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Hybrid Quantum-Classical Neural Networks for Discovering Hidden Floquet Topological Invariants

Lee Cheng, Chang Lei

Published Jul 14, 2026• 0 citations

Abstract

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.

DOI: 10.14293/pr2199.004106.v1

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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.

Floquet engineering offers an unparalleled platform for realizing novel non-equilibrium topological phases. However, the unique structure of Floquet systems, which includes multiple quasienergy gaps, poses a significant challenge to classification using conventional analytical methods. We propose a novel unsupervised machine learning framework that employs a kernel defined in momentum-time (

\boldsymbol{k},t

) space, constructed directly from Floquet-Bloch eigenstates. This approach is intrinsically data-driven and requires no prior knowledge of the underlying topological invariants, providing a fundamental advantage over prior methods that rely on abstract concepts like the micromotion operator or homotopic transformations. Crucially, this work successfully reveals the intrinsic topological characteristics encoded within the Floquet eigenstates themselves. We demonstrate that our method robustly and simultaneously identifies the topological invariants associated with both the

0

-gap and the

π

-gap across various symmetry classes (1D AIII, 1D D, and 2D A), establishing a robust methodology for the systematic classification and discovery of complex non-equilibrium topological matter.

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.

Abstract Floquet engineering provides a unique platform for realizing novel non-equilibrium topological phases. However, the intrinsic multi-gap structure of Floquet systems poses significant challenges for their classification using conventional analytical methods. Here, we propose an unsupervised machine learning framework that employs a kernel defined directly in momentum-time ( k , t ) space, constructed from Floquet–Bloch states. This data-driven approach does not require prior knowledge of topological invariants. By using a compact computational procedure that works directly with the system’s eigenstates, the method can identify topologically distinct phases associated with both the 0 -gap and the π -gap across various symmetry classes (1D AIII, 1D D, and 2D A). Our results demonstrate that key topological characteristics of Floquet systems are directly accessible from their eigenstates in ( k , t ) space, providing a tool for the systematic classification and study of non-equilibrium topological phases without a priori invariant information.

The conventional characterization of periodically driven systems usually necessitates the time-domain information beyond Floquet bands, hence lacking universal and direct schemes of measuring Floquet topological invariants. Here we propose a unified theory, based on quantum quenches, to characterize generic d-dimensional Floquet topological phases in which the topological invariants are constructed with only minimal information of the static Floquet bands. For a d-dimensional phase that is initially static and trivial, we introduce the quench dynamics by suddenly turning on the periodic driving. We show that the quench dynamics exhibits emergent topological patterns in (d-1)-dimensional momentum subspaces where Floquet bands cross, from which the Floquet topological invariants are directly obtained. This result provides a simple and unified characterization in which one can extract the number of conventional and anomalous Floquet boundary modes and identify the topologically protected singularities in the phase bands. These applications are illustrated with one- and two-dimensional models that are readily accessible in cold-atom experiments. Our study opens a new framework for the characterization of Floquet topological phases.

We investigate the dynamical characterization theory for periodically driven systems in which Floquet topology can be fully detected by emergent topological patterns of quench dynamics in momentum subspaces called band-inversion surfaces. We improve the results of a recent work [Zhang et al., Phys. Rev. Lett. 125, 183001 (2020)] and propose a more flexible scheme to characterize a generic class of

d

-dimensional Floquet topological phases classified by

\mathbb{Z}

-valued invariants by applying a quench along an arbitrary spin-polarization axis. Our basic idea is that by disassembling the Floquet system into multiple static subsystems that are periodic in quasienergy, a full characterization of Floquet topological phases reduces to identifying a series of bulk topological invariants for time-independent Hamiltonians, which greatly enhances the convenience and flexibility of the measurement. We illustrate the scheme by numerically analyzing two experimentally realizable models in two and three dimensions, respectively, and adopting two different but equivalent viewpoints to examine the dynamical characterization. Finally, considering the imperfection of experiment, we demonstrate that the present scheme can also be applied to a general situation where the initial state is not completely polarized. This study provides an immediately implementable approach for dynamically classifying Floquet topological phases in ultracold atoms or other quantum simulators.