The conflict between trainability and expressibility is a key challenge in variational quantum computing and quantum machine learning. Resolving this conflict necessitates designing specific quantum neural networks (QNN) tailored for specific problems, which urgently needs a general and efficient method to predict the learnability of QNNs without costly training. In this work, we demonstrate a simple and efficient metric for learnability by comparing the fluctuations of the given training landscape with standard learnable landscapes. This metric shows surprising effectiveness in predicting learnability as it unifies the effects of insufficient expressibility, barren plateaus, bad local minima, and overparametrization. Importantly, it can be estimated efficiently on classical computers via Clifford sampling without actual training on quantum devices. We conduct extensive numerical experiments to validate its effectiveness regarding physical and random Hamiltonians. We also prove a compact lower bound for the metric in locally scrambled circuits as analytical guidance. Our findings enable efficient predictions of learnability, allowing fast selection of suitable QNN architectures for a given problem without training, which can greatly improve the efficiency especially when access to quantum devices is limited.
The conflict between trainability and expressibility is a key challenge in variational quantum computing and quantum machine learning. Resolving this conflict necessitates designing specific quantum neural networks (QNN) tailored for specific problems, which urgently needs a general and efficient method to predict the learnability of QNNs without costly training. In this work, we demonstrate a simple and efficient metric for learnability by comparing the fluctuations of the given training landscape with standard learnable landscapes. This metric shows surprising effectiveness in predicting learnability as it unifies the effects of insufficient expressibility, barren plateaus, bad local minima, and overparametrization. Importantly, it can be estimated efficiently on classical computers via Clifford sampling without actual training on quantum devices. We conduct extensive numerical experiments to validate its effectiveness regarding physical and random Hamiltonians. We also prove a compact lower bound for the metric in locally scrambled circuits as analytical guidance. Our findings enable efficient predictions of learnability, allowing fast selection of suitable QNN architectures for a given problem without training, which can greatly improve the efficiency especially when access to quantum devices is limited.
Balancing trainability and expressibility is a central challenge in variational quantum computing, and quantum architecture search (QAS) plays a pivotal role by automatically designing problem-specific parameterized circuits that address this trade-off. In this work, we introduce a scalable, training-free QAS framework that efficiently explores and evaluates quantum circuits through landscape fluctuation analysis. This analysis captures key characteristics of the cost function landscape, enabling accurate prediction of circuit learnability without costly training. By combining this metric with a streamlined two-level search strategy, our approach identifies high-performance, large-scale circuits with higher accuracy and fewer gates. We further demonstrate the practicality and scalability of our method, achieving significantly lower classical resource consumption compared to prior work. Notably, our framework attains robust performance on a challenging 50-qubit quantum many-body simulation, highlighting its potential for addressing complex quantum problems.
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.
Abstract Quantum neural networks (QNNs) are widely used as trainable models for solving variational problems, where their ability to represent complex functions directly determines performance. However, accurately quantifying this expressivity remains a major challenge, limiting the understanding of the expressive power of QNNs. Here, we introduce the effective rank κ as a quantitative measure of expressivity. We show that the expressivity of a QNN is determined by the interplay of three factors: the dataset, the measurement operators, and the circuit structure. Unlike conventional metrics based on parameter sampling or entanglement structure, κ captures the number of independent parameters that effectively contribute to the model output, providing an operational and data-dependent characterization of expressivity while avoiding costly parameter sampling. We demonstrate that κ can reach its theoretical maximum 4 n − 1 for an n -qubit system when the circuit architecture, input distribution, and measurement protocol are jointly optimized. Leveraging this insight, we employ κ as a design objective within a reinforcement learning framework with a self-attention transformer agent to automatically discover highly expressive circuit architectures. Our results show that the proposed approach efficiently identifies high-performing circuit configurations, outperforming heuristic designs and random search in terms of sample efficiency. By bridging theoretical characterization with automated design, our approach provides a practical, scalable, and computationally efficient route for constructing expressive quantum circuits, advancing the development of quantum machine learning models.
Abstract Barren plateaus are the main obstacle to training quantum machine-learning models. On a barren plateau, the cost-function gradient shrinks exponentially as the circuit grows, and training stalls. Today the only reliable test is to measure it. That means sampling many gradients for every candidate circuit, which is expensive. We ask a different question. Can a classical model predict trainability from the circuit architecture alone? The architecture here means qubit count, depth, ansatz type, entanglement pattern, entangler gate, and cost-observable locality. We test this directly. We use an exact statevector simulator to build a dataset of \Nrows{} random variational circuits. For each circuit we estimate the gradient variance Var[∂C/∂θ] from \Samples{} random parameter vectors, using the parameter-shift rule. This variance is the label. A gradient-boosted regressor then predicts log10Var with $R^2 = \RsqHold{}$ on held-out architectures. A classifier separates barren from trainable circuits with an area under the ROC curve of \AucHold{}. We also test transfer to larger circuits. We train only on small circuits and test on the two largest unseen sizes (n=11,12). The model still reaches $R^2 = \RsqExtrap{}$. Permutation importance then recovers established theory. Cost-observable locality and entanglement structure matter most. We make no quantum-advantage claim. This is classical machine learning about quantum circuits. It offers a cheap way to screen ans\"atze, and it shows which design choices cause untrainability.
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.