1
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Qian Y, Wang X, Du Y, Wu X, Tao D. The Dilemma of Quantum Neural Networks. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2024; 35:5603-5615. [PMID: 36191113 DOI: 10.1109/tnnls.2022.3208313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/16/2023]
Abstract
The core of quantum machine learning is to devise quantum models with good trainability and low generalization error bounds than their classical counterparts to ensure better reliability and interpretability. Recent studies confirmed that quantum neural networks (QNNs) have the ability to achieve this goal on specific datasets. In this regard, it is of great importance to understand whether these advantages are still preserved on real-world tasks. Through systematic numerical experiments, we empirically observe that current QNNs fail to provide any benefit over classical learning models. Concretely, our results deliver two key messages. First, QNNs suffer from the severely limited effective model capacity, which incurs poor generalization on real-world datasets. Second, the trainability of QNNs is insensitive to regularization techniques, which sharply contrasts with the classical scenario. These empirical results force us to rethink the role of current QNNs and to design novel protocols for solving real-world problems with quantum advantages.
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2
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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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3
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Zhu D, Shen W, Giani A, Ray-Majumder S, Neculaes B, Johri S. Copula-based risk aggregation with trapped ion quantum computers. Sci Rep 2023; 13:18511. [PMID: 37898631 PMCID: PMC10613293 DOI: 10.1038/s41598-023-44151-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2023] [Accepted: 10/04/2023] [Indexed: 10/30/2023] Open
Abstract
Copulas are mathematical tools for modeling joint probability distributions. In the past 60 years they have become an essential analysis tool on classical computers in various fields. The recent finding that copulas can be expressed as maximally entangled quantum states has revealed a promising approach to practical quantum advantages: performing tasks faster, requiring less memory, or, as we show, yielding better predictions. Studying the scalability of this quantum approach as both the precision and the number of modeled variables increase is crucial for its adoption in real-world applications. In this paper, we successfully apply a Quantum Circuit Born Machine (QCBM) based approach to modeling 3- and 4-variable copulas on trapped ion quantum computers. We study the training of QCBMs with different levels of precision and circuit design on a simulator and a state-of-the-art trapped ion quantum computer. We observe decreased training efficacy due to the increased complexity in parameter optimization as the models scale up. To address this challenge, we introduce an annealing-inspired strategy that dramatically improves the training results. In our end-to-end tests, various configurations of the quantum models make a comparable or better prediction in risk aggregation tasks than the standard classical models.
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Affiliation(s)
- Daiwei Zhu
- IonQ Inc., 4505 Campus Drive, College Park, MD, USA.
| | - Weiwei Shen
- GE Research, One Research Circle, Niskayuna, NY, USA
| | | | | | | | - Sonika Johri
- IonQ Inc., 4505 Campus Drive, College Park, MD, USA
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4
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Tian J, Sun X, Du Y, Zhao S, Liu Q, Zhang K, Yi W, Huang W, Wang C, Wu X, Hsieh MH, Liu T, Yang W, Tao D. Recent Advances for Quantum Neural Networks in Generative Learning. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2023; 45:12321-12340. [PMID: 37126624 DOI: 10.1109/tpami.2023.3272029] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
Quantum computers are next-generation devices that hold promise to perform calculations beyond the reach of classical computers. A leading method towards achieving this goal is through quantum machine learning, especially quantum generative learning. Due to the intrinsic probabilistic nature of quantum mechanics, it is reasonable to postulate that quantum generative learning models (QGLMs) may surpass their classical counterparts. As such, QGLMs are receiving growing attention from the quantum physics and computer science communities, where various QGLMs that can be efficiently implemented on near-term quantum machines with potential computational advantages are proposed. In this paper, we review the current progress of QGLMs from the perspective of machine learning. Particularly, we interpret these QGLMs, covering quantum circuit Born machines, quantum generative adversarial networks, quantum Boltzmann machines, and quantum variational autoencoders, as the quantum extension of classical generative learning models. In this context, we explore their intrinsic relations and their fundamental differences. We further summarize the potential applications of QGLMs in both conventional machine learning tasks and quantum physics. Last, we discuss the challenges and further research directions for QGLMs.
