Theory of variational quantum simulation

Xiao Yuan1, Suguru Endo1, Qi Zhao2, Ying Li3, and Simon C. Benjamin1

1Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
2Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
3Graduate School of China Academy of Engineering Physics, Beijing 100193, China

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The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.

Universal quantum computers will eventually solve various classically intractable problems, but the exciting challenge is to demonstrate the first real quantum advantage as soon as possible -- with NISQ (for Noisy Intermediate Scaled Quantum) devices. In this regime, we may only be able to manipulate hundreds or thousands of qubits and the operations will be imperfect (or 'noisy'). With such a limited noisy quantum computer, it is unclear how to demonstrate any quantum advantage in any practical task.

This work solves this problem by exploring hybrid algorithms that only solve the core challenging problem with the quantum hardware and the higher level problem with a classical computer. This can be called the quantum coprocessor model: the quantum device handles only the bits that the conventional computer cannot. By considering different variational principles, we show how to simulate real and imaginary time dynamics of closed and open systems. Our work can thus be applied for solving static problems or simulating the dynamics of chemistry and general many-body physics with near-term quantum computers. These are tasks that, until recently, would have been thought to need a full scale fault-tolerant quantum computer in the more distant future.

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[21] Dan-Bo Zhang and Tao Yin, "Collective optimization for variational quantum eigensolvers", Physical Review A 101 3, 032311 (2020).

[22] Suguru Endo, Jinzhao Sun, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational Quantum Simulation of General Processes", Physical Review Letters 125 1, 010501 (2020).

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[25] Youle Wang, Guangxi Li, and Xin Wang, "Variational Quantum Gibbs State Preparation with a Truncated Taylor Series", Physical Review Applied 16 5, 054035 (2021).

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[37] Weitang Li, Zigeng Huang, Changsu Cao, Yifei Huang, Zhigang Shuai, Xiaoming Sun, Jinzhao Sun, Xiao Yuan, and Dingshun Lv, "Toward practical quantum embedding simulation of realistic chemical systems on near-term quantum computers", Chemical Science 13 31, 8953 (2022).

[38] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderón, Nikita Astrakhantsev, and Giuseppe Carleo, "NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems", SciPost Physics Codebases 7 (2022).

[39] Xiaosi Xu, Simon C. Benjamin, and Xiao Yuan, "Variational Circuit Compiler for Quantum Error Correction", Physical Review Applied 15 3, 034068 (2021).

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[42] Tyler Volkoff and Patrick J Coles, "Large gradients via correlation in random parameterized quantum circuits", Quantum Science and Technology 6 2, 025008 (2021).

[43] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

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[49] Manuel G. Algaba, Mario Ponce-Martinez, Carlos Munuera-Javaloy, Vicente Pina-Canelles, Manish J. Thapa, Bruno G. Taketani, Martin Leib, Inés de Vega, Jorge Casanova, and Hermanni Heimonen, "Co-Design quantum simulation of nanoscale NMR", Physical Review Research 4 4, 043089 (2022).

[50] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian Giacomo Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", Journal of The Royal Society Interface 19 196, 20220541 (2022).

[51] Shi-Xin Zhang, Zhou-Quan Wan, Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Variational Quantum-Neural Hybrid Eigensolver", Physical Review Letters 128 12, 120502 (2022).

[52] Manikandan Kondappan, Manish Chaudhary, Ebubechukwu O. Ilo-Okeke, Valentin Ivannikov, and Tim Byrnes, "Imaginary-time evolution with quantum nondemolition measurements: Multiqubit interactions via measurement nonlinearities", Physical Review A 107 4, 042616 (2023).

[53] Nathaniel Wrobel, Anshumitra Baul, Ka-Ming Tam, and Juana Moreno, "Detecting Quantum Critical Points of Correlated Systems by Quantum Convolutional Neural Network Using Data from Variational Quantum Eigensolver", Quantum Reports 4 4, 574 (2022).

