Hybrid Quantum Neural Networks: Theory, Implementations, and Applications

Abstract

Artificial intelligence has been transformed by deep neural networks, yet the search for new learning architectures continues. Quantum machine learning offers one such direction, and hybrid quantum neural networks, which combine classical neural-network components with quantum information processing units, have emerged as a practical framework for near-term quantum technologies. However, the rapid development of the field across diverse architectures, benchmarks and hardware assumptions makes it difficult to assess the utility of various proposals, identify where genuine advantages may arise, and determine how practitioners can use these models. While recent benchmarks caution that such gains have not yet been demonstrated at scale, theoretical work has identified tasks on which quantum models hold provable advantages, and hybrid approaches have delivered promising results on practical problems using deliberately compact quantum components and substantially fewer trainable parameters. Here, we review hybrid quantum neural networks for the machine-learning and quantum-machine-learning communities. We summarize their main theoretical and methodological foundations, survey some of the most promising architectures developed so far, and examine their implementation challenges and reported performance. By consolidating these perspectives, this review provides a structured view of the state of the field and helps identify promising paths for future research and application-driven development.

This review asks when a quantum component in a classical neural-network pipeline can provide a learning or resource advantage, and how such hybrid quantum neural networks (HQNNs) can be designed, implemented, and evaluated. It surveys theoretical results, architectures, software and hardware platforms, and reported applications in forecasting, classification, physics-informed learning, and planning; it does not present a new algorithm or experimental demonstration. The theoretical discussion connects angle-encoded variational circuits to Fourier models, relates controllability and expressivity to dynamical Lie algebras and Fisher information, and examines barren plateaus and classical simulation or surrogate methods. It emphasizes the tension between trainability and classical hardness: gradients with variance $O(b^{-n})$, for $b>1$, can require exponentially many measurement shots, while several restricted, trainable circuit families admit efficient classical treatment. Reviewed provable learning separations concern quantum data or specially structured, often cryptographic, classical tasks rather than generic datasets. The empirical synthesis reports small-scale parameter reductions and competitive accuracy, predominantly from classical simulation. Examples include 87.9% fewer parameters with 99.21% versus 98.71% accuracy on MNIST, and 71.1–89.6% reductions across five PDE benchmarks with mixed error changes. These are literature results, not a common controlled evaluation. Conversely, a systematic benchmark covering 12 quantum models and 160 binary-classification datasets generally favored classical baselines. Stated limitations include unestablished advantage at scale, task-dependent performance, measurement and hardware overheads, dequantization, and the synthesis and magic-state costs of fault-tolerant implementations. The review calls for matched classical baselines and reporting accuracy alongside runtime, sampling, and energy costs.