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5
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Pan X, Lu Z, Wang W, Hua Z, Xu Y, Li W, Cai W, Li X, Wang H, Song YP, Zou CL, Deng DL, Sun L. Deep quantum neural networks on a superconducting processor. Nat Commun 2023; 14:4006. [PMID: 37414812 DOI: 10.1038/s41467-023-39785-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2022] [Accepted: 06/29/2023] [Indexed: 07/08/2023] Open
Abstract
Deep learning and quantum computing have achieved dramatic progresses in recent years. The interplay between these two fast-growing fields gives rise to a new research frontier of quantum machine learning. In this work, we report an experimental demonstration of training deep quantum neural networks via the backpropagation algorithm with a six-qubit programmable superconducting processor. We experimentally perform the forward process of the backpropagation algorithm and classically simulate the backward process. In particular, we show that three-layer deep quantum neural networks can be trained efficiently to learn two-qubit quantum channels with a mean fidelity up to 96.0% and the ground state energy of molecular hydrogen with an accuracy up to 93.3% compared to the theoretical value. In addition, six-layer deep quantum neural networks can be trained in a similar fashion to achieve a mean fidelity up to 94.8% for learning single-qubit quantum channels. Our experimental results indicate that the number of coherent qubits required to maintain does not scale with the depth of the deep quantum neural network, thus providing a valuable guide for quantum machine learning applications with both near-term and future quantum devices.
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Affiliation(s)
- Xiaoxuan Pan
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Zhide Lu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Weiting Wang
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Ziyue Hua
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Yifang Xu
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Weikang Li
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Weizhou Cai
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Xuegang Li
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Haiyan Wang
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Yi-Pu Song
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China
| | - Chang-Ling Zou
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, Anhui, 230026, China
- Hefei National Laboratory, Hefei, 230088, China
| | - Dong-Ling Deng
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China.
- Hefei National Laboratory, Hefei, 230088, China.
- Shanghai Qi Zhi Institute, No. 701 Yunjin Road, Xuhui District, Shanghai, 200232, China.
| | - Luyan Sun
- Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, 100084, China.
- Hefei National Laboratory, Hefei, 230088, China.
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6
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Markidis S. Programming Quantum Neural Networks on NISQ Systems: An Overview of Technologies and Methodologies. ENTROPY (BASEL, SWITZERLAND) 2023; 25:e25040694. [PMID: 37190482 PMCID: PMC10138272 DOI: 10.3390/e25040694] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/01/2023] [Revised: 04/03/2023] [Accepted: 04/04/2023] [Indexed: 05/17/2023]
Abstract
Noisy Intermediate-Scale Quantum (NISQ) systems and associated programming interfaces make it possible to explore and investigate the design and development of quantum computing techniques for Machine Learning (ML) applications. Among the most recent quantum ML approaches, Quantum Neural Networks (QNN) emerged as an important tool for data analysis. With the QNN advent, higher-level programming interfaces for QNN have been developed. In this paper, we survey the current state-of-the-art high-level programming approaches for QNN development. We discuss target architectures, critical QNN algorithmic components, such as the hybrid workflow of Quantum Annealers and Parametrized Quantum Circuits, QNN architectures, optimizers, gradient calculations, and applications. Finally, we overview the existing programming QNN frameworks, their software architecture, and associated quantum simulators.
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7
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Jerbi S, Fiderer LJ, Poulsen Nautrup H, Kübler JM, Briegel HJ, Dunjko V. Quantum machine learning beyond kernel methods. Nat Commun 2023; 14:517. [PMID: 36720861 PMCID: PMC9889392 DOI: 10.1038/s41467-023-36159-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2022] [Accepted: 01/18/2023] [Indexed: 02/02/2023] Open
Abstract
Machine learning algorithms based on parametrized quantum circuits are prime candidates for near-term applications on noisy quantum computers. In this direction, various types of quantum machine learning models have been introduced and studied extensively. Yet, our understanding of how these models compare, both mutually and to classical models, remains limited. In this work, we identify a constructive framework that captures all standard models based on parametrized quantum circuits: that of linear quantum models. In particular, we show using tools from quantum information theory how data re-uploading circuits, an apparent outlier of this framework, can be efficiently mapped into the simpler picture of linear models in quantum Hilbert spaces. Furthermore, we analyze the experimentally-relevant resource requirements of these models in terms of qubit number and amount of data needed to learn. Based on recent results from classical machine learning, we prove that linear quantum models must utilize exponentially more qubits than data re-uploading models in order to solve certain learning tasks, while kernel methods additionally require exponentially more data points. Our results provide a more comprehensive view of quantum machine learning models as well as insights on the compatibility of different models with NISQ constraints.
