An initialization strategy for addressing barren plateaus in parametrized quantum circuits

Edward Grant; Leonard Wossnig; Mateusz Ostaszewski; Marcello Benedetti

doi:10.22331/q-2019-12-09-214

An initialization strategy for addressing barren plateaus in parametrized quantum circuits

Edward Grant¹, Leonard Wossnig¹, Mateusz Ostaszewski², and Marcello Benedetti³

¹Rahko Limited & Department of Computer Science, University College London
²Institute of Theoretical and Applied Informatics, Polish Academy of Sciences
³Cambridge Quantum Computing Limited & Department of Computer Science, University College London

Published:	2019-12-09, volume 3, page 214
Eprint:	arXiv:1903.05076v3
Doi:	https://doi.org/10.22331/q-2019-12-09-214
Citation:	Quantum 3, 214 (2019).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Parametrized quantum circuits initialized with random initial parameter values are characterized by barren plateaus where the gradient becomes exponentially small in the number of qubits. In this technical note we theoretically motivate and empirically validate an initialization strategy which can resolve the barren plateau problem for practical applications. The technique involves randomly selecting some of the initial parameter values, then choosing the remaining values so that the circuit is a sequence of shallow blocks that each evaluates to the identity. This initialization limits the effective depth of the circuits used to calculate the first parameter update so that they cannot be stuck in a barren plateau at the start of training. In turn, this makes some of the most compact ansätze usable in practice, which was not possible before even for rather basic problems. We show empirically that variational quantum eigensolvers and quantum neural networks initialized using this strategy can be trained using a gradient based method.

► BibTeX data

@article{Grant2019initialization,
  doi = {10.22331/q-2019-12-09-214},
  url = {https://doi.org/10.22331/q-2019-12-09-214},
  title = {An initialization strategy for addressing barren plateaus in parametrized quantum circuits},
  author = {Grant, Edward and Wossnig, Leonard and Ostaszewski, Mateusz and Benedetti, Marcello},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {214},
  month = dec,
  year = {2019}
}

► References

[1] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature communications, 9(1):4812, 2018. https://doi.org/10.1038/s41467-018-07090-4.
https://doi.org/10.1038/s41467-018-07090-4

[2] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O’brien. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5:4213, 2014. https://doi.org/10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213

[3] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M Chow, and Jay M Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549(7671):242, 2017. https://doi.org/10.1038/nature23879.
https://doi.org/10.1038/nature23879

[4] Maria Schuld, Alex Bocharov, Krysta Svore, and Nathan Wiebe. Circuit-centric quantum classifiers. arXiv preprint arXiv:1804.00633, 2018.
arXiv:1804.00633

[5] Edward Grant, Marcello Benedetti, Shuxiang Cao, Andrew Hallam, Joshua Lockhart, Vid Stojevic, Andrew G Green, and Simone Severini. Hierarchical quantum classifiers. npj Quantum Information, 4(1):65, 2018. https://doi.org/10.1038/s41534-018-0116-9.
https://doi.org/10.1038/s41534-018-0116-9

[6] Hongxiang Chen, Leonard Wossnig, Simone Severini, Hartmut Neven, and Masoud Mohseni. Universal discriminative quantum neural networks. arXiv preprint arXiv:1805.08654, 2018.
arXiv:1805.08654

[7] Dominic Verdon. Unitary 2-designs, variational quantum eigensolvers, and barren plateaus. https://qitheory.blogs.bristol.ac.uk/files/2019/02/barrenplateausblogpost-1xqcazi.pdf, 2019. [Online; accessed 13-March-2019].
https://qitheory.blogs.bristol.ac.uk/files/2019/02/barrenplateausblogpost-1xqcazi.pdf

[8] Zbigniew Puchała and Jaroslaw Adam Miszczak. Symbolic integration with respect to the haar measure on the unitary groups. Bulletin of the Polish Academy of Sciences Technical Sciences, 65(1):21–27, 2017. https://doi.org/10.1515/bpasts-2017-0003.
https://doi.org/10.1515/bpasts-2017-0003

[9] Andris Ambainis and Joseph Emerson. Quantum t-designs: t-wise independence in the quantum world. In Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07), pages 129–140. IEEE, 2007. https://doi.org/10.1109/CCC.2007.26.
https://doi.org/10.1109/CCC.2007.26

[10] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
arXiv:1412.6980

Cited by

[1] Manpreet Singh Jattana, Fengping Jin, Hans De Raedt, and Kristel Michielsen, "Improved Variational Quantum Eigensolver Via Quasidynamical Evolution", Physical Review Applied 19 2, 024047 (2023).

