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Journal of Physics A Mathematical and Theoretical

Hybrid quantum–classical learning of nonlinear entanglement witnesses via continuous-variable quantum neural networks

Mohammad Rezaei Shokouh, Hossein Davoodi Yeganeh

Published Oct 6, 2026• 0 citations
Publisher

Abstract

Abstract Characterizing entanglement in high-dimensional continuous-variable (CV) systems remains a formidable challenge, particularly when full quantum state tomography is prohibitively resource-intensive. In this work, a hybrid quantum–classical framework is introduced that learns nonlinear entanglement witnesses directly from sparse measurement data via CV quantum neural networks (CV-QNNs). The architecture combines variational interferometers, squeezing operations, and non-Gaussian Kerr nonlinearities with a classical neural head to output a scalar witness value. To rigorously isolate the information-theoretic sample-complexity advantage, a Coherence Trap benchmark is devised in which genuinely entangled and separable states are engineered to exhibit identical photon-number statistics, thereby rendering them indistinguishable to classical classifiers operating on intensity histograms. Numerical simulations show that while classical models fail to classify these states without full tomography, the CV-QNN achieves 100 % accuracy by extracting phase-dependent quantum correlations prior to measurement. To address scalability and optimization under realistic finite-shot measurement noise, a variance-reduced simultaneous perturbation stochastic approximation algorithm is employed, demonstrating sustained convergence and consistent performance across the primary tested cutoffs up to a Fock space cutoff of d = 5 for the 3 -mode CV system, alongside a proof-of-concept indication of algorithmic trainability at d = 7 . Moreover, the generalization of this framework to the detection of bound entangled states with zero negativity is demonstrated, establishing a fundamental advantage in sample complexity over standard linear witnesses. Robustness to photon loss and statistical noise is further quantified under realistic finite-shot measurement conditions, confirming the model’s stability in noisy environments. Collectively, these findings provide a rigorous proof-of-principle for the necessity of non-Gaussian operations and demonstrate a clear information-theoretic advantage in data-driven quantum state characterization.

DOI
10.1088/1751-8121/aeb0e6

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Abstract Characterizing entanglement in high-dimensional continuous-variable (CV) systems remains a formidable challenge, particularly when full quantum state tomography is prohibitively resource-intensive. In this work, a hybrid quantum–classical framework is introduced that learns nonlinear entanglement witnesses directly from sparse measurement data via CV quantum neural networks (CV-QNNs). The architecture combines variational interferometers, squeezing operations, and non-Gaussian Kerr nonlinearities with a classical neural head to output a scalar witness value. To rigorously isolate the information-theoretic sample-complexity advantage, a Coherence Trap benchmark is devised in which genuinely entangled and separable states are engineered to exhibit identical photon-number statistics, thereby rendering them indistinguishable to classical classifiers operating on intensity histograms. Numerical simulations show that while classical models fail to classify these states without full tomography, the CV-QNN achieves 100 % accuracy by extracting phase-dependent quantum correlations prior to measurement. To address scalability and optimization under realistic finite-shot measurement noise, a variance-reduced simultaneous perturbation stochastic approximation algorithm is employed, demonstrating sustained convergence and consistent performance across the primary tested cutoffs up to a Fock space cutoff of d = 5 for the 3 -mode CV system, alongside a proof-of-concept indication of algorithmic trainability at d = 7 . Moreover, the generalization of this framework to the detection of bound entangled states with zero negativity is demonstrated, establishing a fundamental advantage in sample complexity over standard linear witnesses. Robustness to photon loss and statistical noise is further quantified under realistic finite-shot measurement conditions, confirming the model’s stability in noisy environments. Collectively, these findings provide a rigorous proof-of-principle for the necessity of non-Gaussian operations and demonstrate a clear information-theoretic advantage in data-driven quantum state characterization.
A major challenge in quantum information is characterizing entanglement, for which entanglement witnesses offer effective means of detecting quantum correlations. We introduce a hybrid quantum-classical framework that learns a nonlinear entanglement witness directly from quantum data using continuous-variable quantum neural networks (CV-QNNs). Our architecture combines variational interferometers, squeezers and non-Gaussian Kerr gates with a small classical neural head to output a scalar witness value. Numerical simulations were conducted on two- and three-mode families, including Gaussian and non-Gaussian states in both pure and mixed forms. We observed over 99% classification accuracy and a robust performance gap compared to strong classical baselines, especially when scaling from two to three modes. Robustness to photon loss is further quantified under a finite number of measurement shots. On the theory side, we show that when the quantum measurement stage is informationally complete, the hybrid model can approximate any continuous witness-like functional on compact sets of states.Our findings highlight CV-QNNs as a promising framework for data-driven quantum state characterization and propose specific benchmarks where near-term photonic platforms offer tangible advantages.
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In this paper, classical and continuous variable (CV) quantum neural network hybrid multiclassifiers are presented using the MNIST dataset. The combination of cutoff dimension and probability measurement method in the CV model allows a quantum circuit to produce output vectors of size equal to n raised to the power of n where n represents cutoff dimension and m, the number of qumodes. They are then translated as one-hot encoded labels, padded with an appropriate number of zeros. The total of eight different classifiers are built using 2,3,...,8 qumodes, based on the binary classifier architecture proposed in Continuous variable quantum neural networks. The displacement gate and the Kerr gate in the CV model allow for the bias addition and nonlinear activation components of classical neural networks to quantum. The classifiers are composed of a classical feedforward neural network, a quantum data encoding circuit, and a CV quantum neural network circuit. On a truncated MNIST dataset of 600 samples, a 4 qumode hybrid classifier achieves 100% training accuracy.
Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources (the challenging piece for physical implementation) originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick-Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.
In this paper, classical and continuous variable (CV) quantum neural network hybrid multi-classifiers are presented using the MNIST dataset. Currently available classifiers can classify only up to two classes. The proposed architecture allows networks to classify classes up to nm classes, where n represents cutoff dimension and m the number of qumodes on photonic quantum computers. The combination of cutoff dimension and probability measurement method in the CV model allows a quantum circuit to produce output vectors of size nm. They are then interpreted as one-hot encoded labels, padded with nm - 10 zeros. The total of seven different classifiers is built using 2, 3, …, 6, and 8-qumodes on photonic quantum computing simulators, based on the binary classifier architecture proposed in “Continuous variable quantum neural networks” [1]. They are composed of a classical feed-forward neural network, a quantum data encoding circuit, and a CV quantum neural network circuit. On a truncated MNIST dataset of 600 samples, a 4-qumode hybrid classifier achieves 100% training accuracy.
Continuous-variable (CV) quantum computing has shown great potential for building neural network models. These neural networks can have different levels of quantum-classical hybridization depending on the complexity of the problem. Previous work on CV neural network protocols required the implementation of non-Gaussian operators in the network. These operators were used to introduce non-linearity, an essential feature of neural networks. However, these protocols are hard to execute experimentally. We built a CV hybrid quantum-classical neural network protocol that can be realized experimentally with current photonic quantum hardware. Our protocol uses Gaussian gates only with the addition of ancillary qumodes. We implemented non-linearity through repeat-until-success measurements on ancillary qumodes. To test our neural network, we studied canonical machine learning and quantum computer problems in a supervised learning setting -- state preparation, curve fitting, and classification problems. We achieved high fidelity in state preparation of single-photon (99.9%), cat (99.8%), and Gottesman-Kitaev-Preskill (93.9%) states, a well-fitted curve in the presence of noise at a cost of less than 1%, and more than 95% accuracy in classification problems. These results bode well for real-world applications of CV quantum neural networks.