, implying the automatic regularization to avoid the overfitting issue in the quantum circuit learning scenario. This observation is supported by the theory on PAC learnability, which proves that VC dimension is upper bounded due to the locality and unitarity of the hardware-efficient ansatz. Our study provides supporting evidence for automatic regularization by unitarity to suppress overfitting and guidelines for possible performance improvement under hardware constraints.
Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model's overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.
Variational Quantum Circuits (VQCs) have emerged as a cornerstone of hybrid quantum–classical algorithms designed to harness the computational potential of near-term quantum devices. By combining parameterized quantum gates with classical optimization, VQCs provide a flexible framework for tackling machine learning, chemistry, and optimization problems intractable for classical methods. This review comprehensively overviews VQC design principles, ansatz structures, optimization strategies, and real-world applications. Furthermore, we discuss fundamental challenges such as barren plateaus, the expressibility–trainability trade-off, and current noisy intermediate-scale quantum (NISQ) hardware limitations. Finally, we highlight emerging directions that could enable scalable, noise-resilient, and physically interpretable variational quantum models for future quantum computing applications
Abstract Quantum neural network (QNN), or equivalently, the variational quantum circuits with a gradient-based classical optimizer, has been broadly applied to many experimental proposals for noisy intermediate scale quantum (NISQ) devices. However, the learning capability of QNN remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate noise introduced by NISQ machines. In this study, we theoretically explore the learnability of QNN from the perspective of the trainability and generalization. Particularly, we derive the convergence performance of QNN under the NISQ setting, and identify classes of computationally hard concepts that can be efficiently learned by QNN. Our results demonstrate that large gate noise, few quantum measurements, and deep circuit depth will lead to poor convergence rates of QNN towards the empirical risk minimization. Moreover, we prove that any concept class, which is efficiently learnable by a restricted quantum statistical query (QSQ) learning model, can also be efficiently learned by QNN. Since the restricted QSQ learning model can tackle certain problems such as parity learning with a runtime speedup, our result suggests that QNN established on NISQ devices will retain the quantum advantage. Our work provides the theoretical guidance for developing advanced QNNs and opens up avenues for exploring quantum advantages using NISQ devices.
We consider the learnability of the quantum neural network (QNN) built on the variational hybrid quantum-classical scheme, which remains largely unknown due to the non-convex optimization landscape, the measurement error, and the unavoidable gate errors introduced by noisy intermediate-scale quantum (NISQ) machines. Our contributions in this paper are multi-fold. First, we derive the utility bounds of QNN towards empirical risk minimization, and show that large gate noise, few quantum measurements, and deep circuit depth will lead to the poor utility bounds. This result also applies to the variational quantum circuits with gradient-based classical optimization, and can be of independent interest. We then prove that QNN can be treated as a differentially private (DP) model. Thirdly, we show that if a concept class can be efficiently learned by QNN, then it can also be effectively learned by QNN even with gate noise. This result implies the same learnability of QNN whether it is implemented on noiseless or noisy quantum machines. We last exhibit that the quantum statistical query (QSQ) model can be effectively simulated by noisy QNN. Since the QSQ model can tackle certain tasks with runtime speedup, our result suggests that the modified QNN implemented on NISQ devices will retain the quantum advantage. Numerical simulations support the theoretical results.
Quantum computers hold unprecedented potentials for machine learning applications. Here, we prove that physical quantum circuits are PAC (probably approximately correct) learnable on a quantum computer via empirical risk minimization: to learn a quantum circuit with at most nc gates and each gate acting on a constant number of qubits, the sample complexity is bounded by O~(nc+1). In particular, we explicitly construct a family of variational quantum circuits with O(nc+1) elementary gates arranged in a fixed pattern, which can represent all physical quantum circuits consisting of at most nc elementary gates. Our results provide a valuable guide for quantum machine learning in both theory and experiment.