1
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Wang Y, Liu J. A comprehensive review of quantum machine learning: from NISQ to fault tolerance. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2024; 87:116402. [PMID: 39321817 DOI: 10.1088/1361-6633/ad7f69] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2024] [Accepted: 09/25/2024] [Indexed: 09/27/2024]
Abstract
Quantum machine learning, which involves running machine learning algorithms on quantum devices, has garnered significant attention in both academic and business circles. In this paper, we offer a comprehensive and unbiased review of the various concepts that have emerged in the field of quantum machine learning. This includes techniques used in Noisy Intermediate-Scale Quantum (NISQ) technologies and approaches for algorithms compatible with fault-tolerant quantum computing hardware. Our review covers fundamental concepts, algorithms, and the statistical learning theory pertinent to quantum machine learning.
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Affiliation(s)
- Yunfei Wang
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, MD 20742, United States of America
- Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, United States of America
| | - Junyu Liu
- Department of Computer Science, The University of Pittsburgh, Pittsburgh, PA 15260, United States of America
- Department of Computer Science, The University of Chicago, Chicago, IL 60637, United States of America
- Pritzker School of Molecular Engineering, The University of Chicago, Chicago, IL 60637, United States of America
- Kadanoff Center for Theoretical Physics, The University of Chicago, Chicago, IL 60637, United States of America
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2
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Hangleiter D, Gullans MJ. Bell Sampling from Quantum Circuits. PHYSICAL REVIEW LETTERS 2024; 133:020601. [PMID: 39073933 DOI: 10.1103/physrevlett.133.020601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2023] [Accepted: 05/31/2024] [Indexed: 07/31/2024]
Abstract
A central challenge in the verification of quantum computers is benchmarking their performance as a whole and demonstrating their computational capabilities. In this Letter, we find a universal model of quantum computation, Bell sampling, that can be used for both of those tasks and thus provides an ideal stepping stone toward fault tolerance. In Bell sampling, we measure two copies of a state prepared by a quantum circuit in the transversal Bell basis. We show that the Bell samples are classically intractable to produce and at the same time constitute what we call a "circuit shadow": from the Bell samples we can efficiently extract information about the quantum circuit preparing the state, as well as diagnose circuit errors. In addition to known properties that can be efficiently extracted from Bell samples, we give several new and efficient protocols: an estimator of state fidelity, an error-mitigated estimator of Pauli expectation values, a test for the depth of a circuit, and an algorithm to estimate a lower bound on the number of T gates in the circuit. With some additional measurements, the latter algorithm can be used to learn a full description of states prepared by circuits with low T count.
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3
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Thanasilp S, Wang S, Cerezo M, Holmes Z. Exponential concentration in quantum kernel methods. Nat Commun 2024; 15:5200. [PMID: 38890282 PMCID: PMC11189509 DOI: 10.1038/s41467-024-49287-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2022] [Accepted: 05/31/2024] [Indexed: 06/20/2024] Open
Abstract
Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model's parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the input data. We identify four sources that can lead to concentration including expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.
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Affiliation(s)
- Supanut Thanasilp
- Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore.
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.
- Chula Intelligent and Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand.
| | | | - M Cerezo
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
- Quantum Science Center, Oak Ridge, TN, USA
| | - Zoë Holmes
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA.
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4
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Chen S, Oh C, Zhou S, Huang HY, Jiang L. Tight Bounds on Pauli Channel Learning without Entanglement. PHYSICAL REVIEW LETTERS 2024; 132:180805. [PMID: 38759184 DOI: 10.1103/physrevlett.132.180805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2023] [Revised: 04/08/2024] [Accepted: 04/15/2024] [Indexed: 05/19/2024]
Abstract
Quantum entanglement is a crucial resource for learning properties from nature, but a precise characterization of its advantage can be challenging. In this Letter, we consider learning algorithms without entanglement to be those that only utilize states, measurements, and operations that are separable between the main system of interest and an ancillary system. Interestingly, we show that these algorithms are equivalent to those that apply quantum circuits on the main system interleaved with mid-circuit measurements and classical feedforward. Within this setting, we prove a tight lower bound for Pauli channel learning without entanglement that closes the gap between the best-known upper and lower bound. In particular, we show that Θ(2^{n}ϵ^{-2}) rounds of measurements are required to estimate each eigenvalue of an n-qubit Pauli channel to ϵ error with high probability when learning without entanglement. In contrast, a learning algorithm with entanglement only needs Θ(ϵ^{-2}) copies of the Pauli channel. The tight lower bound strengthens the foundation for an experimental demonstration of entanglement-enhanced advantages for Pauli noise characterization.
