Hybrid Quantum-Classical Neural Networks for Recognizing Quantum Phases
Colin Scarato, Johannes Knörzer, Markus K. Hoffmann, Leon C. Sander, Luca Hofele, Shengpu Wang, Kilian Hanke, Ashay Sathe, Dominic Hagmann, Alexander Flasby, Michael J. Hartmann, Petr Zapletal, Andreas Wallraff, Christoph Hellings
quant-phSubmitted Jun 26, 2026• Announced Jun 29, 2026• 0 citations
Identifying quantum phases of matter is key to understanding strongly correlated materials, but remains a challenging task for both conventional computers and current quantum processors. Here, we introduce and implement a hybrid quantum-classical neural network for quantum phase recognition by combining a hardware-efficient parameterized quantum circuit and a feedforward neural network. We jointly train both components with superconducting quantum hardware in the optimization loop, to experimentally demonstrate a classifier for the quantum phases of surface code lattices with up to 4x4 sites in a magnetic field. To learn nonlocal features of the topological phase, we train the hybrid neural network to distinguish topological ground states of the surface code from a featureless ensemble of product states. This allows the trained classifier to distinguish topological ground states from randomly chosen product states, even when subjected to any single-qubit Pauli error. The classifier reaches accuracies above 85% in single-shot measurements, and above 99% when averaging over ten measurements. We expect hybrid neural networks such as the one presented here to be a promising approach for characterizing quantum states in scenarios where classical methods exhibit an unfavorable scaling of sample complexity.
Identifying quantum phases of matter is key to understanding strongly correlated materials, but remains a challenging task for both conventional computers and current quantum processors. Here, we introduce and implement a hybrid quantum-classical neural network for quantum phase recognition by combining a hardware-efficient parameterized quantum circuit and a feedforward neural network. We jointly train both components with superconducting quantum hardware in the optimization loop, to experimentally demonstrate a classifier for the quantum phases of surface code lattices with up to 4x4 sites in a magnetic field. To learn nonlocal features of the topological phase, we train the hybrid neural network to distinguish topological ground states of the surface code from a featureless ensemble of product states. This allows the trained classifier to distinguish topological ground states from randomly chosen product states, even when subjected to any single-qubit Pauli error. The classifier reaches accuracies above 85% in single-shot measurements, and above 99% when averaging over ten measurements. We expect hybrid neural networks such as the one presented here to be a promising approach for characterizing quantum states in scenarios where classical methods exhibit an unfavorable scaling of sample complexity.
With increasing maturity of quantum computers, standard methods for characterizing global properties of their output quantum states via direct measurements and classical post-processing are becoming increasingly impractical due to large measurement costs. Although quantum neural networks could directly process quantum states to identify underlying characteristics with reduced measurement efforts, they often require deep quantum circuits that cannot be implemented on existing devices. To overcome these challenges, we introduce a hybrid quantum-classical neural network that consists of a shallow parameterized quantum circuit, measurements, and a classical neural network. The parameterized quantum circuit performs a nonlocal transformation of the measurement basis, which is jointly trained with the classical neural network to maximize the statistical distance between data obtained by measuring different quantum states. Using supervised learning, we demonstrate that the hybrid neural network distinguishes the topological phase of the surface code from a symmetry-enriched topological phase and random product states. Moreover, this hybrid neural network reduces both inference and training sample complexities of recognizing the topological phase by approximately one order of magnitude compared to a classical neural network trained on randomized Pauli measurements. As this hybrid neural network features a shallow quantum circuit that can be readily implemented on existing quantum computers, it enables the efficient characterization of complex quantum states.
Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.
