Quantum variational algorithms have been one of major applications of quantum computing with current quantum devices. There are recent attempts to establish the foundation for these algorithms. A possible approach is to characterize the training dynamics with quantum neural tangent kernel. In this work, we construct the kernel for two models, Quantun Ensemble and Quantum Neural Network, and show the convergence of these models in the limit of infinitely many qubits. We also show applications of the kernel limit in regression tasks.
Quantum variational algorithms have been one of major applications of quantum computing with current quantum devices. There are recent attempts to establish the foundation for these algorithms. A possible approach is to characterize the training dynamics with quantum neural tangent kernel. In this work, we construct the kernel for two models, Quantun Ensemble and Quantum Neural Network, and show the convergence of these models in the limit of infinitely many qubits. We also show applications of the kernel limit in regression tasks.
A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learning when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural networks, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a non-negligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sublinear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over-parameterization. The new dynamics capture the change of convergence rate during training and implies that the range of measurements is crucial to the fast QNN convergence.
In this work, we study the Neural Tangent Kernel (NTK) of Matrix Product States (MPS) and the convergence of its NTK in the infinite bond dimensional limit. We prove that the NTK of MPS asymptotically converges to a constant matrix during the gradient descent (training) process (and also the initialization phase) as the bond dimensions of MPS go to infinity by the observation that the variation of the tensors in MPS asymptotically goes to zero during training in the infinite limit. By showing the positive-definiteness of the NTK of MPS, the convergence of MPS during the training in the function space (space of functions represented by MPS) is guaranteed without any extra assumptions of the data set. We then consider the settings of (supervised) Regression with Mean Square Error (RMSE) and (unsupervised) Born Machines (BM) and analyze their dynamics in the infinite bond dimensional limit. The ordinary differential equations (ODEs) which describe the dynamics of the responses of MPS in the RMSE and BM are derived and solved in the closed-form. For the Regression, we consider Mercer Kernels (Gaussian Kernels) and find that the evolution of the mean of the responses of MPS follows the largest eigenvalue of the NTK. Due to the orthogonality of the kernel functions in BM, the evolution of different modes (samples) decouples and the "characteristic time" of convergence in training is obtained.
Abstract Predicting the performance of a Quantum Neural Network (QNN) on a given dataset before its training can save critical resources. However, obtaining such a priori information is, in general, very challenging. The Quantum Neural Tangent Kernel (QNTK) has recently emerged as a tool to mathematically describe the behavior of QNNs. Here, we provide practical guidelines for using the QNTK to diagnose the performance of a given QNN model. In particular, we link QNTK spectral features to the training dynamics, and we show how a first-order kernel approximation can predict the expected inference performance. By analyzing the interplay between the number of model parameters and of training samples, we identify a mismatch at the interpolation threshold: while the QNTK exhibits a double descent in generalization error, we observe that QNNs do not show this behavior in practice. This suggests that QNTK diagnostics are mostly reliable for overparametrized models well above the interpolation regime. Extensive numerical simulations across different architectures and datasets validate our approach. Our results demonstrate that QNTK diagnostics yields useful information about QNN behavior for sufficiently deep circuits, and enable detecting and addressing potential shortcomings in model design.
Stochastic contextual bandits are fundamental for sequential decision-making but pose significant challenges for existing neural network-based algorithms, particularly when scaling to quantum neural networks (QNNs) due to issues such as massive over-parameterization, computational instability, and the barren plateau phenomenon. This paper introduces the Quantum Neural Tangent Kernel-Upper Confidence Bound (QNTK-UCB) algorithm, a novel algorithm that leverages the Quantum Neural Tangent Kernel (QNTK) to address these limitations.
By freezing the QNN at a random initialization and utilizing its static QNTK as a kernel for ridge regression, QNTK-UCB bypasses the unstable training dynamics inherent in explicit parameterized quantum circuit training while fully exploiting the unique quantum inductive bias. For a time horizon T and K actions, our theoretical analysis reveals a significantly improved parameter scaling of Ω((TK)3) for QNTK-UCB, a substantial reduction compared to Ω((TK)8) required by classical NeuralUCB algorithms for similar regret guarantees. Empirical evaluations on non-linear synthetic benchmarks and quantum-native variational quantum eigensolver tasks demonstrate QNTK-UCB's superior sample efficiency in low-data regimes. This work highlights how the inherent properties of QNTK provide implicit regularization and a sharper spectral decay, paving the way for achieving ``quantum advantage'' in online learning.
Abstract Recently, quantum neural networks or quantum–classical neural networks (qcNN) have been actively studied, as a possible alternative to the conventional classical neural network (cNN), but their practical and theoretically-guaranteed performance is still to be investigated. In contrast, cNNs and especially deep cNNs, have acquired several solid theoretical basis; one of those basis is the neural tangent kernel (NTK) theory, which can successfully explain the mechanism of various desirable properties of cNNs, particularly the global convergence in the training process. In this paper, we study a class of qcNN composed of a quantum data-encoder followed by a cNN. The quantum part is randomly initialized according to unitary 2-designs, which is an effective feature extraction process for quantum states, and the classical part is also randomly initialized according to Gaussian distributions; then, in the NTK regime where the number of nodes of the cNN becomes infinitely large, the output of the entire qcNN becomes a nonlinear function of the so-called projected quantum kernel. That is, the NTK theory is used to construct an effective quantum kernel, which is in general nontrivial to design. Moreover, NTK defined for the qcNN is identical to the covariance matrix of a Gaussian process, which allows us to analytically study the learning process. These properties are investigated in thorough numerical experiments; particularly, we demonstrate that the qcNN shows a clear advantage over fully classical NNs and qNNs for the problem of learning the quantum data-generating process.
Quantum tangent kernel methods provide an efficient approach to analyzing the performance of quantum machine learning models in the infinite-width limit, which is of crucial importance in designing appropriate circuit architectures for certain learning tasks. Recently, they have been adapted to describe the convergence rate of training errors in quantum neural networks in an analytical manner. Here, we study the connections between the expressibility and value concentration of quantum tangent kernel models. In particular, for global loss functions, we rigorously prove that high expressibility of both the global and local quantum encodings can lead to exponential concentration of quantum tangent kernel values to zero. Whereas for local loss functions, such issue of exponential concentration persists owing to the high expressibility, but can be partially mitigated. We further carry out extensive numerical simulations to support our analytical theories. Our discoveries unveil a fundamental feature of quantum neural tangent kernels, indicating that the issue of their concentration cannot be bypassed merely by transitioning to a local encoding scheme while maintaining high expressibility. This offers valuable insights for the design of wide quantum variational circuit models in practical applications.