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Affiliation(s)
- Senrui Chen
- Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA
| | - Changhun Oh
- Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
| | - Sisi Zhou
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
- Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
| | - Hsin-Yuan Huang
- Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Liang Jiang
- Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA
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5
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Wang X, Du Y, Tu Z, Luo Y, Yuan X, Tao D. Transition role of entangled data in quantum machine learning. Nat Commun 2024; 15:3716. [PMID: 38697959 PMCID: PMC11066002 DOI: 10.1038/s41467-024-47983-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2023] [Accepted: 04/17/2024] [Indexed: 05/05/2024] Open
Abstract
Entanglement serves as the resource to empower quantum computing. Recent progress has highlighted its positive impact on learning quantum dynamics, wherein the integration of entanglement into quantum operations or measurements of quantum machine learning (QML) models leads to substantial reductions in training data size, surpassing a specified prediction error threshold. However, an analytical understanding of how the entanglement degree in data affects model performance remains elusive. In this study, we address this knowledge gap by establishing a quantum no-free-lunch (NFL) theorem for learning quantum dynamics using entangled data. Contrary to previous findings, we prove that the impact of entangled data on prediction error exhibits a dual effect, depending on the number of permitted measurements. With a sufficient number of measurements, increasing the entanglement of training data consistently reduces the prediction error or decreases the required size of the training data to achieve the same prediction error. Conversely, when few measurements are allowed, employing highly entangled data could lead to an increased prediction error. The achieved results provide critical guidance for designing advanced QML protocols, especially for those tailored for execution on early-stage quantum computers with limited access to quantum resources.
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Affiliation(s)
- Xinbiao Wang
- Institute of Artificial Intelligence, School of Computer Science, Wuhan University, Hubei, 430072, China
- National Engineering Research Center for Multimedia Software, Wuhan University, Hubei, 430072, China
- JD Explore Academy, Beijing, 101111, China
| | - Yuxuan Du
- JD Explore Academy, Beijing, 101111, China.
- School of Computer Science and Engineering, Nanyang Technological University, Singapore, 639798, Singapore.
| | - Zhuozhuo Tu
- School of Computer Science, Faculty of Engineering, University of Sydney, Sydney, NSW, 2008, Australia
| | - Yong Luo
- Institute of Artificial Intelligence, School of Computer Science, Wuhan University, Hubei, 430072, China.
- National Engineering Research Center for Multimedia Software, Wuhan University, Hubei, 430072, China.
| | - Xiao Yuan
- Center on Frontiers of Computing Studies, Peking University, Beijing, 100871, China
- School of Computer Science, Peking University, Beijing, 100871, China
| | - Dacheng Tao
- School of Computer Science and Engineering, Nanyang Technological University, Singapore, 639798, Singapore.
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6
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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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7
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Lewis L, Huang HY, Tran VT, Lehner S, Kueng R, Preskill J. Improved machine learning algorithm for predicting ground state properties. Nat Commun 2024; 15:895. [PMID: 38291046 PMCID: PMC10828424 DOI: 10.1038/s41467-024-45014-7] [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: 03/26/2023] [Accepted: 01/08/2024] [Indexed: 02/01/2024] Open
Abstract
Finding the ground state of a quantum many-body system is a fundamental problem in quantum physics. In this work, we give a classical machine learning (ML) algorithm for predicting ground state properties with an inductive bias encoding geometric locality. The proposed ML model can efficiently predict ground state properties of an n-qubit gapped local Hamiltonian after learning from only [Formula: see text] data about other Hamiltonians in the same quantum phase of matter. This improves substantially upon previous results that require [Formula: see text] data for a large constant c. Furthermore, the training and prediction time of the proposed ML model scale as [Formula: see text] in the number of qubits n. Numerical experiments on physical systems with up to 45 qubits confirm the favorable scaling in predicting ground state properties using a small training dataset.
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Affiliation(s)
- Laura Lewis
- California Institute of Technology, Pasadena, CA, USA
- University of Cambridge, Cambridge, UK
| | - Hsin-Yuan Huang
- California Institute of Technology, Pasadena, CA, USA.
- Massachusetts Institute of Technology, Cambridge, MA, USA.