Quantum convolutional neural networks (QCNNs) are quantum circuits for characterizing complex quantum states. They have been proposed for recognizing quantum phases of matter at low sampling cost and have been designed for condensed matter systems in one dimension. Here we construct a QCNN that can perform phase recognition in two dimensions and correctly identify the phase transition from a toric code phase with Z 2 topological order to the paramagnetic phase. The network also exhibits a noise threshold up to which the topological order is recognized. Furthermore, it captures correlations between all stabilizer elements of the toric code, which cannot be accessed by direct measurements. This increases the threshold for errors leading to such correlations and allows for correctly identifying the topological phase in the presence of strong correlated errors. Our work generalizes phase recognition with QCNNs to higher spatial dimensions and intrinsic topological order, where exploration and characterization via classical numerics become challenging.
The discovery of topological phases in non-equilibrium quantum systems, particularly those driven by periodic fields (Floquet systems), holds immense potential for next-generation quantum technologies. However, identifying and classifying these phases is notoriously challenging, as the associated topological invariants are often non-local, abstract, and lack straightforward experimental observables. Traditional theoretical methods typically rely on idealized models and exact symmetries, which break down in the presence of disorder, interactions, or finite-size effects inherent to real materials. Here, we propose a novel paradigm for the discovery of hidden Floquet topological invariants using hybrid quantum-classical neural networks. Our framework synergistically combines a parameterized quantum circuit, acting as a variational ansatz for the Floquet quasienergy states, with a classical neural network that learns to map the system's time-resolved dynamics onto a robust, data-driven topological classifier. By training the hybrid model on simulated measurement data—such as time-of-flight images or spin-resolved photoelectron spectra—the classical network distills the essential, often obscured, features of the system's unitary evolution without prior knowledge of the underlying Hamiltonian. Crucially, we demonstrate that this approach can uncover novel invariants for anomalous Floquet topological phases that are invisible to conventional frequency-domain analyses. Furthermore, the trained network provides a practical, noise-resilient protocol for real-time classification, enabling the experimental detection of topological transitions solely from short-time dynamics. This work establishes a powerful, generalizable framework for accelerating the discovery and characterization of emergent topological order in complex driven quantum matter, bridging the gap between abstract topological theory and experimental reality.
Deep learning is one of the most successful and far-reaching strategies used in machine learning today. However, the scale and utility of neural networks is still greatly limited by the current hardware used to train them. These concerns have become increasingly pressing as conventional computers are soon expected to approach the physical limitations that will slow their performance improvements in the near future. For these reasons, scientists have begun to explore alternative computing platforms, like quantum computers, for training neural networks. In recent years, variational quantum circuits have emerged as one of the most successful approaches to quantum deep learning on noisy intermediate scale quantum devices. We propose a hybrid quantum-classical neural network architecture where each neuron is a variational quantum circuit. We empirically analyze the performance of this hybrid neural network on a series of binary classification data sets using a simulated IBM universal quantum computer and a state-of-the-art IBM universal quantum computer. On the simulated hardware, we observe that the hybrid neural network achieves around 10% higher classification accuracy and 20% better minimization of the cost function than an individual variational quantum circuit. On the quantum hardware, we observe that each model only performs well when the qubit and gate count is sufficiently small.
Image classification is a fundamental task in deep learning, and recent advances in quantum computing have generated significant interest in quantum neural networks. Traditionally, Convolutional Neural Networks (CNNs) are employed to extract image features, while Multilayer Perceptrons (MLPs) handle decision making. However, parameterized quantum circuits offer the potential to capture complex image features and define sophisticated decision boundaries. In this paper, we present a novel Hybrid Quantum–Classical Neural Network (H-QNN) for image classification, and demonstrate its effectiveness using the MNIST dataset. Our model combines quantum computing with classical supervised learning to enhance classification accuracy and computational efficiency. In this study, we detail the architecture of the H-QNN, emphasizing its capability in feature learning and image classification. Experimental results demonstrate that the proposed H-QNN model outperforms conventional deep learning methods in various training scenarios, showcasing its effectiveness in high-dimensional image classification tasks. Additionally, we explore the broader applicability of hybrid quantum–classical approaches in other domains. Our findings contribute to the growing body of work in quantum machine learning, and underscore the potential of quantum-enhanced models for image recognition and classification.