1
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Gil-Fuster E, Eisert J, Bravo-Prieto C. Understanding quantum machine learning also requires rethinking generalization. Nat Commun 2024; 15:2277. [PMID: 38480684 PMCID: PMC10938005 DOI: 10.1038/s41467-024-45882-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2023] [Accepted: 02/06/2024] [Indexed: 03/17/2024] Open
Abstract
Quantum machine learning models have shown successful generalization performance even when trained with few data. In this work, through systematic randomization experiments, we show that traditional approaches to understanding generalization fail to explain the behavior of such quantum models. Our experiments reveal that state-of-the-art quantum neural networks accurately fit random states and random labeling of training data. This ability to memorize random data defies current notions of small generalization error, problematizing approaches that build on complexity measures such as the VC dimension, the Rademacher complexity, and all their uniform relatives. We complement our empirical results with a theoretical construction showing that quantum neural networks can fit arbitrary labels to quantum states, hinting at their memorization ability. Our results do not preclude the possibility of good generalization with few training data but rather rule out any possible guarantees based only on the properties of the model family. These findings expose a fundamental challenge in the conventional understanding of generalization in quantum machine learning and highlight the need for a paradigm shift in the study of quantum models for machine learning tasks.
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Affiliation(s)
- Elies Gil-Fuster
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany
- Fraunhofer Heinrich Hertz Institute, Berlin, Germany
| | - Jens Eisert
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany.
- Fraunhofer Heinrich Hertz Institute, Berlin, Germany.
- Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, Germany.
| | - Carlos Bravo-Prieto
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany.
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2
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Chalkiadakis A, Theocharakis M, Barmparis GD, Tsironis GP. Quantum neural networks for the discovery and implementation of quantum error-correcting codes. CHAOS (WOODBURY, N.Y.) 2023; 33:113127. [PMID: 37988608 DOI: 10.1063/5.0157940] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/12/2023] [Accepted: 10/27/2023] [Indexed: 11/23/2023]
Abstract
We implement and use quantum neural networks that exploit bit-flip quantum error-correcting codes that correct bit-flip errors in arbitrary logical qubit states. We introduce conjugate layer quantum autoencoders and use them in order to restore states impacted by amplitude damping through the utilization of an approximative four-qubit error-correcting codeword. Our specific implementation avoids barren plateaus of the cost function and improves the training time. Moreover, we propose a strategy that allows one to discover new encryption protocols tailored for specific quantum channels. This is exemplified by learning to generate logical qubits explicitly for the bit-flip channel. Our modified quantum neural networks consistently outperform the standard implementations across all tasks.
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Affiliation(s)
- A Chalkiadakis
- Department of Physics, University of Crete, Heraklion 70013, Greece
| | - M Theocharakis
- Department of Physics, University of Crete, Heraklion 70013, Greece
| | - G D Barmparis
- Department of Physics, University of Crete, Heraklion 70013, Greece
| | - G P Tsironis
- Department of Physics, University of Crete, Heraklion 70013, Greece
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
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3
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Ryu JY, Elala E, Rhee JKK. Quantum Graph Neural Network Models for Materials Search. MATERIALS (BASEL, SWITZERLAND) 2023; 16:4300. [PMID: 37374486 DOI: 10.3390/ma16124300] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2023] [Revised: 06/03/2023] [Accepted: 06/05/2023] [Indexed: 06/29/2023]
Abstract
Inspired by classical graph neural networks, we discuss a novel quantum graph neural network (QGNN) model to predict the chemical and physical properties of molecules and materials. QGNNs were investigated to predict the energy gap between the highest occupied and lowest unoccupied molecular orbitals of small organic molecules. The models utilize the equivariantly diagonalizable unitary quantum graph circuit (EDU-QGC) framework to allow discrete link features and minimize quantum circuit embedding. The results show QGNNs can achieve lower test loss compared to classical models if a similar number of trainable variables are used, and converge faster in training. This paper also provides a review of classical graph neural network models for materials research and various QGNNs.