- Google Quantum AI, Venice, CA, USA.
| | | | | | | | - John Preskill
- California Institute of Technology, Pasadena, CA, USA
- AWS Center for Quantum Computing, Pasadena, CA, USA
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8
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Du Y, Yang Y, Tao D, Hsieh MH. Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification. PHYSICAL REVIEW LETTERS 2023; 131:140601. [PMID: 37862647 DOI: 10.1103/physrevlett.131.140601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 08/17/2023] [Indexed: 10/22/2023]
Abstract
Quantum neural networks (QNNs) have become an important tool for understanding the physical world, but their advantages and limitations are not fully understood. Some QNNs with specific encoding methods can be efficiently simulated by classical surrogates, while others with quantum memory may perform better than classical classifiers. Here we systematically investigate the problem-dependent power of quantum neural classifiers (QCs) on multiclass classification tasks. Through the analysis of expected risk, a measure that weighs the training loss and the generalization error of a classifier jointly, we identify two key findings: first, the training loss dominates the power rather than the generalization ability; second, QCs undergo a U-shaped risk curve, in contrast to the double-descent risk curve of deep neural classifiers. We also reveal the intrinsic connection between optimal QCs and the Helstrom bound and the equiangular tight frame. Using these findings, we propose a method that exploits loss dynamics of QCs to estimate the optimal hyperparameter settings yielding the minimal risk. Numerical results demonstrate the effectiveness of our approach to explain the superiority of QCs over multilayer Perceptron on parity datasets and their limitations over convolutional neural networks on image datasets. Our work sheds light on the problem-dependent power of QNNs and offers a practical tool for evaluating their potential merit.
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Affiliation(s)
- Yuxuan Du
- JD Explore Academy, Beijing 10010, China
| | - Yibo Yang
- JD Explore Academy, Beijing 10010, China
- King Abdullah University of Science and Technology, Thuwal 4700, Kingdom of Saudi Arabia
| | - Dacheng Tao
- JD Explore Academy, Beijing 10010, China
- Sydney AI Centre, School of Computer Science, The University of Sydney, New South Wales 2008, Australia
| | - Min-Hsiu Hsieh
- Hon Hai (Foxconn) Research Institute, Taipei 114699, Taiwan
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9
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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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10
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Chen S, Cotler J, Huang HY, Li J. The complexity of NISQ. Nat Commun 2023; 14:6001. [PMID: 37752125 PMCID: PMC10522708 DOI: 10.1038/s41467-023-41217-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2023] [Accepted: 08/25/2023] [Indexed: 09/28/2023] Open
Abstract
The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class NISQ, which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, NISQ, and fault-tolerant quantum computation to solve some problems based on modifications of Simon's problems. We then consider the power of NISQ for three well-studied problems. For unstructured search, we prove that NISQ cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that NISQ only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that NISQ is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits.
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Affiliation(s)
- Sitan Chen
- Department of Electrical Engineering and Computer Science, University of California, Berkeley, Berkeley, CA, USA.
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, USA.
| | - Jordan Cotler
- Society of Fellows, Harvard University, Cambridge, MA, USA.
| | - Hsin-Yuan Huang
- Institute for Quantum Information and Matter, CAltech, Pasadena, CA, USA.
- Department of Computing and Mathematical Sciences, CAltech, Pasadena, CA, USA.
| | - Jerry Li
- Microsoft Research AI, Redmond, WA, USA.
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11
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Barhoumi M, Liu J, Lefkidis G, Hübner W. Ultrafast control of laser-induced spin-dynamics scenarios on two-dimensional Ni3@C63H54 magnetic system. J Chem Phys 2023; 159:084304. [PMID: 37638625 DOI: 10.1063/5.0158160] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2023] [Accepted: 08/07/2023] [Indexed: 08/29/2023] Open
Abstract
The concept of building logically functional networks employing spintronics or magnetic heterostructures is becoming more and more popular today. Incorporating logical segments into a circuit needs physical bonds between the magnetic molecules or clusters involved. In this framework, we systematically study ultrafast laser-induced spin-manipulation scenarios on a closed system of three carbon chains to which three Ni atoms are attached. After the inclusion of spin-orbit coupling and an external magnetic field, different ultrafast spin dynamics scenarios involving spin-flip and long-distance spin-transfer processes are achieved by various appropriately well-tailored time-resolved laser pulses within subpicosecond timescales. We additionally study the various effects of an external magnetic field on spin-flip and spin-transfer processes. Moreover, we obtain spin-dynamics processes induced by a double laser pulse, rather than a single one. We suggest enhancing the spatial addressability of spin-flip and spin-transfer processes. The findings presented in this article will improve our knowledge of the magnetic properties of carbon-based magnetic molecular structures. They also support the relevant experimental realization of spin dynamics and their potential applications in future molecular spintronics devices.