[2] Chiara Leadbeater, Louis Sharrock, Brian Coyle, and Marcello Benedetti, "F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits", Entropy 23 10, 1281 (2021).

[3] Andrew Arrasmith, Zoë Holmes, M Cerezo, and Patrick J Coles, "Equivalence of quantum barren plateaus to cost concentration and narrow gorges", Quantum Science and Technology 7 4, 045015 (2022).

[4] Lucas Friedrich and Jonas Maziero, "Avoiding barren plateaus with classical deep neural networks", Physical Review A 106 4, 042433 (2022).

[5] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, "Noise-induced barren plateaus in variational quantum algorithms", Nature Communications 12 1, 6961 (2021).

[6] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, "Experimental quantum learning of a spectral decomposition", Physical Review Research 3 3, 033200 (2021).

[7] Ho Lun Tang, V.O. Shkolnikov, George S. Barron, Harper R. Grimsley, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Qubit-ADAPT-VQE: An Adaptive Algorithm for Constructing Hardware-Efficient Ansätze on a Quantum Processor", PRX Quantum 2 2, 020310 (2021).

[8] Alexey Uvarov, Jacob D. Biamonte, and Dmitry Yudin, "Variational quantum eigensolver for frustrated quantum systems", Physical Review B 102 7, 075104 (2020).

[9] Max Wilson, Rachel Stromswold, Filip Wudarski, Stuart Hadfield, Norm M. Tubman, and Eleanor G. Rieffel, "Optimizing quantum heuristics with meta-learning", Quantum Machine Intelligence 3 1, 13 (2021).

[10] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Long-time simulations for fixed input states on quantum hardware", npj Quantum Information 8 1, 135 (2022).

[11] Oriel Kiss, Francesco Tacchino, Sofia Vallecorsa, and Ivano Tavernelli, "Quantum neural networks force fields generation", Machine Learning: Science and Technology 3 3, 035004 (2022).

[12] Zoë Holmes, Andrew Arrasmith, Bin Yan, Patrick J. Coles, Andreas Albrecht, and Andrew T. Sornborger, "Barren Plateaus Preclude Learning Scramblers", Physical Review Letters 126 19, 190501 (2021).

[13] Kun Wang, Zhixin Song, Xuanqiang Zhao, Zihe Wang, and Xin Wang, "Detecting and quantifying entanglement on near-term quantum devices", npj Quantum Information 8 1, 52 (2022).

[14] Kathleen E. Hamilton, Emily Lynn, Vicente Leyton-Ortega, Swarnadeep Majumder, and Raphael C. Pooser, 2021 IEEE/ACM International Conference On Computer Aided Design (ICCAD) 1 (2021) ISBN:978-1-6654-4507-8.

[15] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan, "Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation", Journal of the Physical Society of Japan 90 3, 032001 (2021).

[16] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles, "Cost function dependent barren plateaus in shallow parametrized quantum circuits", Nature Communications 12 1, 1791 (2021).

[17] Lucas Slattery, Benjamin Villalonga, and Bryan K. Clark, "Unitary block optimization for variational quantum algorithms", Physical Review Research 4 2, 023072 (2022).

[18] 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).

[19] Oriel Kiss, Michele Grossi, Pavel Lougovski, Federico Sanchez, Sofia Vallecorsa, and Thomas Papenbrock, "Quantum computing of the Li6 nucleus via ordered unitary coupled clusters", Physical Review C 106 3, 034325 (2022).

[20] 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).

[21] Tobias Haug, Kishor Bharti, and M.S. Kim, "Capacity and Quantum Geometry of Parametrized Quantum Circuits", PRX Quantum 2 4, 040309 (2021).