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Affiliation(s)
- Ju-Young Ryu
- School of Electrical Engineering & ITRC of Quantum Computing for AI, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
- Qunova Computing, Incorporated, 193 Munji-ro, Yuseong-gu, Daejeon 34051, Republic of Korea
| | - Eyuel Elala
- School of Electrical Engineering & ITRC of Quantum Computing for AI, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
- Qunova Computing, Incorporated, 193 Munji-ro, Yuseong-gu, Daejeon 34051, Republic of Korea
| | - June-Koo Kevin Rhee
- School of Electrical Engineering & ITRC of Quantum Computing for AI, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
- Qunova Computing, Incorporated, 193 Munji-ro, Yuseong-gu, Daejeon 34051, Republic of Korea
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4
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Zhou MG, Liu ZP, Yin HL, Li CL, Xu TK, Chen ZB. Quantum Neural Network for Quantum Neural Computing. RESEARCH (WASHINGTON, D.C.) 2023; 6:0134. [PMID: 37223480 PMCID: PMC10202373 DOI: 10.34133/research.0134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/21/2022] [Accepted: 04/11/2023] [Indexed: 05/25/2023]
Abstract
Neural networks have achieved impressive breakthroughs in both industry and academia. How to effectively develop neural networks on quantum computing devices is a challenging open problem. Here, we propose a new quantum neural network model for quantum neural computing using (classically controlled) single-qubit operations and measurements on real-world quantum systems with naturally occurring environment-induced decoherence, which greatly reduces the difficulties of physical implementations. Our model circumvents the problem that the state-space size grows exponentially with the number of neurons, thereby greatly reducing memory requirements and allowing for fast optimization with traditional optimization algorithms. We benchmark our model for handwritten digit recognition and other nonlinear classification tasks. The results show that our model has an amazing nonlinear classification ability and robustness to noise. Furthermore, our model allows quantum computing to be applied in a wider context and inspires the earlier development of a quantum neural computer than standard quantum computers.
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Affiliation(s)
- Min-Gang Zhou
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Zhi-Ping Liu
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Hua-Lei Yin
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Chen-Long Li
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
| | - Tong-Kai Xu
- MatricTime Digital Technology Co. Ltd., Nanjing 211899, China
| | - Zeng-Bing Chen
- National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
- MatricTime Digital Technology Co. Ltd., Nanjing 211899, China
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5
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Liu J, Najafi K, Sharma K, Tacchino F, Jiang L, Mezzacapo A. Analytic Theory for the Dynamics of Wide Quantum Neural Networks. PHYSICAL REVIEW LETTERS 2023; 130:150601. [PMID: 37115896 DOI: 10.1103/physrevlett.130.150601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2022] [Revised: 11/11/2022] [Accepted: 03/02/2023] [Indexed: 06/19/2023]
Abstract
Parametrized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parametrized quantum circuits in the limit of a large number of qubits and variational parameters. Then, we find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. Finally, we validate our analytic results with numerical experiments.
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Affiliation(s)
- Junyu Liu
- Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA
- Chicago Quantum Exchange, Chicago, Illinois 60637, USA
- Kadanoff Center for Theoretical Physics, The University of Chicago, Chicago, Illinois 60637, USA
| | - Khadijeh Najafi
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
| | - Kunal Sharma
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
- Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, Maryland 20742, USA
| | | | - Liang Jiang
- Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA
- Chicago Quantum Exchange, Chicago, Illinois 60637, USA
| | - Antonio Mezzacapo
- IBM Quantum, IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA
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6
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Gillman E, Carollo F, Lesanovsky I. Using (1+1)D quantum cellular automata for exploring collective effects in large-scale quantum neural networks. Phys Rev E 2023; 107:L022102. [PMID: 36932502 DOI: 10.1103/physreve.107.l022102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2022] [Accepted: 01/13/2023] [Indexed: 06/18/2023]
Abstract
Central to the field of quantum machine learning is the design of quantum perceptrons and neural network architectures. A key question in this regard is the impact of quantum effects on the way such models process information. Here, we establish a connection between (1+1)D quantum cellular automata, which implement a discrete nonequilibrium quantum many-body dynamics through successive applications of local quantum gates, and quantum neural networks (QNNs), which process information by feeding it through perceptrons interconnecting adjacent layers. Exploiting this link, we construct a class of QNNs that are highly structured-aiding both interpretability and helping to avoid trainability issues in machine learning tasks-yet can be connected rigorously to continuous-time Lindblad dynamics. We further analyze the universal properties of an example case, identifying a change of critical behavior when quantum effects are varied, showing their potential impact on the collective dynamical behavior underlying information processing in large-scale QNNs.