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Affiliation(s)
- Mohamed Barhoumi
- Department of Physics, Rheinland-Pfälzische Technische Universität Kaiserslautern (RPTU) Kaiserslautern-Landau, P.O. Box 3049, 67653 Kaiserslautern, Germany
| | - Jing Liu
- Institute of Theoretical Chemistry, Ulm University, 89081 Ulm, Germany
| | - Georgios Lefkidis
- Department of Physics, Rheinland-Pfälzische Technische Universität Kaiserslautern (RPTU) Kaiserslautern-Landau, P.O. Box 3049, 67653 Kaiserslautern, Germany
| | - Wolfgang Hübner
- Department of Physics, Rheinland-Pfälzische Technische Universität Kaiserslautern (RPTU) Kaiserslautern-Landau, P.O. Box 3049, 67653 Kaiserslautern, Germany
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12
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Liu YT, Wang K, Liu YD, Wang DS. A Survey of Universal Quantum von Neumann Architecture. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1187. [PMID: 37628217 PMCID: PMC10453143 DOI: 10.3390/e25081187] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2023] [Revised: 07/31/2023] [Accepted: 08/03/2023] [Indexed: 08/27/2023]
Abstract
The existence of universal quantum computers has been theoretically well established. However, building up a real quantum computer system not only relies on the theory of universality, but also needs methods to satisfy requirements on other features, such as programmability, modularity, scalability, etc. To this end, here we study the recently proposed model of quantum von Neumann architecture by putting it in a practical and broader setting, namely, the hierarchical design of a computer system. We analyze the structures of quantum CPU and quantum control units and draw their connections with computational advantages. We also point out that a recent demonstration of our model would require less than 20 qubits.
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Affiliation(s)
- Yuan-Ting Liu
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Kai Wang
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Yuan-Dong Liu
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Dong-Sheng Wang
- CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
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13
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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 PMCID: PMC10325994 DOI: 10.1038/s41467-023-39785-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [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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14
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Caro MC, Huang HY, Ezzell N, Gibbs J, Sornborger AT, Cincio L, Coles PJ, Holmes Z. Out-of-distribution generalization for learning quantum dynamics. Nat Commun 2023; 14:3751. [PMID: 37407571 PMCID: PMC10322910 DOI: 10.1038/s41467-023-39381-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/17/2022] [Accepted: 06/09/2023] [Indexed: 07/07/2023] Open
Abstract
Generalization bounds are a critical tool to assess the training data requirements of Quantum Machine Learning (QML). Recent work has established guarantees for in-distribution generalization of quantum neural networks (QNNs), where training and testing data are drawn from the same data distribution. However, there are currently no results on out-of-distribution generalization in QML, where we require a trained model to perform well even on data drawn from a different distribution to the training distribution. Here, we prove out-of-distribution generalization for the task of learning an unknown unitary. In particular, we show that one can learn the action of a unitary on entangled states having trained only product states. Since product states can be prepared using only single-qubit gates, this advances the prospects of learning quantum dynamics on near term quantum hardware, and further opens up new methods for both the classical and quantum compilation of quantum circuits.
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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.
- Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin, Germany.
- Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA.
| | - Hsin-Yuan Huang
- Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA
- Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
| | - Nicholas Ezzell
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
- Department of Physics & Astronomy, University of Southern California, Los Angeles, CA, USA
| | - Joe Gibbs
- Department of Physics, University of Surrey, Guildford, GU2 7XH, UK
- AWE, Aldermaston, Reading, RG7 4PR, UK
| | | | - Lukasz Cincio
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
| | - Patrick J Coles
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
- Normal Computing Corporation, New York, NY, USA
| | - Zoë Holmes
- Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA
- Institute of Physics, Ecole Polytechnique Fédéderale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
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15
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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 PMCID: PMC10304445 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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16
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White GAL, Modi K, Hill CD. Filtering Crosstalk from Bath Non-Markovianity via Spacetime Classical Shadows. PHYSICAL REVIEW LETTERS 2023; 130:160401. [PMID: 37154634 DOI: 10.1103/physrevlett.130.160401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2022] [Accepted: 03/02/2023] [Indexed: 05/10/2023]
Abstract
From an open system perspective non-Markovian effects due to a nearby bath or neighboring qubits are dynamically equivalent. However, there is a conceptual distinction to account for: neighboring qubits may be controlled. We combine recent advances in non-Markovian quantum process tomography with the framework of classical shadows to characterize spatiotemporal quantum correlations. Observables here constitute operations applied to the system, where the free operation is the maximally depolarizing channel. Using this as a causal break, we systematically erase causal pathways to narrow down the progenitors of temporal correlations. We show that one application of this is to filter out the effects of crosstalk and probe only non-Markovianity from an inaccessible bath. It also provides a lens on spatiotemporally spreading correlated noise throughout a lattice from common environments. We demonstrate both examples on synthetic data. Owing to the scaling of classical shadows, we can erase arbitrarily many neighboring qubits at no extra cost. Our procedure is thus efficient and amenable to systems even with all-to-all interactions.