[22] Jinfeng Zeng, Chenfeng Cao, Chao Zhang, Pengxiang Xu, and Bei Zeng, "A variational quantum algorithm for Hamiltonian diagonalization", Quantum Science and Technology 6 4, 045009 (2021).

[23] Francesco Tacchino, Panagiotis Barkoutsos, Chiara Macchiavello, Ivano Tavernelli, Dario Gerace, and Daniele Bajoni, "Quantum implementation of an artificial feed-forward neural network", Quantum Science and Technology 5 4, 044010 (2020).

[24] Carlos Ortiz Marrero, Mária Kieferová, and Nathan Wiebe, "Entanglement-Induced Barren Plateaus", PRX Quantum 2 4, 040316 (2021).

[25] N. N. Hegade, P. Chandarana, K. Paul, Xi Chen, F. Albarrán-Arriagada, and E. Solano, "Portfolio optimization with digitized counterdiabatic quantum algorithms", Physical Review Research 4 4, 043204 (2022).

[26] Iordanis Kerenidis and Alessandro Luongo, "Classification of the MNIST data set with quantum slow feature analysis", Physical Review A 101 6, 062327 (2020).

[27] Joonho Kim, Jaedeok Kim, and Dario Rosa, "Universal effectiveness of high-depth circuits in variational eigenproblems", Physical Review Research 3 2, 023203 (2021).

[28] Shiro Tamiya, Sho Koh, and Yuya O. Nakagawa, "Calculating nonadiabatic couplings and Berry's phase by variational quantum eigensolvers", Physical Review Research 3 2, 023244 (2021).

[29] Tomislav Piskor, Jan-Michael Reiner, Sebastian Zanker, Nicolas Vogt, Michael Marthaler, Frank K. Wilhelm, and Florian G. Eich, "Using gradient-based algorithms to determine ground-state energies on a quantum computer", Physical Review A 105 6, 062415 (2022).

[30] Christa Zoufal, Ryan V. Mishmash, Nitin Sharma, Niraj Kumar, Aashish Sheshadri, Amol Deshmukh, Noelle Ibrahim, Julien Gacon, and Stefan Woerner, "Variational quantum algorithm for unconstrained black box binary optimization: Application to feature selection", Quantum 7, 909 (2023).

[31] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, "Parameterized quantum circuits as machine learning models", Quantum Science and Technology 4 4, 043001 (2019).

[32] Liam Madden, Albert Akhriev, and Andrea Simonetto, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 492 (2022) ISBN:978-1-6654-9113-6.

[33] 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).

[34] Marco Maronese, Claudio Destri, and Enrico Prati, "Quantum activation functions for quantum neural networks", Quantum Information Processing 21 4, 128 (2022).

[35] Francesco Tacchino, Stefano Mangini, Panagiotis Kl. Barkoutsos, Chiara Macchiavello, Dario Gerace, Ivano Tavernelli, and Daniele Bajoni, "Variational Learning for Quantum Artificial Neural Networks", IEEE Transactions on Quantum Engineering 2, 1 (2021).

[36] Carlos Bravo-Prieto, Diego García-Martín, and José I. Latorre, "Quantum singular value decomposer", Physical Review A 101 6, 062310 (2020).

[37] 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).

[38] Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Variational fast forwarding for quantum simulation beyond the coherence time", npj Quantum Information 6 1, 82 (2020).

[39] A V Uvarov and J D Biamonte, "On barren plateaus and cost function locality in variational quantum algorithms", Journal of Physics A: Mathematical and Theoretical 54 24, 245301 (2021).

[40] Vladimir Vargas-Calderón, Fabio A. González, and Herbert Vinck-Posada, "Optimisation-free density estimation and classification with quantum circuits", Quantum Machine Intelligence 4 2, 16 (2022).

[41] Wenyang Qian, Robert Basili, Soham Pal, Glenn Luecke, and James P. Vary, "Solving hadron structures using the basis light-front quantization approach on quantum computers", Physical Review Research 4 4, 043193 (2022).

[42] Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee, "Quantum machine learning: from physics to software engineering", Advances in Physics: X 8 1, 2165452 (2023).