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Affiliation(s)
- Edward Gillman
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Federico Carollo
- Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
| | - Igor Lesanovsky
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom
- Institut für Theoretische Physik, Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
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7
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Liu Z, Yu LW, Duan LM, Deng DL. Presence and Absence of Barren Plateaus in Tensor-Network Based Machine Learning. PHYSICAL REVIEW LETTERS 2022; 129:270501. [PMID: 36638302 DOI: 10.1103/physrevlett.129.270501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/2021] [Accepted: 11/22/2022] [Indexed: 06/17/2023]
Abstract
Tensor networks are efficient representations of high-dimensional tensors with widespread applications in quantum many-body physics. Recently, they have been adapted to the field of machine learning, giving rise to an emergent research frontier that has attracted considerable attention. Here, we study the trainability of tensor-network based machine learning models by exploring the landscapes of different loss functions, with a focus on the matrix product states (also called tensor trains) architecture. In particular, we rigorously prove that barren plateaus (i.e., exponentially vanishing gradients) prevail in the training process of the machine learning algorithms with global loss functions. Whereas, for local loss functions the gradients with respect to variational parameters near the local observables do not vanish as the system size increases. Therefore, the barren plateaus are absent in this case and the corresponding models could be efficiently trainable. Our results reveal a crucial aspect of tensor-network based machine learning in a rigorous fashion, which provide a valuable guide for both practical applications and theoretical studies in the future.
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Affiliation(s)
- Zidu Liu
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
| | - Li-Wei Yu
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
- Theoretical Physics Division, Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People's Republic of China
| | - L-M Duan
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
| | - Dong-Ling Deng
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
- Shanghai Qi Zhi Institute, 41th Floor, AI Tower, No. 701 Yunjin Road, Xuhui District, Shanghai 200232, China
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8
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Xu H, Li B. Pseudo almost periodic solutions for Clifford-valued neutral-type fuzzy neural networks with multi-proportional delay and D operator1. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2022. [DOI: 10.3233/jifs-221017] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
In this paper, a class of Clifford-valued neutral fuzzy neural-type networks with proportional delay and D operator and whose self feedback coefficients are also Clifford numbers are considered. By using the Banach fixed point theorem and some differential inequality techniques, we directly study the existence and global asymptotic stability of pseudo almost periodic solutions by not decomposing the considered Clifford-valued systems into real-valued systems. Finally, two examples are given to illustrate our main results. Our results of this paper are new.
