Trainability and generalization are two key issues in quantum machine learning on the basis of variational algorithms. It has been demonstrated in [1] that with the reduced-domain initialization strategy, the trainability of variational parameters can be greatly enhanced for complex objectives and general deep circuits. In this paper, we further explore the generalization of variational quantum learning under the reduced-domain initialization. We theoretically derive the upper bound of the generalization error and demonstrate the enhancement as compared to that under the usual parameter initialization methods. Thus, our results clearly show that the reduced-domain initialization can push the boundary of applicability of deep variational quantum learning toward practical applications.
Trainability and generalization are two key issues in quantum machine learning on the basis of variational algorithms. It has been demonstrated in [1] that with the reduced-domain initialization strategy, the trainability of variational parameters can be greatly enhanced for complex objectives and general deep circuits. In this paper, we further explore the generalization of variational quantum learning under the reduced-domain initialization. We theoretically derive the upper bound of the generalization error and demonstrate the enhancement as compared to that under the usual parameter initialization methods. Thus, our results clearly show that the reduced-domain initialization can push the boundary of applicability of deep variational quantum learning toward practical applications.
Parameterized quantum circuits (PQCs) have been widely used as a machine learning model to explore the potential of achieving quantum advantages for various tasks. However, training PQCs is notoriously challenging owing to the phenomenon of plateaus and/or the existence of (exponentially) many spurious local minima. To enhance trainability, in this work we propose an efficient parameter initialization strategy with theoretical guarantees. We prove that, by reducing the initial domain of each parameter inversely proportional to the square root of the circuit depth, the magnitude of the cost gradient decays at most polynomially with respect to the qubit count and circuit depth. Our theoretical results are substantiated through numerical simulations of variational quantum eigensolver tasks. Moreover, we demonstrate that the reduced-domain initialization strategy can protect specific quantum neural networks from exponentially many spurious local minima. Our results highlight the significance of an appropriate parameter initialization strategy, offering insights to enhance the trainability and convergence of variational quantum algorithms.
Variational quantum circuits have been widely employed in quantum simulation and quantum machine learning in recent years. However, quantum circuits with random structures have poor trainability due to the exponentially vanishing gradient with respect to the circuit depth and the qubit number. This result leads to a general standpoint that deep quantum circuits would not be feasible for practical tasks. In this work, we propose an initialization strategy with theoretical guarantees for the vanishing gradient problem in general deep quantum circuits. Specifically, we prove that under proper Gaussian initialized parameters, the norm of the gradient decays at most polynomially when the qubit number and the circuit depth increase. Our theoretical results hold for both the local and the global observable cases, where the latter was believed to have vanishing gradients even for very shallow circuits. Experimental results verify our theoretical findings in the quantum simulation and quantum chemistry.
Abstract Variational quantum linear solvers (VQLSs) are a promising class of hybrid quantum--classical algorithms for solving linear systems on near-term quantum devices. However, the performance of VQLSs is often impeded by barren plateaus, particularly when using randomly initialized variational quantum circuits (VQCs). To mitigate this issue, we propose \textit{PVLS}, a GNN-based parameter initialization framework that improves both convergence speed and final solution quality. By reformulating the linear system Ax=b as a graph with A encoded in the edges and b as node features, PVLS learns to predict effective initial VQC parameters. Our method is trained on thousands of randomly generated matrices with varying dimensions (n∈[4,10]), using optimized VQC parameters as ground-truth labels. On unseen test instances, PVLS reduces the initial cost by an average of 81.3\% and the final loss by 71\% compared to random initialization. PVLS also accelerates convergence, reducing the number of optimization steps by more than 60\% on average. We further evaluate PVLS on ten real-world sparse matrices, demonstrating its generalization capability and robustness. Our results highlight the utility of machine-learned priors in improving the trainability of VQLSs and alleviating optimization challenges in variational quantum algorithms.
Recent advancements in machine learning have revolutionised research across various fields. Despite their success, conventional learning techniques are hindered by their significant computational resources and energy requirements. Prompted by recent experimental breakthroughs in quantum computing, variational quantum machine learning (QML) – machine learning integrated with variational quantum circuits (VQCs) – has emerged as a promising alternative. Nonetheless, the theoretical framework underpinning the advantages of variational QML is still rudimentary. Specifically, the training of VQCs faces several challenges, such as the barren plateau problem, where the gradient diminishes exponentially with an increasing qubits. A related issue arises in variational QML training, where the convergence rate is exponentially small. In this thesis, we present theoretically guaranteed solutions to these challenges. First, we construct innovative circuit architectures to address the vanishing gradient problem in deep VQCs. We propose quantum controlled-layer and quantum ResNet structures, demonstrating that the expected gradient norm's lower bound is unaffected by the increase in qubits and circuit depth. Next, we introduce an initialization strategy to mitigate the vanishing gradient issue in general deep quantum circuits. We prove that Gaussian-initialized parameters ensure the gradient norm's decay rate remains inversely polynomial despite the increase in qubit numbers and circuit depth. Finally, we propose a novel and effective theory for analysing the training of quantum neural networks with moderate depths. We prove that, under certain randomness conditions in the circuits and datasets, training converges linearly with a rate inversely proportional to the dataset size. Our approach surpasses previous results, achieving exponentially larger convergence rates with modest depth, or conversely, requiring exponentially less depth for equivalent rates.
In the era of noisy intermediate-scale quantum (NISQ), variational quantum circuits (VQCs) have been widely applied in various domains, demonstrating the potential advantages of quantum circuits over classical models. Similar to classic models, VQCs can be optimized by various gradient-based methods. However, the optimization may get stuck in barren plateaus initially or trapped in saddle points during training. These gradient-related issues can severely impact the trainability of VQCs. In this work, we propose a strategy that regularizes model parameters with prior knowledge of the training data and Gaussian noise diffusion. We conduct ablation studies to verify the effectiveness of our strategy across four public datasets and demonstrate that our method can improve the trainability of VQCs against the above-mentioned gradient issues.
The quest for successful variational quantum machine learning (QML) relies on the design of suitable parametrized quantum circuits (PQCs), as analogues to neural networks in classical machine learning. Successful QML models must fulfill the properties of trainability and non-dequantization, among others. Recent works have highlighted an intricate interplay between trainability and dequantization of such models, which is still unresolved. In this work we contribute to this debate from the perspective of machine learning, proving a number of results identifying, among others when trainability and non-dequantization are not mutually exclusive. We begin by providing a number of new somewhat broader definitions of the relevant concepts, compared to what is found in other literature, which are operationally motivated, and consistent with prior art. With these precise definitions given and motivated, we then study the relation between trainability and dequantization of variational QML. Next, we also discuss the degrees of "variationalness" of QML models, where we distinguish between models like the hardware efficient ansatz and quantum kernel methods. Finally, we introduce recipes for building PQC-based QML models which are both trainable and nondequantizable, and corresponding to different degrees of variationalness. We do not address the practical utility for such models. Our work however does point toward a way forward for finding more general constructions, for which finding applications may become feasible.