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Affiliation(s)
- Sofiene Jerbi
- grid.5771.40000 0001 2151 8122Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
| | - Lukas J. Fiderer
- grid.5771.40000 0001 2151 8122Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
| | - Hendrik Poulsen Nautrup
- grid.5771.40000 0001 2151 8122Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
| | - Jonas M. Kübler
- grid.419534.e0000 0001 1015 6533Max Planck Institute for Intelligent Systems, Tübingen, Germany
| | - Hans J. Briegel
- grid.5771.40000 0001 2151 8122Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria
| | - Vedran Dunjko
- grid.5132.50000 0001 2312 1970Leiden University, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
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8
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Ding L, Spector L. Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits. ENTROPY (BASEL, SWITZERLAND) 2023; 25:93. [PMID: 36673234 PMCID: PMC9857551 DOI: 10.3390/e25010093] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/16/2022] [Revised: 12/27/2022] [Accepted: 12/28/2022] [Indexed: 06/17/2023]
Abstract
Recent work on hybrid quantum-classical machine learning systems has demonstrated success in utilizing parameterized quantum circuits (PQCs) to solve the challenging reinforcement learning (RL) tasks, with provable learning advantages over classical systems, e.g., deep neural networks. While existing work demonstrates and exploits the strength of PQC-based models, the design choices of PQC architectures and the interactions between different quantum circuits on learning tasks are generally underexplored. In this work, we introduce a Multi-objective Evolutionary Architecture Search framework for parameterized quantum circuits (MEAS-PQC), which uses a multi-objective genetic algorithm with quantum-specific configurations to perform efficient searching of optimal PQC architectures. Experimental results show that our method can find architectures that have superior learning performance on three benchmark RL tasks, and are also optimized for additional objectives including reductions in quantum noise and model size. Further analysis of patterns and probability distributions of quantum operations helps identify performance-critical design choices of hybrid quantum-classical learning systems.
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Affiliation(s)
- Li Ding
- Manning College of Information & Computer Sciences, University of Massachusetts Amherst, Amherst, MA 01002, USA
| | - Lee Spector
- Manning College of Information & Computer Sciences, University of Massachusetts Amherst, Amherst, MA 01002, USA
- Department of Computer Science, Amherst College, Amherst, MA 01002, USA
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9
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Zhao W, Wang Y, Qu Y, Ma H, Wang S. Binary Classification Quantum Neural Network Model Based on Optimized Grover Algorithm. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1783. [PMID: 36554188 PMCID: PMC9777537 DOI: 10.3390/e24121783] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/26/2022] [Revised: 11/23/2022] [Accepted: 11/29/2022] [Indexed: 06/17/2023]
Abstract
We focus on the problem that the Grover algorithm is not suitable for the completely unknown proportion of target solutions. Considering whether the existing quantum classifier used by the current quantum neural network (QNN) to complete the classification task can solve the problem of the classical classifier, this paper proposes a binary quantum neural network classifical model based on an optimized Grover algorithm based on partial diffusion. Trial and error is adopted to extend the partial diffusion quantum search algorithm with the known proportion of target solutions to the unknown state, and to apply the characteristics of the supervised learning of the quantum neural network to binary classify the classified data. Experiments show that the proposed method can effectively retrieve quantum states with similar features. The test accuracy of BQM retrieval under the depolarization noise at the 20th period can reach 97% when the depolarization rate is 0.1. It improves the retrieval accuracy by about 4% and 10% compared with MSE and BCE in the same environment.
