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Machine Learning-Enhanced Entanglement Characterization in Bi-partite Ququart Systems

Rithvik Sreekantham, Ravindra Pratap Singh, Shashi Prabhakar

Published Jun 10, 2025• 0 citations
Publisher

Abstract

Abstract We present a systematic comparative analysis of machine learning and traditional approaches for quantum entanglement characterization in bi-partite ququart systems. Quantifying entanglement traditionally requires full quantum state tomography, necessitating $N_{mub}^2D^2N_{cop}$ measurements, where $N_{mub}$ is the number of mutually unbiased bases, $D$ is the subsystem dimensionality, and $N_{cop}$ is the number of identical copies. For ququart systems with $D=4$, this translates to hundreds of distinct measurement settings, each requiring multiple copies ($N_{cop}$ typically in thousands)—resulting in hundreds of thousands of total measurements in practice. Our research demonstrates that neural networks achieve comparable or superior accuracy with orders of magnitude fewer measurement settings. We evaluate three architectures—Multi-Layer Perceptron (MLP), Convolutional Neural Network (CNN), and Transformer—against Maximum Likelihood Estimation (MLE) and Bayesian methods across measurement counts ranging from 10 to 400. Neural approaches achieve up to 1650× faster computation times compared to traditional methods while maintaining competitive accuracy. At 100 measurements, the Transformer achieves Mean Squared Error (MSE) of $1.49 \times 10^{-1}$, while MLE yields $2.04 \times 10^{-1}$—over 10× higher error—despite taking 178 times longer. All neural methods in our study show error reduction scaling as approximately $1/\sqrt{N}$ with increased measurements. While this observed scaling might be influenced by our specific architectures, it provides important practical guidance for experimental design. The Transformer architecture demonstrates exceptional sample efficiency, achieving with 100 measurements what traditional methods require 250-400 measurements to accomplish—a significant advantage for resource-constrained quantum experiments. This research provides a viable pathway for real-time entanglement characterization in higher-dimensional quantum systems where traditional methods become computationally prohibitive.