[54] Xiao Yuan, Jinzhao Sun, Junyu Liu, Qi Zhao, and You Zhou, "Quantum Simulation with Hybrid Tensor Networks", Physical Review Letters 127 4, 040501 (2021).

[55] Alexander Miessen, Pauline J. Ollitrault, and Ivano Tavernelli, "Quantum algorithms for quantum dynamics: A performance study on the spin-boson model", Physical Review Research 3 4, 043212 (2021).

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[57] Zi-Jian Zhang, Jinzhao Sun, Xiao Yuan, and Man-Hong Yung, "Low-Depth Hamiltonian Simulation by an Adaptive Product Formula", Physical Review Letters 130 4, 040601 (2023).

[58] Suguru Endo, Iori Kurata, and Yuya O. Nakagawa, "Calculation of the Green's function on near-term quantum computers", Physical Review Research 2 3, 033281 (2020).

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[61] Jacob L. Beckey, M. Cerezo, Akira Sone, and Patrick J. Coles, "Variational quantum algorithm for estimating the quantum Fisher information", Physical Review Research 4 1, 013083 (2022).

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[63] Keisuke Matsumoto, Yuta Shingu, Suguru Endo, Shiro Kawabata, Shohei Watabe, Tetsuro Nikuni, Hideaki Hakoshima, and Yuichiro Matsuzaki, "Calculation of Gibbs partition function with imaginary time evolution on near-term quantum computers", Japanese Journal of Applied Physics 61 4, 042002 (2022).

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[66] R. Sagastizabal, S. P. Premaratne, B. A. Klaver, M. A. Rol, V. Negîrneac, M. S. Moreira, X. Zou, S. Johri, N. Muthusubramanian, M. Beekman, C. Zachariadis, V. P. Ostroukh, N. Haider, A. Bruno, A. Y. Matsuura, and L. DiCarlo, "Variational preparation of finite-temperature states on a quantum computer", npj Quantum Information 7 1, 130 (2021).

[67] Hideyuki Miyahara and Vwani Roychowdhury, "Ansatz-Independent Variational Quantum Classifiers and the Price of Ansatz", Scientific Reports 12 1, 19520 (2022).

[68] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, "Noisy intermediate-scale quantum algorithm for semidefinite programming", Physical Review A 105 5, 052445 (2022).

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[74] Jingwei Wen, Chao Zheng, Zhiguo Huang, and Ling Qian, "Iteration-free digital quantum simulation of imaginary-time evolution based on the approximate unitary expansion", Europhysics Letters 141 6, 68001 (2023).

[75] Johanna Barzen, Quantum Computing in the Arts and Humanities 1 (2022) ISBN:978-3-030-95537-3.

[76] Taichi Kosugi, Yusuke Nishiya, Hirofumi Nishi, and Yu-ichiro Matsushita, "Imaginary-time evolution using forward and backward real-time evolution with a single ancilla: First-quantized eigensolver algorithm for quantum chemistry", Physical Review Research 4 3, 033121 (2022).

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[78] Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, and Sabre Kais, "Quantum machine learning for chemistry and physics", Chemical Society Reviews 51 15, 6475 (2022).

[79] Jacopo Rizzo, Francesco Libbi, Francesco Tacchino, Pauline J. Ollitrault, Nicola Marzari, and Ivano Tavernelli, "One-particle Green's functions from the quantum equation of motion algorithm", Physical Review Research 4 4, 043011 (2022).

[80] Lorenzo Leone, Salvatore F. E. Oliviero, Stefano Piemontese, Sarah True, and Alioscia Hamma, "Retrieving information from a black hole using quantum machine learning", Physical Review A 106 6, 062434 (2022).

[81] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo, "An efficient quantum algorithm for the time evolution of parameterized circuits", Quantum 5, 512 (2021).