[43] Pinaki Sen, Amandeep Singh Bhatia, Kamalpreet Singh Bhangu, Ahmed Elbeltagi, and Thippa Reddy Gadekallu, "Variational quantum classifiers through the lens of the Hessian", PLOS ONE 17 1, e0262346 (2022).

[44] Hiroyuki Tezuka, Kouhei Nakaji, Takahiko Satoh, and Naoki Yamamoto, "Grover search revisited: Application to image pattern matching", Physical Review A 105 3, 032440 (2022).

[45] Yu Pan, Yifan Tong, Shibei Xue, and Guofeng Zhang, "Efficient depth selection for the implementation of noisy quantum approximate optimization algorithm", Journal of the Franklin Institute 359 18, 11273 (2022).

[46] Tyler Volkoff, Zoë Holmes, and Andrew Sornborger, "Universal Compiling and (No-)Free-Lunch Theorems for Continuous-Variable Quantum Learning", PRX Quantum 2 4, 040327 (2021).

[47] Alejandro Sopena, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López, "Algebraic Bethe Circuits", Quantum 6, 796 (2022).

[48] Roy J. Garcia, Chen Zhao, Kaifeng Bu, and Arthur Jaffe, "Barren plateaus from learning scramblers with local cost functions", Journal of High Energy Physics 2023 1, 90 (2023).

[49] Lewis W. Anderson, Martin Kiffner, Panagiotis Kl. Barkoutsos, Ivano Tavernelli, Jason Crain, and Dieter Jaksch, "Coarse-grained intermolecular interactions on quantum processors", Physical Review A 105 6, 062409 (2022).

[50] Yusen Wu and Jingbo B Wang, "Estimating Gibbs partition function with quantum Clifford sampling", Quantum Science and Technology 7 2, 025006 (2022).

[51] Weikang Li and Dong-Ling Deng, "Recent advances for quantum classifiers", Science China Physics, Mechanics & Astronomy 65 2, 220301 (2022).

[52] Kosuke Mitarai, Yasunari Suzuki, Wataru Mizukami, Yuya O. Nakagawa, and Keisuke Fujii, "Quadratic Clifford expansion for efficient benchmarking and initialization of variational quantum algorithms", Physical Review Research 4 3, 033012 (2022).

[53] Michael L. Wall, Paraj Titum, Gregory Quiroz, Michael Foss-Feig, and Kaden R. A. Hazzard, "Tensor-network discriminator architecture for classification of quantum data on quantum computers", Physical Review A 105 6, 062439 (2022).

[54] Maxine T. Khumalo, Hazel A. Chieza, Krupa Prag, and Matthew Woolway, "An investigation of IBM quantum computing device performance on combinatorial optimisation problems", Neural Computing and Applications (2022).

[55] Dmitry A. Fedorov, Bo Peng, Niranjan Govind, and Yuri Alexeev, "VQE method: a short survey and recent developments", Materials Theory 6 1, 2 (2022).

[56] David A. Herrera-Martí, "Policy Gradient Approach to Compilation of Variational Quantum Circuits", Quantum 6, 797 (2022).

[57] Sukin Sim, Jonathan Romero, Jérôme F Gonthier, and Alexander A Kunitsa, "Adaptive pruning-based optimization of parameterized quantum circuits", Quantum Science and Technology 6 2, 025019 (2021).

[58] Chen Zhao and Xiao-Shan Gao, "QDNN: deep neural networks with quantum layers", Quantum Machine Intelligence 3 1, 15 (2021).

[59] Dylan G Stone and Carlo Bradac, "Machine and quantum learning for diamond-based quantum applications", Materials for Quantum Technology 3 1, 012001 (2023).

[60] Barnaby van Straaten and Bálint Koczor, "Measurement Cost of Metric-Aware Variational Quantum Algorithms", PRX Quantum 2 3, 030324 (2021).

[61] P. Chandarana, N. N. Hegade, K. Paul, F. Albarrán-Arriagada, E. Solano, A. del Campo, and Xi Chen, "Digitized-counterdiabatic quantum approximate optimization algorithm", Physical Review Research 4 1, 013141 (2022).

[62] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational quantum state eigensolver", npj Quantum Information 8 1, 113 (2022).

[63] Ryan LaRose and Brian Coyle, "Robust data encodings for quantum classifiers", Physical Review A 102 3, 032420 (2020).

