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npj Quantum Information

Direct entanglement detection of quantum systems using machine learning

Yulei Huang, Liangyu Che, Chao Wei, Feng Xu, Xinfang Nie, Jun Li, Dawei Lu, Tao Xin

Published Feb 20, 2025• 18 citations
Publisher

Abstract

Entanglement plays a crucial role in advancing quantum technologies and exploring quantum many-body simulations. Here, we introduce a protocol aided by neural networks for measuring entanglement in both equilibrium and non-equilibrium states of local Hamiltonians, with a favorable amount of training data. Our numerical simulations across various Hamiltonian models and qubit configurations reveal that this approach can predict comprehensive entanglement metrics, such as Rényi entropy, for up to 100 qubits using only single-qubit and two-qubit Pauli measurements. Excitingly, future entanglement dynamics beyond the measurement window can be predicted based solely on previous single-qubit traces. Experimentally, we utilize a nuclear spin quantum processor and a neural network to measure entanglement in the ground and dynamical states of a one-dimensional spin chain. The results demonstrate the feasibility of our method in practical experiments. Therefore, our approach offers a promising method for experimentally measuring entanglement in systems with dozens to hundreds of qubits.

DOI
10.1038/s41534-025-00970-w

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Entanglement plays a crucial role in advancing quantum technologies and exploring quantum many-body simulations. Here, we introduce a protocol aided by neural networks for measuring entanglement in both equilibrium and non-equilibrium states of local Hamiltonians, with a favorable amount of training data. Our numerical simulations across various Hamiltonian models and qubit configurations reveal that this approach can predict comprehensive entanglement metrics, such as Rényi entropy, for up to 100 qubits using only single-qubit and two-qubit Pauli measurements. Excitingly, future entanglement dynamics beyond the measurement window can be predicted based solely on previous single-qubit traces. Experimentally, we utilize a nuclear spin quantum processor and a neural network to measure entanglement in the ground and dynamical states of a one-dimensional spin chain. The results demonstrate the feasibility of our method in practical experiments. Therefore, our approach offers a promising method for experimentally measuring entanglement in systems with dozens to hundreds of qubits.
Entanglement is a key property in the development of quantum technologies and in the study of quantum many-body simulations. However, entanglement measurement typically requires quantum full-state tomography (FST). Here we present a neural network-assisted protocol for measuring entanglement in equilibrium and non-equilibrium states of local Hamiltonians. Instead of FST, it can learn comprehensive entanglement quantities from single-qubit or two-qubit Pauli measurements, such as Rényi entropy, partially-transposed (PT) moments, and coherence. It is also exciting that our neural network is able to learn the future entanglement dynamics using only single-qubit traces from the previous time. In addition, we perform experiments using a nuclear spin quantum processor and train an adoptive neural network to study entanglement in the ground and dynamical states of a one-dimensional spin chain. Quantum phase transitions (QPT) are revealed by measuring static entanglement in ground states, and the entanglement dynamics beyond measurement time is accurately estimated in dynamical states. These precise results validate our neural network. Our work will have a wide range of applications in quantum many-body systems, from quantum phase transitions to intriguing non-equilibrium phenomena such as quantum thermalization.
Entanglement entropy is an essential metric for characterizing quantum many-body systems, but its numerical evaluation for neural network representations of quantum states has so far been inefficient and demonstrated only for the restricted Boltzmann machine architecture. Here we estimate generalized R\'enyi entropies of autoregressive neural quantum states with up to N=256N=256N=256 spins using quantum Monte Carlo methods. A naive ``direct sampling'' approach performs well for low-order R\'enyi entropies but fails for larger orders when benchmarked on a one-dimensional Heisenberg model. We therefore propose an improved ``conditional sampling'' method exploiting the autoregressive structure of the network ansatz, which outperforms direct sampling and facilitates calculations of higher-order R\'enyi entropies in both one- and two-dimensional Heisenberg models. Access to higher-order R\'enyi entropies allows for an approximation of the von Neumann entropy as well as extraction of the single-copy entanglement. Both methods elucidate the potential of neural network quantum states in quantum Monte Carlo studies of entanglement entropy for many-body systems.
Abstract With the rapid growth of controllable qubits in recent years, experimental multipartite entangled states can be created with high fidelity in various moderate‐ and large‐scale physical systems. However, the characterization of multipartite entanglement structure remains a formidable challenge, as traditionally it requires exponential number of local measurements to realize the identification. Machine learning is demonstrated to be an efficient tool to detect the underlying entanglement structure for ideal states, but it has non‐negligible underperformance when tackling imperfect experimental data in reality. Here, a modified classifier based on feed‐forward neural network to predict experimental entanglement structure in terms of entanglement intactness and depth is proposed. By preprocessing the input data, the classifier maintains efficiency and reliability against experimental noises, with the accuracy being enhanced from 69.7% to 91.2% for 6‐qubit entangled states in spin systems. This method is anticipated to shed light on future studies of entanglement structure, in particular when the number of controlled qubits reaches explosive growth in practice.
Quantum entanglement acts as a crucial part in quantum computation and quantum information, hence quantifying unknown entanglement is an important task. Due to the fact that the amount of entanglement cannot be achieved directly by measuring any physical observables, it remains an open problem to quantify entanglement experimentally. In this work, we provide an effective way to quantify entanglement for the unknown quantum states via artificial neural networks. By choosing the expectation values of measurements as input features and the values of entanglement measures as labels, we train artificial neural network models to predict the entanglement for new quantum states accurately. Our method does not require the full information about unknown quantum states, which highlights the effectiveness and versatility of machine learning in exploring quantum entanglement.
Entanglement is a central resource in quantum information and quantum technologies, yet its characterization remains challenging due to both theoretical complexity and measurement requirements. Machine learning has emerged as a promising alternative, enabling entanglement characterization from incomplete measurement data, however model interpretability remains a challenge. In this work, we introduce a unified and interpretable framework for SLOCC entanglement classification of two- and three-qubit states, encompassing both pure and mixed states. We train dense and convolutional neural networks on Pauli-measurement outcomes, provide design guidelines for each architecture, and systematically compare their performance across types of states. To interpret the models, we compute Shapley values to quantify the contribution of each measurement, analyze measurement-importance patterns across different systems, and use these insights to guide a measurement-reduction scheme. Accuracy-versus-measurement curves and comparisons with analytical entanglement criteria demonstrate the minimal resources required for reliable classification and highlight both the capabilities and limitations of Shapley-based interpretability when using machine learning models for entanglement detection and classification.
Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose a machine-learning-assisted scheme to measure the entanglement between arbitrary subsystems of size N_{A} and N_{B}, with O(N_{A}+N_{B}) measurements, and without any prior knowledge of the state. The method exploits a neural network to learn the unknown, nonlinear function relating certain measurable moments and the logarithmic negativity. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many-body systems in both equilibrium and nonequilibrium regimes.