[82] Bálint Koczor and Simon C. Benjamin, "Quantum natural gradient generalized to noisy and nonunitary circuits", Physical Review A 106 6, 062416 (2022).

[83] Shichuan Xue, Yizhi Wang, Junwei Zhan, Yaxuan Wang, Ru Zeng, Jiangfang Ding, Weixu Shi, Yong Liu, Yingwen Liu, Anqi Huang, Guangyao Huang, Chunlin Yu, Dongyang Wang, Xiang Fu, Xiaogang Qiang, Ping Xu, Mingtang Deng, Xuejun Yang, and Junjie Wu, "Variational Entanglement-Assisted Quantum Process Tomography with Arbitrary Ancillary Qubits", Physical Review Letters 129 13, 133601 (2022).

[84] Kaoru Mizuta, Mikiya Fujii, Shigeki Fujii, Kazuhide Ichikawa, Yutaka Imamura, Yukihiro Okuno, and Yuya O. Nakagawa, "Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials", Physical Review Research 3 4, 043121 (2021).

[85] Brian Doolittle, R. Thomas Bromley, Nathan Killoran, and Eric Chitambar, "Variational Quantum Optimization of Nonlocality in Noisy Quantum Networks", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[86] Qing-Xing Xie, Yi Song, and Yan Zhao, "Power of the Sine Hamiltonian Operator for Estimating the Eigenstate Energies on Quantum Computers", Journal of Chemical Theory and Computation 18 12, 7586 (2022).

[87] Corey Jason Trahan, Mark Loveland, Noah Davis, and Elizabeth Ellison, "A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods", Entropy 25 4, 580 (2023).

[88] Alexander Miessen, Pauline J. Ollitrault, Francesco Tacchino, and Ivano Tavernelli, "Quantum algorithms for quantum dynamics", Nature Computational Science 3 1, 25 (2022).

[89] Stuart M. Harwood, Dimitar Trenev, Spencer T. Stober, Panagiotis Barkoutsos, Tanvi P. Gujarati, Sarah Mostame, and Donny Greenberg, "Improving the Variational Quantum Eigensolver Using Variational Adiabatic Quantum Computing", ACM Transactions on Quantum Computing 3 1, 1 (2022).

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[91] Chenfeng Cao and Xin Wang, "Noise-Assisted Quantum Autoencoder", Physical Review Applied 15 5, 054012 (2021).

[92] Trevor Keen, Thomas Maier, Steven Johnston, and Pavel Lougovski, "Quantum-classical simulation of two-site dynamical mean-field theory on noisy quantum hardware", Quantum Science and Technology 5 3, 035001 (2020).

[93] Nikolay V. Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M. Anisimov, Andrew T. Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A. Dub, "Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in the Variational Quantum Eigensolver", PRX Quantum 2 2, 020337 (2021).

[94] Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, "Error mitigation with Clifford quantum-circuit data", Quantum 5, 592 (2021).

[95] Alexander M. Czajka, Zhong-Bo Kang, Henry Ma, and Fanyi Zhao, "Quantum simulation of chiral phase transitions", Journal of High Energy Physics 2022 8, 209 (2022).

[96] Jin-Min Liang, Qiao-Qiao Lv, Zhi-Xi Wang, and Shao-Ming Fei, "Assisted quantum simulation of open quantum systems", iScience 26 4, 106306 (2023).

[97] Kaoru Mizuta, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii, "Local Variational Quantum Compilation of Large-Scale Hamiltonian Dynamics", PRX Quantum 3 4, 040302 (2022).

[98] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", SciPost Physics 10 2, 040 (2021).

[99] Zhimin He, Junjian Su, Chuangtao Chen, Minghua Pan, and Haozhen Situ, "Search space pruning for quantum architecture search", The European Physical Journal Plus 137 4, 491 (2022).