[64] Peng-Fei Zhou, Rui Hong, and Shi-Ju Ran, "Automatically differentiable quantum circuit for many-qubit state preparation", Physical Review A 104 4, 042601 (2021).

[65] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo, "Diagnosing Barren Plateaus with Tools from Quantum Optimal Control", Quantum 6, 824 (2022).

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

[67] Xu-Dan Xie, Xingyu Guo, Hongxi Xing, Zheng-Yuan Xue, Dan-Bo Zhang, and Shi-Liang Zhu, "Variational thermal quantum simulation of the lattice Schwinger model", Physical Review D 106 5, 054509 (2022).

[68] Michelle Chalupnik, Hans Melo, Yuri Alexeev, and Alexey Galda, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 97 (2022) ISBN:978-1-6654-9113-6.

[69] Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, Yong-Jian Han, and Guo-Ping Guo, "Mitigating barren plateaus with transfer-learning-inspired parameter initializations", New Journal of Physics 25 1, 013039 (2023).

[70] 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).

[71] James Dborin, Fergus Barratt, Vinul Wimalaweera, Lewis Wright, and Andrew G Green, "Matrix product state pre-training for quantum machine learning", Quantum Science and Technology 7 3, 035014 (2022).

[72] 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).

[73] Philip Easom-Mccaldin, Ahmed Bouridane, Ammar Belatreche, and Richard Jiang, "On Depth, Robustness and Performance Using the Data Re-Uploading Single-Qubit Classifier", IEEE Access 9, 65127 (2021).

[74] Xiaozhen Ge, Re-Bing Wu, and Herschel Rabitz, "The optimization landscape of hybrid quantum–classical algorithms: From quantum control to NISQ applications", Annual Reviews in Control 54, 314 (2022).

[75] Ran-Yi-Liu Chen, Ben-Chi Zhao, Zhi-Xin Song, Xuan-Qiang Zhao, Kun Wang, and Xin Wang, "Hybrid quantum-classical algorithms: Foundation, design and applications", Acta Physica Sinica 70 21, 210302 (2021).

[76] 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).

[77] Ernesto Campos, Aly Nasrallah, and Jacob Biamonte, "Abrupt transitions in variational quantum circuit training", Physical Review A 103 3, 032607 (2021).

[78] Zidu Liu, Li-Wei Yu, L.-M. Duan, and Dong-Ling Deng, "Presence and Absence of Barren Plateaus in Tensor-Network Based Machine Learning", Physical Review Letters 129 27, 270501 (2022).

[79] Andrea Skolik, Jarrod R. McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib, "Layerwise learning for quantum neural networks", Quantum Machine Intelligence 3 1, 5 (2021).

[80] Martín Larocca, Frédéric Sauvage, Faris M. Sbahi, Guillaume Verdon, Patrick J. Coles, and M. Cerezo, "Group-Invariant Quantum Machine Learning", PRX Quantum 3 3, 030341 (2022).

[81] Fabio Valerio Massoli, Lucia Vadicamo, Giuseppe Amato, and Fabrizio Falchi, "A Leap among Quantum Computing and Quantum Neural Networks: A Survey", ACM Computing Surveys 55 5, 1 (2023).

[82] Gian-Luca R Anselmetti, David Wierichs, Christian Gogolin, and Robert M Parrish, "Local, expressive, quantum-number-preserving VQE ansätze for fermionic systems", New Journal of Physics 23 11, 113010 (2021).

[83] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[84] B Jaderberg, L W Anderson, W Xie, S Albanie, M Kiffner, and D Jaksch, "Quantum self-supervised learning", Quantum Science and Technology 7 3, 035005 (2022).

[85] Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Neural predictor based quantum architecture search", Machine Learning: Science and Technology 2 4, 045027 (2021).

[86] M Cerezo and Patrick J Coles, "Higher order derivatives of quantum neural networks with barren plateaus", Quantum Science and Technology 6 3, 035006 (2021).

[87] Chen Zhao and Xiao-Shan Gao, "Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus", Quantum 5, 466 (2021).