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Affiliation(s)
- Wenlin Zhao
- School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266520, China
| | - Yinuo Wang
- School of Science, Qingdao University of Technology, Qingdao 266520, China
| | - Yingjie Qu
- School of Science, Qingdao University of Technology, Qingdao 266520, China
| | - Hongyang Ma
- School of Science, Qingdao University of Technology, Qingdao 266520, China
| | - Shumei Wang
- School of Science, Qingdao University of Technology, Qingdao 266520, China
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10
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Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer. Nat Commun 2022; 13:5743. [PMID: 36220831 PMCID: PMC9553922 DOI: 10.1038/s41467-022-33335-4] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2022] [Accepted: 09/14/2022] [Indexed: 11/08/2022] Open
Abstract
The famous, yet unsolved, Fermi-Hubbard model for strongly-correlated electronic systems is a prominent target for quantum computers. However, accurately representing the Fermi-Hubbard ground state for large instances may be beyond the reach of near-term quantum hardware. Here we show experimentally that an efficient, low-depth variational quantum algorithm with few parameters can reproduce important qualitative features of medium-size instances of the Fermi-Hubbard model. We address 1 × 8 and 2 × 4 instances on 16 qubits on a superconducting quantum processor, substantially larger than previous work based on less scalable compression techniques, and going beyond the family of 1D Fermi-Hubbard instances, which are solvable classically. Consistent with predictions for the ground state, we observe the onset of the metal-insulator transition and Friedel oscillations in 1D, and antiferromagnetic order in both 1D and 2D. We use a variety of error-mitigation techniques, including symmetries of the Fermi-Hubbard model and a recently developed technique tailored to simulating fermionic systems. We also introduce a new variational optimisation algorithm based on iterative Bayesian updates of a local surrogate model.
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11
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Zhang H, Wan L, Haug T, Mok WK, Paesani S, Shi Y, Cai H, Chin LK, Karim MF, Xiao L, Luo X, Gao F, Dong B, Assad S, Kim MS, Laing A, Kwek LC, Liu AQ. Resource-efficient high-dimensional subspace teleportation with a quantum autoencoder. SCIENCE ADVANCES 2022; 8:eabn9783. [PMID: 36206336 PMCID: PMC9544333 DOI: 10.1126/sciadv.abn9783] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/12/2022] [Accepted: 08/23/2022] [Indexed: 06/16/2023]
Abstract
Quantum autoencoders serve as efficient means for quantum data compression. Here, we propose and demonstrate their use to reduce resource costs for quantum teleportation of subspaces in high-dimensional systems. We use a quantum autoencoder in a compress-teleport-decompress manner and report the first demonstration with qutrits using an integrated photonic platform for future scalability. The key strategy is to compress the dimensionality of input states by erasing redundant information and recover the initial states after chip-to-chip teleportation. Unsupervised machine learning is applied to train the on-chip autoencoder, enabling the compression and teleportation of any state from a high-dimensional subspace. Unknown states are decompressed at a high fidelity (~0.971), obtaining a total teleportation fidelity of ~0.894. Subspace encodings hold great potential as they support enhanced noise robustness and increased coherence. Laying the groundwork for machine learning techniques in quantum systems, our scheme opens previously unidentified paths toward high-dimensional quantum computing and networking.