[100] Hedayat Alghassi, Amol Deshmukh, Noelle Ibrahim, Nicolas Robles, Stefan Woerner, and Christa Zoufal, "A variational quantum algorithm for the Feynman-Kac formula", Quantum 6, 730 (2022).

[101] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan, "Overlapped grouping measurement: A unified framework for measuring quantum states", Quantum 7, 896 (2023).

[102] Conor Mc Keever and Michael Lubasch, "Classically optimized Hamiltonian simulation", Physical Review Research 5 2, 023146 (2023).

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[104] Mario Motta and Julia E. Rice, "Emerging quantum computing algorithms for quantum chemistry", WIREs Computational Molecular Science 12 3(2022).

[105] Pauline J. Ollitrault, Alexander Miessen, and Ivano Tavernelli, "Molecular Quantum Dynamics: A Quantum Computing Perspective", Accounts of Chemical Research 54 23, 4229 (2021).

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[107] Yuhan Huang, Qingyu Li, Xiaokai Hou, Rebing Wu, Man-Hong Yung, Abolfazl Bayat, and Xiaoting Wang, "Robust resource-efficient quantum variational ansatz through an evolutionary algorithm", Physical Review A 105 5, 052414 (2022).

[108] Laura Gentini, Alessandro Cuccoli, and Leonardo Banchi, "Variational Adiabatic Gauge Transformation on Real Quantum Hardware for Effective Low-Energy Hamiltonians and Accurate Diagonalization", Physical Review Applied 18 3, 034025 (2022).

[109] Yuchen Guo and Shuo Yang, "Quantum Error Mitigation via Matrix Product Operators", PRX Quantum 3 4, 040313 (2022).

[110] Rishabh Gupta, Manas Sajjan, Raphael D. Levine, and Sabre Kais, "Variational approach to quantum state tomography based on maximal entropy formalism", Physical Chemistry Chemical Physics 24 47, 28870 (2022).

[111] Tobias Haug and M. S. Kim, "Natural parametrized quantum circuit", Physical Review A 106 5, 052611 (2022).

[112] Chee Kong Lee, Pranay Patil, Shengyu Zhang, and Chang Yu Hsieh, "Neural-network variational quantum algorithm for simulating many-body dynamics", Physical Review Research 3 2, 023095 (2021).

[113] Sergey Bravyi, Oliver Dial, Jay M. Gambetta, Darío Gil, and Zaira Nazario, "The future of quantum computing with superconducting qubits", Journal of Applied Physics 132 16, 160902 (2022).

[114] Bálint Koczor, "Exponential Error Suppression for Near-Term Quantum Devices", Physical Review X 11 3, 031057 (2021).

[115] Kunal Sharma, M. Cerezo, Zoë Holmes, Lukasz Cincio, Andrew Sornborger, and Patrick J. Coles, "Reformulation of the No-Free-Lunch Theorem for Entangled Datasets", Physical Review Letters 128 7, 070501 (2022).

[116] Kishor Bharti, "Fisher Information: A Crucial Tool for NISQ Research", Quantum Views 5, 61 (2021).

[117] Zhenhuan Liu, Pei Zeng, You Zhou, and Mile Gu, "Characterizing correlation within multipartite quantum systems via local randomized measurements", Physical Review A 105 2, 022407 (2022).

[118] Francesco Libbi, Jacopo Rizzo, Francesco Tacchino, Nicola Marzari, and Ivano Tavernelli, "Effective calculation of the Green's function in the time domain on near-term quantum processors", Physical Review Research 4 4, 043038 (2022).

[119] Weitang Li, Jiajun Ren, Sainan Huai, Tianqi Cai, Zhigang Shuai, and Shengyu Zhang, "Efficient quantum simulation of electron-phonon systems by variational basis state encoder", Physical Review Research 5 2, 023046 (2023).

[120] Kian Hwee Lim, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, "Fast-forwarding with NISQ processors without feedback loop", Quantum Science and Technology 7 1, 015001 (2022).