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Affiliation(s)
- Huili Xu
- School of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, Yunnan, China
| | - Bing Li
- School of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, Yunnan, China
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9
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Huerta Alderete C, Gordon MH, Sauvage F, Sone A, Sornborger AT, Coles PJ, Cerezo M. Inference-Based Quantum Sensing. PHYSICAL REVIEW LETTERS 2022; 129:190501. [PMID: 36399750 DOI: 10.1103/physrevlett.129.190501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2022] [Revised: 09/13/2022] [Accepted: 10/03/2022] [Indexed: 06/16/2023]
Abstract
In a standard quantum sensing (QS) task one aims at estimating an unknown parameter θ, encoded into an n-qubit probe state, via measurements of the system. The success of this task hinges on the ability to correlate changes in the parameter to changes in the system response R(θ) (i.e., changes in the measurement outcomes). For simple cases the form of R(θ) is known, but the same cannot be said for realistic scenarios, as no general closed-form expression exists. In this Letter, we present an inference-based scheme for QS. We show that, for a general class of unitary families of encoding, R(θ) can be fully characterized by only measuring the system response at 2n+1 parameters. This allows us to infer the value of an unknown parameter given the measured response, as well as to determine the sensitivity of the scheme, which characterizes its overall performance. We show that inference error is, with high probability, smaller than δ, if one measures the system response with a number of shots that scales only as Ω(log^{3}(n)/δ^{2}). Furthermore, the framework presented can be broadly applied as it remains valid for arbitrary probe states and measurement schemes, and, even holds in the presence of quantum noise. We also discuss how to extend our results beyond unitary families. Finally, to showcase our method we implement it for a QS task on real quantum hardware, and in numerical simulations.
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Affiliation(s)
- C Huerta Alderete
- Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- Quantum Science Center, Oak Ridge, Tennessee 37931, USA
| | - Max Hunter Gordon
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid 28049, Spain
| | - Frédéric Sauvage
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Akira Sone
- Aliro Technologies, Inc, Boston, Massachusetts 02135, USA
| | - Andrew T Sornborger
- Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- Quantum Science Center, Oak Ridge, Tennessee 37931, USA
| | - Patrick J Coles
- Quantum Science Center, Oak Ridge, Tennessee 37931, USA
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - M Cerezo
- Information Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
- Quantum Science Center, Oak Ridge, Tennessee 37931, USA
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10
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Cerezo M, Verdon G, Huang HY, Cincio L, Coles PJ. Challenges and opportunities in quantum machine learning. NATURE COMPUTATIONAL SCIENCE 2022; 2:567-576. [PMID: 38177473 DOI: 10.1038/s43588-022-00311-3] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2022] [Accepted: 08/04/2022] [Indexed: 01/06/2024]
Abstract
At the intersection of machine learning and quantum computing, quantum machine learning has the potential of accelerating data analysis, especially for quantum data, with applications for quantum materials, biochemistry and high-energy physics. Nevertheless, challenges remain regarding the trainability of quantum machine learning models. Here we review current methods and applications for quantum machine learning. We highlight differences between quantum and classical machine learning, with a focus on quantum neural networks and quantum deep learning. Finally, we discuss opportunities for quantum advantage with quantum machine learning.
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Affiliation(s)
- M Cerezo
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA
- Quantum Science Center, Oak Ridge, TN, USA
| | - Guillaume Verdon
- X, Mountain View, CA, USA
- Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada
| | - Hsin-Yuan Huang
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA
- Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA
| | - Lukasz Cincio
- Quantum Science Center, Oak Ridge, TN, USA
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Patrick J Coles
- Quantum Science Center, Oak Ridge, TN, USA.
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA.
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11
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Caro MC, Huang HY, Cerezo M, Sharma K, Sornborger A, Cincio L, Coles PJ. Generalization in quantum machine learning from few training data. Nat Commun 2022; 13:4919. [PMID: 35995777 PMCID: PMC9395350 DOI: 10.1038/s41467-022-32550-3] [Citation(s) in RCA: 19] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2022] [Accepted: 08/04/2022] [Indexed: 11/19/2022] Open
Abstract
Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number N of training data points. We show that the generalization error of a quantum machine learning model with T trainable gates scales at worst as [Formula: see text]. When only K ≪ T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to [Formula: see text]. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.
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Affiliation(s)
- Matthias C Caro
- Department of Mathematics, Technical University of Munich, Garching, Germany.
- Munich Center for Quantum Science and Technology (MCQST), Munich, Germany.
| | - Hsin-Yuan Huang
- Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA
- Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
| | - M Cerezo
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Kunal Sharma
- Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, 20742, USA
| | - Andrew Sornborger
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
- Quantum Science Center, Oak Ridge, TN, 37931, USA
| | - Lukasz Cincio
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Patrick J Coles
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
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