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Affiliation(s)
- G A L White
- School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
| | - K Modi
- School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
- Centre for Quantum Technology, Transport for New South Wales, Sydney, New South Wales 2000, Australia
| | - C D Hill
- School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia
- School of Mathematics and Statistics, University of Melbourne, Parkville, Victoria, 3010, Australia
- Silicon Quantum Computing, The University of New South Wales, Sydney, New South Wales 2052, Australia
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17
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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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18
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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] [Grants] [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
- Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020, Innsbruck, Austria.
| | - Lukas J Fiderer
- Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020, Innsbruck, Austria
| | - Hendrik Poulsen Nautrup
- Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020, Innsbruck, Austria
| | - Jonas M Kübler
- Max Planck Institute for Intelligent Systems, Tübingen, Germany
| | - Hans J Briegel
- Institute for Theoretical Physics, University of Innsbruck, Technikerstr. 21a, A-6020, Innsbruck, Austria
| | - Vedran Dunjko
- Leiden University, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands
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19
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Anschuetz ER, Kiani BT. Quantum variational algorithms are swamped with traps. Nat Commun 2022; 13:7760. [PMID: 36522354 PMCID: PMC9755303 DOI: 10.1038/s41467-022-35364-5] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2022] [Accepted: 11/30/2022] [Indexed: 12/23/2022] Open
Abstract
One of the most important properties of classical neural networks is how surprisingly trainable they are, though their training algorithms typically rely on optimizing complicated, nonconvex loss functions. Previous results have shown that unlike the case in classical neural networks, variational quantum models are often not trainable. The most studied phenomenon is the onset of barren plateaus in the training landscape of these quantum models, typically when the models are very deep. This focus on barren plateaus has made the phenomenon almost synonymous with the trainability of quantum models. Here, we show that barren plateaus are only a part of the story. We prove that a wide class of variational quantum models-which are shallow, and exhibit no barren plateaus-have only a superpolynomially small fraction of local minima within any constant energy from the global minimum, rendering these models untrainable if no good initial guess of the optimal parameters is known. We also study the trainability of variational quantum algorithms from a statistical query framework, and show that noisy optimization of a wide variety of quantum models is impossible with a sub-exponential number of queries. Finally, we numerically confirm our results on a variety of problem instances. Though we exclude a wide variety of quantum algorithms here, we give reason for optimism for certain classes of variational algorithms and discuss potential ways forward in showing the practical utility of such algorithms.
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Affiliation(s)
- Eric R Anschuetz
- MIT Center for Theoretical Physics, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA.
| | - Bobak T Kiani
- MIT Department of Electrical Engineering and Computer Science, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA.
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20
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Huggins WJ, Wan K, McClean J, O'Brien TE, Wiebe N, Babbush R. Nearly Optimal Quantum Algorithm for Estimating Multiple Expectation Values. PHYSICAL REVIEW LETTERS 2022; 129:240501. [PMID: 36563264 DOI: 10.1103/physrevlett.129.240501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2022] [Revised: 08/30/2022] [Accepted: 09/30/2022] [Indexed: 06/17/2023]
Abstract
Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of state preparations that scales with the target error ϵ as O(1/ϵ). In this Letter, we address the task of estimating the expectation values of M different observables, each to within additive error ϵ, with the same 1/ϵ dependence. We describe an approach that leverages Gilyén et al.'s quantum gradient estimation algorithm to achieve O(sqrt[M]/ϵ) scaling up to logarithmic factors, regardless of the commutation properties of the M observables. We prove that this scaling is worst-case optimal in the high-precision regime if the state preparation is treated as a black box, even when the operators are mutually commuting. We highlight the flexibility of our approach by presenting several generalizations, including a strategy for accelerating the estimation of a collection of dynamic correlation functions.