[121] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, "Quantum Natural Gradient", Quantum 4, 269 (2020).

[122] Chufan Lyu, Xiaoyu Tang, Junning Li, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Variational quantum simulation of long-range interacting systems", New Journal of Physics 25 5, 053022 (2023).

[123] D. Zeuch and N. E. Bonesteel, "Efficient two-qubit pulse sequences beyond CNOT", Physical Review B 102 7, 075311 (2020).

[124] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Trainability of Dissipative Perceptron-Based Quantum Neural Networks", Physical Review Letters 128 18, 180505 (2022).

[125] Luca Crippa, Francesco Tacchino, Mario Chizzini, Antonello Aita, Michele Grossi, Alessandro Chiesa, Paolo Santini, Ivano Tavernelli, and Stefano Carretta, "Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers", Magnetochemistry 7 8, 117 (2021).

[126] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu, "Experimental Quantum State Measurement with Classical Shadows", Physical Review Letters 127 20, 200501 (2021).

[127] Steffen Backes, Yuta Murakami, Shiro Sakai, and Ryotaro Arita, "Dynamical mean-field theory for the Hubbard-Holstein model on a quantum device", Physical Review B 107 16, 165155 (2023).

[128] Yong-Xin Yao, Niladri Gomes, Feng Zhang, Cai-Zhuang Wang, Kai-Ming Ho, Thomas Iadecola, and Peter P. Orth, "Adaptive Variational Quantum Dynamics Simulations", PRX Quantum 2 3, 030307 (2021).

[129] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, "Variational quantum algorithms", Nature Reviews Physics 3 9, 625 (2021).

[130] Filipe Fontanela, Antoine Jacquier, and Mugad Oumgari, "Short Communication: A Quantum Algorithm for Linear PDEs Arising in Finance", SIAM Journal on Financial Mathematics 12 4, SC98 (2021).

[131] Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, "NISQ Algorithm for Hamiltonian simulation via truncated Taylor series", SciPost Physics 12 4, 122 (2022).

[132] Niladri Gomes, Anirban Mukherjee, Feng Zhang, Thomas Iadecola, Cai‐Zhuang Wang, Kai‐Ming Ho, Peter P. Orth, and Yong‐Xin Yao, "Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation", Advanced Quantum Technologies 4 12, 2100114 (2021).

[133] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, "Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus", PRX Quantum 3 1, 010313 (2022).

[134] Laura Gentini, Alessandro Cuccoli, Stefano Pirandola, Paola Verrucchi, and Leonardo Banchi, "Noise-resilient variational hybrid quantum-classical optimization", Physical Review A 102 5, 052414 (2020).

[135] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational quantum algorithms for trace distance and fidelity estimation", Quantum Science and Technology 7 1, 015019 (2022).

[136] Tobias Haug and Kishor Bharti, "Generalized quantum assisted simulator", Quantum Science and Technology 7 4, 045019 (2022).

[137] Zidu Liu, L.-M. Duan, and Dong-Ling Deng, "Solving quantum master equations with deep quantum neural networks", Physical Review Research 4 1, 013097 (2022).

[138] Kenji Kubo, Yuya O. Nakagawa, Suguru Endo, and Shota Nagayama, "Variational quantum simulations of stochastic differential equations", Physical Review A 103 5, 052425 (2021).

[139] Leonardo Banchi and Gavin E. Crooks, "Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule", Quantum 5, 386 (2021).

[140] Joseph C. Aulicino, Trevor Keen, and Bo Peng, "State preparation and evolution in quantum computing: A perspective from Hamiltonian moments", International Journal of Quantum Chemistry 122 5(2022).

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The above citations are from Crossref's cited-by service (last updated successfully 2023-06-08 14:56:41) and SAO/NASA ADS (last updated successfully 2023-06-08 14:56:42). The list may be incomplete as not all publishers provide suitable and complete citation data.

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