Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 3, June 2024
Page(s) 193 - 194
DOI https://doi.org/10.1051/wujns/2024293193
Published online 03 July 2024

Quantum entanglement is a key resource for achieving superiority of quantum computing. Currently, scientists are extensively focusing on how to integrate quantum entanglement into various components of quantum machine learning (QML) models, aiming to surpass the performance of traditional machine learning models. Notable successes include the use of entangled measurements[1-3] and entangled channels[4], which have been shown to reduce query complexity or improve the prediction precision for specified QML tasks. Quantum entangled data, capable of encoding more information compared to classical data of the same size, is recognized for its potential to achieve quantum advantages. Nevertheless, the impact of the entanglement degree in quantum data on model performance remains a challenging and unresolved research question.

Recently, Wang and collaborators[5] (Nat Commun 2024,15, 3716) rigorously analyzed the impact of the entanglement degree of quantum data, the number and type of measurements, and the size of training dataset on the prediction error of QML models by establishing the quantum no-free-lunch (NFL) theorem. They consider tasks of quantum dynamics learning with entangled data, which is fundamental to many advanced quantum algorithms, as illustrated in Fig.1. While conventional understanding suggests that entanglement mainly confers benefits to QML in terms of sample complexity[6,7], this study demonstrates that the effect of the degree of entanglement on prediction error exhibits a dual effect, that is, whether quantum entanglement improves performance depends on the number of measurements allowed. With a sufficient number of measurements, increasing the entanglement degree in quantum data can effectively reduce the prediction error, or decrease the size of the training dataset needed to achieve the same level of prediction error. Conversely, using highly entangled data may increase the prediction error when only a few measurements are permitted.

thumbnail Fig.1 Illustration of quantum NFL setting with the entangled data[5]

The goal of the quantum learner is to learn a unitary that can accurately predict the output of the target unitary under a fixed observable , where the subscript refers to the quantum system in which the operator act on. The learning process is as follows. (a) A total number of entangled bipartite quantum states living in Hilbert space ( denotes the reference system) are taken as inputs, dubbed entangled data. (b) Quantum learner proceeds incoherent learning. The entangled data separately interacts with the target unitary (agnostic) and the candidate hypothesis extracted from the same Hypothesis set . (c) The quantum learner is restricted to leverage the finite measured outcomes of the observable O on the output states of and to conduct learning. (d) A classical computer is exploited to infer that best estimates according to the measurement outcomes. For example, in the case of variational quantum algorithms, the classical computer serves as an optimizer to update the tunable parameters of the ansatz . (e) The learned unitary is used to predict the output of unseen quantum states in Hilbert space under the evolution of the target unitary and the measurement of . A large Schmidt rank can enhance the prediction accuracy when combined with a large number of measurements , but may lead to a decrease in accuracy when m is small.

This research provides practical guidance for designing advanced quantum learning protocols, especially those tailored for current quantum computers with limited quantum computing resources.

On the theoretical side, the results of Wang and collaborators are an important reminder of the entangled data utilization, suggesting that extracting classical information from entangled data is difficult although they can store more information than unentangled data. An important research direction motivated by this study is inquiring whether there exists a similar transition role when exploiting entangled measurements which has more powerful information-extraction capabilities from entangled data and has been shown to achieve an exponential advantage under many specific learning tasks[1-3].

References

  1. Huang H Y, Kueng R, Preskill J. Information-theoretic bounds on quantum advantage in machine learning[J]. Physical Review Letters, 2021, 126(19): 190505. [CrossRef] [PubMed] [Google Scholar]
  2. Chen S T, Cotler J, Huang H Y, et al. Exponential separations between learning with and without quantum memory[C]//2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). New York: IEEE, 2022: 574-585. [Google Scholar]
  3. Huang H Y, Broughton M, Cotler J, et al. Quantum advantage in learning from experiments[J]. Science, 2022, 376(6598): 1182-1186. [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  4. Hao S H, Shi H W, Li W, et al. Entanglement-assisted communication surpassing the ultimate classical capacity[J]. Physical Review Letters, 2021, 126(25): 250501. [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
  5. Wang X B, Du Y X, Tu Z Z, et al. Transition role of entangled data in quantum machine learning[J]. Nature Communications, 2024, 15(1): 3716. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  6. Sharma K, Cerezo M, Holmes Z, et al. Reformulation of the no-free-lunch theorem for entangled datasets[J]. Physical Review Letters, 2022, 128(7): 070501. [CrossRef] [PubMed] [Google Scholar]
  7. Zhao H M, Lewis L, Kannan I, et al. Learning quantum states and unitaries of bounded gate complexity[EB/OL]. [2023-05-10]. https://arxiv.org/abs/2310.19882. □ [Google Scholar]

© Wuhan University 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

All Figures

thumbnail Fig.1 Illustration of quantum NFL setting with the entangled data[5]

The goal of the quantum learner is to learn a unitary that can accurately predict the output of the target unitary under a fixed observable , where the subscript refers to the quantum system in which the operator act on. The learning process is as follows. (a) A total number of entangled bipartite quantum states living in Hilbert space ( denotes the reference system) are taken as inputs, dubbed entangled data. (b) Quantum learner proceeds incoherent learning. The entangled data separately interacts with the target unitary (agnostic) and the candidate hypothesis extracted from the same Hypothesis set . (c) The quantum learner is restricted to leverage the finite measured outcomes of the observable O on the output states of and to conduct learning. (d) A classical computer is exploited to infer that best estimates according to the measurement outcomes. For example, in the case of variational quantum algorithms, the classical computer serves as an optimizer to update the tunable parameters of the ansatz . (e) The learned unitary is used to predict the output of unseen quantum states in Hilbert space under the evolution of the target unitary and the measurement of . A large Schmidt rank can enhance the prediction accuracy when combined with a large number of measurements , but may lead to a decrease in accuracy when m is small.

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