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Affiliation(s)
- Hui Zhang
- Quantum Science and Engineering Centre (QSec), Nanyang Technological University, 50 Nanyang Ave., Singapore 639798, Singapore
| | - Lingxiao Wan
- Quantum Science and Engineering Centre (QSec), Nanyang Technological University, 50 Nanyang Ave., Singapore 639798, Singapore
| | - Tobias Haug
- Quantum Optics and Laser Science, Imperial College London, Exhibition Road, London SW72AZ, UK
| | - Wai-Keong Mok
- Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore 117543, Singapore
| | - Stefano Paesani
- Center for Hybrid Quantum Networks (Hy-Q), Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
- Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1TH, UK
| | - Yuzhi Shi
- Institute of Precision Optical Engineering, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
| | - Hong Cai
- Institute of Microelectronics, A*STAR (Agency for Science, Technology and Research), Singapore 138634, Singapore
| | - Lip Ket Chin
- Quantum Science and Engineering Centre (QSec), Nanyang Technological University, 50 Nanyang Ave., Singapore 639798, Singapore
| | - Muhammad Faeyz Karim
- Quantum Science and Engineering Centre (QSec), Nanyang Technological University, 50 Nanyang Ave., Singapore 639798, Singapore
| | - Limin Xiao
- School of Information Science and Technology, Fudan University, Shanghai 200433, China
| | - Xianshu Luo
- Advanced Micro Foundry, 11 Science Park Road, Singapore 117685 Singapore
| | - Feng Gao
- Advanced Micro Foundry, 11 Science Park Road, Singapore 117685 Singapore
| | - Bin Dong
- Advanced Micro Foundry, 11 Science Park Road, Singapore 117685 Singapore
| | - Syed Assad
- Department of Quantum Science, Centre for Quantum Computation and Communication Technology, The Australian National University, Canberra, ACT 2600, Australia
| | - M. S. Kim
- Quantum Optics and Laser Science, Imperial College London, Exhibition Road, London SW72AZ, UK
| | - Anthony Laing
- Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1TH, UK
| | - Leong Chuan Kwek
- Quantum Science and Engineering Centre (QSec), Nanyang Technological University, 50 Nanyang Ave., Singapore 639798, Singapore
- Centre for Quantum Technologies, National University of Singapore, Block S15, 3 Science Drive 2, Singapore 117543, Singapore
- National Institute of Education, 1 Nanyang Walk, Singapore 637616 Singapore
| | - Ai Qun Liu
- Quantum Science and Engineering Centre (QSec), Nanyang Technological University, 50 Nanyang Ave., Singapore 639798, Singapore
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12
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Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity. MATHEMATICS 2022. [DOI: 10.3390/math10060940] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
In quantum and quantum-inspired machine learning, a key step is to embed the data in the quantum space known as Hilbert space. Studying quantum kernel function, which defines the distances among the samples in the Hilbert space, belongs to the fundamental topics in this direction. In this work, we propose a tunable quantum-inspired kernel function (QIKF) named rescaled logarithmic fidelity (RLF) and a non-parametric algorithm for the semi-supervised learning in the quantum space. The rescaling takes advantage of the non-linearity of the kernel to tune the mutual distances of samples in the Hilbert space, and meanwhile avoids the exponentially-small fidelities between quantum many-qubit states. Being non-parametric excludes the possible effects from the variational parameters, and evidently demonstrates the properties of the kernel itself. Our results on the hand-written digits (MNIST dataset) and movie reviews (IMDb dataset) support the validity of our method, by comparing with the standard fidelity as the QIKF as well as several well-known non-parametric algorithms (naive Bayes classifiers, k-nearest neighbors, and spectral clustering). High accuracy is demonstrated, particularly for the unsupervised case with no labeled samples and the few-shot cases with small numbers of labeled samples. With the visualizations by t-stochastic neighbor embedding, our results imply that the machine learning in the Hilbert space complies with the principles of maximal coding rate reduction, where the low-dimensional data exhibit within-class compressibility, between-class discrimination, and overall diversity. The proposed QIKF and semi-supervised algorithm can be further combined with the parametric models such as tensor networks, quantum circuits, and quantum neural networks.
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13
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Du Y, Tu Z, Yuan X, Tao D. Efficient Measure for the Expressivity of Variational Quantum Algorithms. PHYSICAL REVIEW LETTERS 2022; 128:080506. [PMID: 35275658 DOI: 10.1103/physrevlett.128.080506] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2021] [Revised: 01/26/2022] [Accepted: 01/31/2022] [Indexed: 06/14/2023]
Abstract
The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.
