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.
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.
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.
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 Nmub2D2Ncop measurements, where Nmub is the number of mutually unbiased bases, D is the subsystem dimensionality, and Ncop is the number of identical copies. For ququart systems with D=4, this translates to hundreds of distinct measurement settings, each requiring multiple copies (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 1.49×10−1, while MLE yields 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/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.
We use a meta-learning neural-network approach to predict measurement outcomes of a quantum state in arbitrary local bases and thus carry out an approximate quantum state tomography. Each stage of this procedure can be performed efficiently, allowing it to be used effectively on large systems. We demonstrate this approach on the most recent noisy intermediate-scale IBM Quantum devices, achieving an accurate generative model for a 6-qubit state's measurement outcomes with only 100 random measurement settings as opposed to the 729 settings required for full tomography. This reduction in the required number of measurements scales favourably, with around 200 measurement settings yielding good results for a 10 qubit state that would require 59,049 settings for full quantum state tomography. This reduction in the number of measurement settings coupled with the efficiency of the procedure could allow for estimations of expectation values and state fidelities in practicable times on current quantum devices. For suitable states, this could then help in increasing the speed of other optimization schemes when attempting to produce states on noisy quantum devices at a scale where traditional maximum likelihood based approaches are impractical.
Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core challenge in understanding, implementing and ultimately using quantum neural networks is that they represent many-body systems with an exponentially large Hilbert space, in combination with a large parameter search space. Moreover, noise -- which is inherent to any quantum measurement -- sets practical limits for the estimation of training loss. Here, we study physics-informed large-scale quantum neural networks that are trained through a finite number of noisy loss function measurements. We show that this architecture permits the construction of nontrivial decision boundaries that enable the classification of quantum states through measuring an order parameter. Our approach can directly process quantum data that is output from quantum simulators and computers and is well suited for implementation on current hardware. Moreover, owed to a close link between the neural network dynamics and the evolution of Markovian open many-body quantum systems, one may expect a certain robustness to noise, which is ubiquitous in the current NISQ era.
Abstract We study the evolution of quantum correlations in Bell, Werner, and maximally entangled mixed states of two qubits subjected to correlated amplitude-damping channels. Our primary focus is to evaluate the robustness of entanglement as a resource for quantum information protocols such as dense coding, teleportation, and Einstein–Podolsky–Rosen (EPR) steering under the influence of noise. In addition, we investigate the behaviour of other quantum correlations, including quantum discord and coherence, and analyze their hierarchy under decoherence. To counteract the detrimental effects of the channels, we apply the weak measurement and quantum measurement reversal (WMR) protocol, comparing the effectiveness of single-qubit and two-qubit WMR techniques. Our results show that the two-qubit WMR protocol significantly outperforms the single-qubit approach in preserving quantum correlations. Furthermore, we employ a neural network model to enhance our analysis of the relationship between different quantum correlation measures during the evolution. Using a MATLAB-based artificial neural network with 80 neurons across three hidden layers and trained with the Levenberg–Marquardt algorithm, we successfully predict trace distance discord from other correlations, achieving low prediction errors. Besides, our analysis of the neural network weights suggests that concurrence and EPR steering have the most positive influence on the accurate discord predictions.
Quantum state classification and entanglement quantification are of significant importance in the fundamental research of quantum information science and various quantum applications. Traditional methods, such as quantum state tomography, face exponential measurement demands with increasing numbers of qubits, necessitating more efficient approaches. Recent work has shown promise in using artificial neural networks (ANNs) for quantum state analysis. However, existing ANNs may falter when confronted with states affected by dephasing noise, especially with limited data and computational resources. In this study, we employ a machine-learning approach to investigate the disentanglement process in two-qubit systems in the presence of dephasing noise. Our findings highlight the limitations of general state-trained ANNs in classifying states under dephasing noise. Specialized ANN algorithms, tailored for classifying states and quantifying entanglement in such noisy environments, demonstrate excellent performance using only a subset of tomographic features.