[88] Sergio Altares-López, Angela Ribeiro, and Juan José García-Ripoll, "Automatic design of quantum feature maps", Quantum Science and Technology 6 4, 045015 (2021).

[89] Antonio A. Mele, Glen B. Mbeng, Giuseppe E. Santoro, Mario Collura, and Pietro Torta, "Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz", Physical Review A 106 6, L060401 (2022).

[90] 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).

[91] Patrick Huembeli and Alexandre Dauphin, "Characterizing the loss landscape of variational quantum circuits", Quantum Science and Technology 6 2, 025011 (2021).

[92] Xin Wang, Zhixin Song, and Youle Wang, "Variational Quantum Singular Value Decomposition", Quantum 5, 483 (2021).

[93] Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin, "Entanglement devised barren plateau mitigation", Physical Review Research 3 3, 033090 (2021).

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

[95] Mateusz Ostaszewski, Edward Grant, and Marcello Benedetti, "Structure optimization for parameterized quantum circuits", Quantum 5, 391 (2021).

[96] Kouhei Nakaji and Naoki Yamamoto, "Expressibility of the alternating layered ansatz for quantum computation", Quantum 5, 434 (2021).

[97] Brian Coyle, Mina Doosti, Elham Kashefi, and Niraj Kumar, "Progress toward practical quantum cryptanalysis by variational quantum cloning", Physical Review A 105 4, 042604 (2022).

[98] 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).

[99] Kouhei Nakaji, Shumpei Uno, Yohichi Suzuki, Rudy Raymond, Tamiya Onodera, Tomoki Tanaka, Hiroyuki Tezuka, Naoki Mitsuda, and Naoki Yamamoto, "Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicators", Physical Review Research 4 2, 023136 (2022).

[100] 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).

[101] Bobak Toussi Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, and Seth Lloyd, "Learning quantum data with the quantum earth mover’s distance", Quantum Science and Technology 7 4, 045002 (2022).

[102] 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).

[103] Jacques Carolan, Masoud Mohseni, Jonathan P. Olson, Mihika Prabhu, Changchen Chen, Darius Bunandar, Murphy Yuezhen Niu, Nicholas C. Harris, Franco N. C. Wong, Michael Hochberg, Seth Lloyd, and Dirk Englund, "Variational quantum unsampling on a quantum photonic processor", Nature Physics 16 3, 322 (2020).

[104] Wenjie Liu, Jiaojiao Zhao, and Qingshan Wu, "A hybrid quantum-classical generative adversarial networks algorithm based on inherited layerwise learning with circle-connectivity circuit", Quantum Information Processing 21 11, 372 (2022).

[105] Jiahao Yao, Lin Lin, and Marin Bukov, "Reinforcement Learning for Many-Body Ground-State Preparation Inspired by Counterdiabatic Driving", Physical Review X 11 3, 031070 (2021).

[106] Michael L. Wall and Giuseppe D'Aguanno, "Tree-tensor-network classifiers for machine learning: From quantum inspired to quantum assisted", Physical Review A 104 4, 042408 (2021).

[107] Stefano Barison, Filippo Vicentini, Ignacio Cirac, and Giuseppe Carleo, "Variational dynamics as a ground-state problem on a quantum computer", Physical Review Research 4 4, 043161 (2022).

[108] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles, "Effect of barren plateaus on gradient-free optimization", Quantum 5, 558 (2021).

[109] Zhimin He, Chuangtao Chen, Lvzhou Li, Shenggen Zheng, and Haozhen Situ, "Quantum Architecture Search with Meta‐Learning", Advanced Quantum Technologies 5 8, 2100134 (2022).

[110] 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).

[111] Brian Coyle, Maxwell Henderson, Justin Chan Jin Le, Niraj Kumar, Marco Paini, and Elham Kashefi, "Quantum versus classical generative modelling in finance", Quantum Science and Technology 6 2, 024013 (2021).

[112] Jonathan Wei Zhong Lau, Kian Hwee Lim, Harshank Shrotriya, and Leong Chuan Kwek, "NISQ computing: where are we and where do we go?", AAPPS Bulletin 32 1, 27 (2022).

[113] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).

[114] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, and Dacheng Tao, "Learnability of Quantum Neural Networks", PRX Quantum 2 4, 040337 (2021).