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Affiliation(s)
- Yuxuan Du
- JD Explore Academy, Beijing 101111, China
| | - Zhuozhuo Tu
- School of Computer Science, Faculty of Engineering, The University of Sydney, Darlington, NSW 2008, Australia
| | - Xiao Yuan
- Center on Frontiers of Computing Studies, Department of Computer Science, Peking University, Beijing 100871, China
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14
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Leadbeater C, Sharrock L, Coyle B, Benedetti M. F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits. ENTROPY (BASEL, SWITZERLAND) 2021; 23:1281. [PMID: 34682005 PMCID: PMC8534817 DOI: 10.3390/e23101281] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/07/2021] [Revised: 09/27/2021] [Accepted: 09/28/2021] [Indexed: 11/16/2022]
Abstract
Generative modelling is an important unsupervised task in machine learning. In this work, we study a hybrid quantum-classical approach to this task, based on the use of a quantum circuit born machine. In particular, we consider training a quantum circuit born machine using f-divergences. We first discuss the adversarial framework for generative modelling, which enables the estimation of any f-divergence in the near term. Based on this capability, we introduce two heuristics which demonstrably improve the training of the born machine. The first is based on f-divergence switching during training. The second introduces locality to the divergence, a strategy which has proved important in similar applications in terms of mitigating barren plateaus. Finally, we discuss the long-term implications of quantum devices for computing f-divergences, including algorithms which provide quadratic speedups to their estimation. In particular, we generalise existing algorithms for estimating the Kullback-Leibler divergence and the total variation distance to obtain a fault-tolerant quantum algorithm for estimating another f-divergence, namely, the Pearson divergence.
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Affiliation(s)
- Chiara Leadbeater
- Cambridge Quantum Computing Limited, London SW1E 6DR, UK; (C.L.); (L.S.); (B.C.)
| | - Louis Sharrock
- Cambridge Quantum Computing Limited, London SW1E 6DR, UK; (C.L.); (L.S.); (B.C.)
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
| | - Brian Coyle
- Cambridge Quantum Computing Limited, London SW1E 6DR, UK; (C.L.); (L.S.); (B.C.)
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK
| | - Marcello Benedetti
- Cambridge Quantum Computing Limited, London SW1E 6DR, UK; (C.L.); (L.S.); (B.C.)
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Anand A, Degroote M, Aspuru-Guzik A. Natural evolutionary strategies for variational quantum computation. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2021. [DOI: 10.1088/2632-2153/abf3ac] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Abstract
Natural evolutionary strategies (NES) are a family of gradient-free black-box optimization algorithms. This study illustrates their use for the optimization of randomly initialized parameterized quantum circuits (PQCs) in the region of vanishing gradients. We show that using the NES gradient estimator the exponential decrease in variance can be alleviated. We implement two specific approaches, the exponential and separable NES, for parameter optimization of PQCs and compare them against standard gradient descent. We apply them to two different problems of ground state energy estimation using variational quantum eigensolver and state preparation with circuits of varying depth and length. We also introduce batch optimization for circuits with larger depth to extend the use of ES to a larger number of parameters. We achieve accuracy comparable to state-of-the-art optimization techniques in all the above cases with a lower number of circuit evaluations. Our empirical results indicate that one can use NES as a hybrid tool in tandem with other gradient-based methods for optimization of deep quantum circuits in regions with vanishing gradients.
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Alcazar J, Leyton-Ortega V, Perdomo-Ortiz A. Classical versus quantum models in machine learning: insights from a finance application. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2020. [DOI: 10.1088/2632-2153/ab9009] [Citation(s) in RCA: 16] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022] Open
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Tranter A, Love PJ, Mintert F, Wiebe N, Coveney PV. Ordering of Trotterization: Impact on Errors in Quantum Simulation of Electronic Structure. ENTROPY 2019. [PMCID: PMC7514563 DOI: 10.3390/e21121218] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
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Affiliation(s)
- Andrew Tranter
- Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA
- Correspondence:
| | - Peter J. Love
- Department of Physics and Astronomy, Tufts University, Medford, MA 02155, USA
| | - Florian Mintert
- Department of Physics, Imperial College London, London SW7 2AZ, UK
| | - Nathan Wiebe
- Department of Physics, University of Washington, Seattle, WA 98105, USA
- Pacific Northwest National Laboratory, Richland, WA 98382, USA
| | - Peter V. Coveney
- Centre for Computational Science, University College London, London WC1H 0AJ, UK
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