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Affiliation(s)
| | - Kianna Wan
- Google Quantum AI, Mountain View, 94043 California, USA
- Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
| | | | | | - Nathan Wiebe
- University of Toronto, Toronto, Ontario ON M5S, Canada
- Pacific Northwest National Laboratory, Richland, 99354 Washington, USA
| | - Ryan Babbush
- Google Quantum AI, Mountain View, 94043 California, USA
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21
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Huang HY, Kueng R, Torlai G, Albert VV, Preskill J. Provably efficient machine learning for quantum many-body problems. Science 2022; 377:eabk3333. [PMID: 36137032 DOI: 10.1126/science.abk3333] [Citation(s) in RCA: 16] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground-state properties of gapped Hamiltonians after learning from other Hamiltonians in the same quantum phase of matter. By contrast, under a widely accepted conjecture, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.
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Affiliation(s)
- Hsin-Yuan Huang
- Institute for Quantum Information and Matter and Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
| | - Richard Kueng
- Institute for Integrated Circuits, Johannes Kepler University, Linz, Austria
| | | | - Victor V Albert
- Joint Center for Quantum Information and Computer Science, National Institute of Standards and Technology and University of Maryland, College Park, MD, USA
| | - John Preskill
- Institute for Quantum Information and Matter and Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA.,AWS Center for Quantum Computing, Pasadena, CA, USA
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22
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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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23
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O’Brien TE, Ioffe LB, Su Y, Fushman D, Neven H, Babbush R, Smelyanskiy V. Quantum computation of molecular structure using data from challenging-to-classically-simulate nuclear magnetic resonance experiments. PRX QUANTUM : A PHYSICAL REVIEW JOURNAL 2022; 3:030345. [PMID: 36624758 PMCID: PMC9825292 DOI: 10.1103/prxquantum.3.030345] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to non-ergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.
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Affiliation(s)
| | - Lev B. Ioffe
- Google Quantum AI, Venice, CA 90291, United States
| | - Yuan Su
- Google Quantum AI, Venice, CA 90291, United States
| | - David Fushman
- Department of Chemistry and Biochemistry, Center for Biomolecular Structure and Organization, University of Maryland, College Park, MD 20742, United States
| | | | - Ryan Babbush
- Google Quantum AI, Venice, CA 90291, United States
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24
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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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25
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Huang HY, Broughton M, Cotler J, Chen S, Li J, Mohseni M, Neven H, Babbush R, Kueng R, Preskill J, McClean JR. Quantum advantage in learning from experiments. Science 2022; 376:1182-1186. [PMID: 35679419 DOI: 10.1126/science.abn7293] [Citation(s) in RCA: 39] [Impact Index Per Article: 19.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
Quantum technology promises to revolutionize how we learn about the physical world. An experiment that processes quantum data with a quantum computer could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. We proved that quantum machines could learn from exponentially fewer experiments than the number required by conventional experiments. This exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis, and learning about physical dynamics. Furthermore, the quantum resources needed for achieving an exponential advantage are quite modest in some cases. Conducting experiments with 40 superconducting qubits and 1300 quantum gates, we demonstrated that a substantial quantum advantage is possible with today's quantum processors.
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Affiliation(s)
- Hsin-Yuan Huang
- Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA.,Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA
| | | | - Jordan Cotler
- Harvard Society of Fellows, Cambridge, MA 02138, USA.,Black Hole Initiative, Cambridge, MA 02138, USA
| | - Sitan Chen
- Department of Electrical Engineering and Computer Science, University of California Berkeley, Berkeley, CA, USA.,Simons Institute for the Theory of Computing, Berkeley, CA, USA
| | - Jerry Li
- Microsoft Research AI, Redmond, WA 98052, USA
| | | | | | | | - Richard Kueng
- Institute for Integrated Circuits, Johannes Kepler University Linz, Austria
| | - John Preskill
- Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA.,Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA, USA.,AWS Center for Quantum Computing, Pasadena, CA 91125, USA
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26
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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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27
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Aharonov D, Cotler J, Qi XL. Quantum algorithmic measurement. Nat Commun 2022; 13:887. [PMID: 35173160 PMCID: PMC8850572 DOI: 10.1038/s41467-021-27922-0] [Citation(s) in RCA: 12] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2021] [Accepted: 12/14/2021] [Indexed: 11/22/2022] Open
Abstract
There has been recent promising experimental and theoretical evidence that quantum computational tools might enhance the precision and efficiency of physical experiments. However, a systematic treatment and comprehensive framework are missing. Here we initiate the systematic study of experimental quantum physics from the perspective of computational complexity. To this end, we define the framework of quantum algorithmic measurements (QUALMs), a hybrid of black box quantum algorithms and interactive protocols. We use the QUALM framework to study two important experimental problems in quantum many-body physics: determining whether a system’s Hamiltonian is time-independent or time-dependent, and determining the symmetry class of the dynamics of the system. We study abstractions of these problems and show for both cases that if the experimentalist can use her experimental samples coherently (in both space and time), a provable exponential speedup is achieved compared to the standard situation in which each experimental sample is accessed separately. Our work suggests that quantum computers can provide a new type of exponential advantage: exponential savings in resources in quantum experiments. Applying the language of computational complexity to study real-world experiments requires a rigorous framework. Here, the authors provide such a framework and establish that there can be an exponential savings in resources if an experimentalist can entangle apparatuses with experimental samples.
