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arXiv:2309.07556 · Physical Review A

Sample-efficient estimation of entanglement entropy through supervised learning

Maximilian Rieger, Moritz Reh, Martin Gärttner

Published Jan 2, 2024• 10 citations
arXivHTMLPDF

Abstract

We explore a supervised machine-learning approach to estimate the entanglement entropy of multiqubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network's estimate and benchmark against the best-known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for nonunitary evolution by training our model on different noise strengths.

DOI
10.1103/physreva.109.012403

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We explore a supervised machine-learning approach to estimate the entanglement entropy of multiqubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network's estimate and benchmark against the best-known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for nonunitary evolution by training our model on different noise strengths.
For an unknown quantum state ρ\rhoρ, a set of measurement outcomes provides partial information. The principe of maximum entropy states that the density matrix which best represents the current knowlege is the one with the maximum entropy. However, the maximum entropy estimation (MEE) is hard to calculate in general. In this work, we develop a method that outputs an approximation to the MEE from limited measurement results by using supervised learning techniques. The method estimates the MEE of theoretical data of several different quantum systems with high fidelities. We also test our method experimentally with a photonic set-up, which exhibits high fidelities (all greater than 99.7%) and robustness against experimental errors. In comparison with previous known optimization algorithms for MEE, our method provides adequate fidelities with high efficiency.
Entanglement, which quantifies nonlocal correlations in quantum mechanics, is the fascinating concept behind much of the aspiration towards quantum technologies. Nevertheless, directly measuring the entanglement of a many-particle system is very challenging. Here, we show that via supervised machine learning using a convolutional neural network, we can infer the entanglement from a measurable observable for a disordered interacting quantum many-particle system. Several structures of neural networks were tested and a convolutional neural network akin to structures used for image and speech recognition performed the best. After training on a set of 500 realizations of disorder, the network was applied to 200 new realizations and its results for the entanglement entropy were compared to a direct computation of the entanglement entropy. Excellent agreement was found, except for several rare regions which in a previous study were identified as belonging to an inclusion of a Griffiths-like quantum phase. Training the network on a test set with different parameters (in the same phase) also works quite well.
Detection of entanglement is an indispensable step to practical quantum computation and communication. Compared with the conventional entanglement witness method based on fidelity, we propose a flexible, machine learning assisted entanglement detection protocol that is robust to different types of noises and sample efficient. In this protocol, an entanglement classifier for a generic entangled state is obtained by training a classical machine learning model with a synthetic dataset. The dataset contains classical features of two types of states and their labels (either entangled or separable). The classical features of a state, which are expectation values of a set of k-local Pauli observables, are estimated sample-efficiently by the classical shadow method. In the numerical simulation, our classifier can detect the entanglement of 4-qubit GHZ states with coherent noise and W states mixed with large white noise, with high accuracy.
Quantum entanglement is a key resource in quantum computing and quantum information processing tasks. However, its quantification remains a major challenge since it cannot be directly extracted from physical observables. To address this issue, we study a few machine-learning based models to estimate the amount of entanglement in two-qubit as well as three-qubit systems. We use measurement outcomes as the input features and entanglement measures as the training labels. Our models predict entanglement without requiring the full state information. This demonstrates the potential of machine learning as an effcient and powerful tool for characterizing quantum entanglement
ABSTRACT Quantum machine learning (QML) aims to leverage the principles of quantum mechanics to speed up the process of solving machine learning problems or improve the quality of solutions. In supervised learning, training samples that are entangled with an auxiliary system were shown to reduce the approximation error of the trained model. However, also the training process is affected by various properties of the supervised learning task, including the model's circuit structure, the used cost function, and noise on the quantum computer. To evaluate the applicability of entanglement in supervised learning, the effect of highly entangled training data on the trainability is investigated in this work. This study shows that for highly expressive models, i.e., models capable of expressing a large number of candidate solutions, the possible improvement of loss function values in constrained neighborhoods during optimization is severely limited when maximally entangled states are used. Furthermore, this finding is experimentally supported by simulating training with Parameterized Quantum Circuits (PQCs). As the expressivity of the PQC increases, this training process becomes more susceptible to loss concentration induced by entangled training data. Lastly, for non‐maximally entangled states, the experiments highlight the fundamental role of entanglement entropy as a predictor for the trainability.
The ability to extract general laws from a few known examples depends on the complexity of the problem and on the amount of training data. In the quantum setting, the learner's generalization performance is further challenged by the destructive nature of quantum measurements that, together with the no-cloning theorem, limits the amount of information that can be extracted from each training sample. In this paper we focus on hybrid quantum learning techniques where classical machine-learning methods are paired with quantum algorithms and show that, in some settings, the uncertainty coming from a few measurement shots can be the dominant source of errors. We identify an instance of this possibly general issue by focusing on the classification of maximally entangled vs. separable states, showing that this toy problem becomes challenging for learners unaware of entanglement theory. Finally, we introduce an estimator based on classical shadows that performs better in the big data, few copy regime. Our results show that the naive application of classical machine-learning methods to the quantum setting is problematic, and that a better theoretical foundation of quantum learning is required.