DOI
10.21203/rs.3.rs-6486345/v1

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Abstract We present a systematic comparative analysis of machine learning and traditional approaches for quantum entanglement characterization in bi-partite ququart systems. Quantifying entanglement traditionally requires full quantum state tomography, necessitating Nmub2D2NcopN_{mub}^2D^2N_{cop}Nmub2​D2Ncop​ measurements, where NmubN_{mub}Nmub​ is the number of mutually unbiased bases, DDD is the subsystem dimensionality, and NcopN_{cop}Ncop​ is the number of identical copies. For ququart systems with D=4D=4D=4, this translates to hundreds of distinct measurement settings, each requiring multiple copies (NcopN_{cop}Ncop​ typically in thousands)—resulting in hundreds of thousands of total measurements in practice. Our research demonstrates that neural networks achieve comparable or superior accuracy with orders of magnitude fewer measurement settings. We evaluate three architectures—Multi-Layer Perceptron (MLP), Convolutional Neural Network (CNN), and Transformer—against Maximum Likelihood Estimation (MLE) and Bayesian methods across measurement counts ranging from 10 to 400. Neural approaches achieve up to 1650× faster computation times compared to traditional methods while maintaining competitive accuracy. At 100 measurements, the Transformer achieves Mean Squared Error (MSE) of 1.49×10−11.49 \times 10^{-1}1.49×10−1, while MLE yields 2.04×10−12.04 \times 10^{-1}2.04×10−1—over 10× higher error—despite taking 178 times longer. All neural methods in our study show error reduction scaling as approximately 1/N1/\sqrt{N}1/N​ with increased measurements. While this observed scaling might be influenced by our specific architectures, it provides important practical guidance for experimental design. The Transformer architecture demonstrates exceptional sample efficiency, achieving with 100 measurements what traditional methods require 250-400 measurements to accomplish—a significant advantage for resource-constrained quantum experiments. This research provides a viable pathway for real-time entanglement characterization in higher-dimensional quantum systems where traditional methods become computationally prohibitive.
Accurate classification of multipartite entanglement in high-dimensional quantum systems is crucial for advancing quantum communication and information processing. However, conventional methods are resource-intensive, and even many machine-learning-based approaches necessitate large training datasets, creating a significant experimental bottleneck for data acquisition. To address this challenge, we propose a hybrid neural network architecture integrating Convolutional and Bidirectional Long Short-Term Memory networks (CNN-BiLSTM). This design leverages CNNs for local feature extraction and BiLSTMs for sequential dependency modeling, enabling robust feature learning from minimal training data. We investigate two fusion paradigms: Architecture 1 (flattening-based) and Architecture 2 (dimensionality-transforming). When trained on only 100 samples, Architecture 2 maintains classification accuracies exceeding 90% for both 3-qubit and 4-qubit systems, demonstrating rapid loss convergence within tens of epochs. Under full-data conditions (400 000 samples), both architectures achieve accuracies above 99.97%. Comparative benchmarks reveal that our CNN-BiLSTM models, especially Architecture 2, consistently outperform standalone CNNs, BiLSTMs, and MLPs in low-data regimes, albeit with increased training time. These results demonstrates that the tailored CNN-BiLSTM fusion significantly alleviates experimental data acquisition burden, offering a practical pathway toward scalable entanglement verification in complex quantum systems.
k-entanglement, including entanglement relative to full separability and genuinely multipartite entanglement, is a fundamental quantum resource in multipartite quantum systems. Its identification and quantification play essential roles in quantum information processing, quantum simulation, and quantum metrology. However, the practical computation of rigorous k-entanglement measures remains highly challenging due to the need for high-dimensional optimization. In this work, we propose a machine-learning-based surrogate framework for approximating the witness-based k-entanglement measure Ew(k,n). The numerical evaluation of the computationally realized quantity E˜w(k,n)(ρ) is reformulated as a supervised regression problem, where the input is the density matrix ρ and the labels are obtained from finite witness databases. The framework combines multilayer perceptrons (MLPs), convolutional neural networks (CNNs), and light gradient boosting machine (LightGBM) through a stacking ensemble. Numerical experiments are performed for 3- and 4-qubit systems as representative demonstrations of the proposed workflow. The results show that the learned models achieve high predictive accuracy in terms of MAE, MSE, and R2, while providing millisecond-level inference for single-state evaluation. Werner state tests serve as symmetric benchmark checks, and an additional four-qubit noisy circuit-generated state family, obtained from finite-depth circuit preparation followed by local amplitude-damping noise, is used as a structured physical test beyond random density matrices. Compared with the optimization-based evaluation, the trained surrogate model significantly reduces the computational time while maintaining accuracy within the tested system sizes and data distributions. These results show that the proposed framework provides an efficient numerical surrogate for rapid approximation of witness-based k-entanglement measures, while extensions to larger systems and experimental data require further validation.
Quantum entanglement is a powerful resource for quantum-enhanced technologies. However, its reliable quantification remains challenging due to the exponential scaling of the Hilbert space with system size, which renders full state tomography infeasible. Moreover, experimentally estimating entanglement typically requires a large number of measurement samples leading to a significant overhead. In this work, we present two models, a feed-forward neural network and an attention-based model, to accurately predict the bipartite second Renyi from projective measurements of quantum states. We benchmark their performance against standard classical shadow estimators and find that the machine-learning approaches achieve higher accuracy and improved sample efficiency across a range of system sizes. Our results demonstrate the potential of machine learning for scalable and efficient estimation of quantum correlations.
Introduction: The scalability of superconducting quantum processors is fundamentally limited by environmental noise and the exponential measurement overhead required for multi-qubit entanglement verification. Methods: In this work, we introduce a hybrid machine learning (ML) framework utilizing convolutional neural networks (CNNs) to simultaneously characterize decoherence channels and verify multipartite entanglement. The model is trained on synthetic noisy syndrome measurement data augmented with empirical SPAM models and applied to a 5-qubit superconducting device, where the neural network acts as a non-linear entanglement witness. Results: sweeps. Furthermore, it successfully verifies genuine multipartite entanglement with a fidelity prediction of 99.1%, while requiring only 12% of the measurement bases demanded by full quantum state tomography (QST). Discussion: Crucially, our methodology bypasses two distinct challenges: it drastically reduces the measurement complexity inherent in multi-qubit entanglement verification, and it inherently filters out state preparation and measurement (SPAM) errors without requiring noise-amplifying matrix inversions. These numerical results demonstrate that ML-assisted protocols provide a highly resource-efficient pathway for the characterization of noisy intermediate-scale quantum (NISQ) devices.
Measuring complex properties in quantum systems, such as measures of quantum entanglement and Bell nonlocality, is inherently challenging. Traditional methods, like quantum state tomography (QST), require a full reconstruction of the density matrix for a given system and demand resources that scale exponentially with system size. We propose an alternative strategy that reduces the required information by combining multicopy measurements with artificial neural networks (ANNs), resulting in a 67% reduction in measurement requirements compared to QST. We have successfully measured two-qubit quantum correlations of Bell states subjected to a depolarizing channel (resulting in Werner states) and an amplitude-damping channel (leading to Horodecki states) using the multicopy approach on real quantum hardware. Our experiments, conducted with transmon qubits on IBMQ quantum processors, quantified the violation of Bell's inequality and the negativity of two-qubit entangled states. We compared these results with those obtained from the standard QST approach and applied a maximum likelihood method to mitigate noise. We trained ANNs to estimate degrees of entanglement and nonlocality measures using optimized sets of projections identified through Shapley's (SHAP) analysis for the Werner and Horodecki states. The ANN output, based on this reduced set of projections, aligns well with expected values and enhances noise robustness. This approach simplifies and improves the error robustness of multicopy measurements, eliminating the need for complex nonlinear equation analysis. It represents a significant advancement in AI-assisted quantum measurements, making the practical implementation on current quantum hardware more feasible. The experimental results demonstrate improved noise robustness on the current noisy intermediate-scale quantum (NISQ) hardware, representing a practical advance in resource-efficient characterization of quantum correlations.
The quantification of the entanglement present in a physical system is of paramount importance for fundamental research and many cutting-edge applications. Now, achieving this goal requires either a priori knowledge on the system or very demanding experimental procedures such as full state tomography or collective measurements. Here, we demonstrate that, by using neural networks, we can quantify the degree of entanglement without the need to know the full description of the quantum state. Our method allows for direct quantification of the quantum correlations using an incomplete set of local measurements. Despite using undersampled measurements, we achieve a quantification error of up to an order of magnitude lower than the state-of-the-art quantum tomography. Furthermore, we achieve this result using networks trained using exclusively simulated data. Last, we derive a method based on a convolutional network input that can accept data from various measurement scenarios and perform, to some extent, independently of the measurement device.