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Affiliation(s)
- Dorit Aharonov
- School of Computer Science and Engineering, The Hebrew University of Jerusalem, The Edmond J. Safra Campus, 9190416, Jerusalem, Israel
| | - Jordan Cotler
- Society of Fellows, Harvard University, Cambridge, MA, 02138, USA. .,Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, USA.
| | - Xiao-Liang Qi
- Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, USA
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28
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Ho WW, Choi S. Exact Emergent Quantum State Designs from Quantum Chaotic Dynamics. PHYSICAL REVIEW LETTERS 2022; 128:060601. [PMID: 35213180 DOI: 10.1103/physrevlett.128.060601] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2021] [Accepted: 01/18/2022] [Indexed: 06/14/2023]
Abstract
We present exact results on a novel kind of emergent random matrix universality that quantum many-body systems at infinite temperature can exhibit. Specifically, we consider an ensemble of pure states supported on a small subsystem, generated from projective measurements of the remainder of the system in a local basis. We rigorously show that the ensemble, derived for a class of quantum chaotic systems undergoing quench dynamics, approaches a universal form completely independent of system details: it becomes uniformly distributed in Hilbert space. This goes beyond the standard paradigm of quantum thermalization, which dictates that the subsystem relaxes to an ensemble of quantum states that reproduces the expectation values of local observables in a thermal mixed state. Our results imply more generally that the distribution of quantum states themselves becomes indistinguishable from those of uniformly random ones, i.e., the ensemble forms a quantum state design in the parlance of quantum information theory. Our work establishes bridges between quantum many-body physics, quantum information and random matrix theory, by showing that pseudorandom states can arise from isolated quantum dynamics, opening up new ways to design applications for quantum state tomography and benchmarking.
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Affiliation(s)
- Wen Wei Ho
- Department of Physics, Stanford University, Stanford, California 94305, USA
| | - Soonwon Choi
- Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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29
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Zhang T, Sun J, Fang XX, Zhang XM, Yuan X, Lu H. Experimental Quantum State Measurement with Classical Shadows. PHYSICAL REVIEW LETTERS 2021; 127:200501. [PMID: 34860036 DOI: 10.1103/physrevlett.127.200501] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2021] [Accepted: 09/30/2021] [Indexed: 06/13/2023]
Abstract
A crucial subroutine for various quantum computing and communication algorithms is to efficiently extract different classical properties of quantum states. In a notable recent theoretical work by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)NPAHAX1745-247310.1038/s41567-020-0932-7], a thrifty scheme showed how to project the quantum state into classical shadows and simultaneously predict M different functions of a state with only O(log_{2}M) measurements, independent of the system size and saturating the information-theoretical limit. Here, we experimentally explore the feasibility of the scheme in the realistic scenario with a finite number of measurements and noisy operations. We prepare a four-qubit GHZ state and show how to estimate expectation values of multiple observables and Hamiltonians. We compare the measurement strategies with uniform, biased, and derandomized classical shadows to conventional ones that sequentially measure each state function exploiting either importance sampling or observable grouping. We next demonstrate the estimation of nonlinear functions using classical shadows and analyze the entanglement of the prepared quantum state. Our experiment verifies the efficacy of exploiting (derandomized) classical shadows and sheds light on efficient quantum computing with noisy intermediate-scale quantum hardware.
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Affiliation(s)
- Ting Zhang
- School of Physics, Shandong University, Jinan 250100, China
| | - Jinzhao Sun
- Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
- Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom
| | - Xiao-Xu Fang
- School of Physics, Shandong University, Jinan 250100, China
| | - Xiao-Ming Zhang
- Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
- Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR, China
| | - Xiao Yuan
- Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China
| | - He Lu
- School of Physics, Shandong University, Jinan 250100, China
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McClean JR. From Molecules to Quantum Computers: A Research Retrospective. Comput Sci Eng 2021. [DOI: 10.1109/mcse.2021.3120703] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
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McClean JR, Rubin NC, Lee J, Harrigan MP, O'Brien TE, Babbush R, Huggins WJ, Huang HY. What the foundations of quantum computer science teach us about chemistry. J Chem Phys 2021; 155:150901. [PMID: 34686056 DOI: 10.1063/5.0060367] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Abstract
With the rapid development of quantum technology, one of the leading applications that has been identified is the simulation of chemistry. Interestingly, even before full scale quantum computers are available, quantum computer science has exhibited a remarkable string of results that directly impact what is possible in a chemical simulation with any computer. Some of these results even impact our understanding of chemistry in the real world. In this Perspective, we take the position that direct chemical simulation is best understood as a digital experiment. While on the one hand, this clarifies the power of quantum computers to extend our reach, it also shows us the limitations of taking such an approach too directly. Leveraging results that quantum computers cannot outpace the physical world, we build to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can deliver their solution, in general, known in computer science as undecidable problems. This has implications for the predictive power of thermodynamic models and topics such as the ergodic hypothesis. However, we argue that this Perspective is not defeatist but rather helps shed light on the success of existing chemical models such as transition state theory, molecular orbital theory, and thermodynamics as models that benefit from data. We contextualize recent results, showing that data-augmented models are a more powerful rote simulation. These results help us appreciate the success of traditional chemical theory and anticipate new models learned from experimental data. Not only can quantum computers provide data for such models, but they can also extend the class and power of models that utilize data in fundamental ways. These discussions culminate in speculation on new ways for quantum computing and chemistry to interact and our perspective on the eventual roles of quantum computers in the future of chemistry.
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Affiliation(s)
- Jarrod R McClean
- Google Quantum AI, 340 Main Street, Venice, California 90291, USA
| | - Nicholas C Rubin
- Google Quantum AI, 340 Main Street, Venice, California 90291, USA
| | - Joonho Lee
- Google Quantum AI, 340 Main Street, Venice, California 90291, USA
| | | | - Thomas E O'Brien
- Google Quantum AI, 340 Main Street, Venice, California 90291, USA
| | - Ryan Babbush
- Google Quantum AI, 340 Main Street, Venice, California 90291, USA
| | | | - Hsin-Yuan Huang
- Institute for Quantum Information and Matter and Department of Computing and Mathematical Sciences, Caltech, Pasadena, California 91125, USA
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Huang HY, Kueng R, Preskill J. Efficient Estimation of Pauli Observables by Derandomization. PHYSICAL REVIEW LETTERS 2021; 127:030503. [PMID: 34328776 DOI: 10.1103/physrevlett.127.030503] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2021] [Accepted: 06/14/2021] [Indexed: 06/13/2023]
Abstract
We consider the problem of jointly estimating expectation values of many Pauli observables, a crucial subroutine in variational quantum algorithms. Starting with randomized measurements, we propose an efficient derandomization procedure that iteratively replaces random single-qubit measurements by fixed Pauli measurements; the resulting deterministic measurement procedure is guaranteed to perform at least as well as the randomized one. In particular, for estimating any L low-weight Pauli observables, a deterministic measurement on only of order log(L) copies of a quantum state suffices. In some cases, for example, when some of the Pauli observables have high weight, the derandomized procedure is substantially better than the randomized one. Specifically, numerical experiments highlight the advantages of our derandomized protocol over various previous methods for estimating the ground-state energies of small molecules.
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Affiliation(s)
- Hsin-Yuan Huang
- Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA
- Department of Computing and Mathematical Sciences, Caltech, Pasadena, California 91125, USA
| | - Richard Kueng
- Institute for Integrated Circuits, Johannes Kepler University Linz, A-4040, Austria
| | - John Preskill
- Institute for Quantum Information and Matter, Caltech, Pasadena, California 91125, USA
- Department of Computing and Mathematical Sciences, Caltech, Pasadena, California 91125, USA
- Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, California 91125, USA
- AWS Center for Quantum Computing, Pasadena, California 91125, USA
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