[115] Alba Cervera-Lierta, Jakob S. Kottmann, and Alán Aspuru-Guzik, "Meta-Variational Quantum Eigensolver: Learning Energy Profiles of Parameterized Hamiltonians for Quantum Simulation", PRX Quantum 2 2, 020329 (2021).

[116] Hongxiang Chen, Max Nusspickel, Jules Tilly, and George H. Booth, "Variational quantum eigensolver for dynamic correlation functions", Physical Review A 104 3, 032405 (2021).

[117] Lucas Friedrich and Jonas Maziero, "Evolution strategies: application in hybrid quantum-classical neural networks", Quantum Information Processing 22 3, 132 (2023).

[118] Toshiaki Koike-Akino, Pu Wang, and Ye Wang, ICC 2022 - IEEE International Conference on Communications 654 (2022) ISBN:978-1-5386-8347-7.

[119] Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, and Dacheng Tao, "Quantum circuit architecture search for variational quantum algorithms", npj Quantum Information 8 1, 62 (2022).

[120] S. Mangini, F. Tacchino, D. Gerace, D. Bajoni, and C. Macchiavello, "Quantum computing models for artificial neural networks", Europhysics Letters 134 1, 10002 (2021).

[121] Yudai Suzuki, Hiroshi Yano, Rudy Raymond, and Naoki Yamamoto, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 1 (2021) ISBN:978-1-6654-1691-7.

[122] Tyson Jones and Simon C. Benjamin, "Robust quantum compilation and circuit optimisation via energy minimisation", Quantum 6, 628 (2022).

[123] Mohannad M. Ibrahim, Hamed Mohammadbagherpoor, Cynthia Rios, Nicholas T. Bronn, and Gregory T. Byrd, "Evaluation of Parameterized Quantum Circuits With Cross-Resonance Pulse-Driven Entanglers", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[124] Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, and Jarrod R. McClean, "Power of data in quantum machine learning", Nature Communications 12 1, 2631 (2021).

[125] Jules Tilly, Glenn Jones, Hongxiang Chen, Leonard Wossnig, and Edward Grant, "Computation of molecular excited states on IBM quantum computers using a discriminative variational quantum eigensolver", Physical Review A 102 6, 062425 (2020).

[126] Zidu Liu, Pei-Xin Shen, Weikang Li, L-M Duan, and Dong-Ling Deng, "Quantum capsule networks", Quantum Science and Technology 8 1, 015016 (2023).

[127] Reza Haghshenas, Johnnie Gray, Andrew C. Potter, and Garnet Kin-Lic Chan, "Variational Power of Quantum Circuit Tensor Networks", Physical Review X 12 1, 011047 (2022).

[128] Ryan LaRose, Arkin Tikku, Étude O’Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational quantum state diagonalization", npj Quantum Information 5 1, 57 (2019).

[129] Alona Sakhnenko, Corey O’Meara, Kumar J. B. Ghosh, Christian B. Mendl, Giorgio Cortiana, and Juan Bernabé-Moreno, "Hybrid classical-quantum autoencoder for anomaly detection", Quantum Machine Intelligence 4 2, 27 (2022).

[130] Shiro Tamiya and Hayata Yamasaki, "Stochastic gradient line Bayesian optimization for efficient noise-robust optimization of parameterized quantum circuits", npj Quantum Information 8 1, 90 (2022).

[131] George S. Barron, Bryan T. Gard, Orien J. Altman, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Preserving Symmetries for Variational Quantum Eigensolvers in the Presence of Noise", Physical Review Applied 16 3, 034003 (2021).

[132] Daniel Herr, Benjamin Obert, and Matthias Rosenkranz, "Anomaly detection with variational quantum generative adversarial networks", Quantum Science and Technology 6 4, 045004 (2021).

[133] Chufan Lyu, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Symmetry enhanced variational quantum spin eigensolver", Quantum 7, 899 (2023).

[134] Zhide Lu, Pei-Xin Shen, and Dong-Ling Deng, "Markovian Quantum Neuroevolution for Machine Learning", Physical Review Applied 16 4, 044039 (2021).

[135] Ankit Kulshrestha and Ilya Safro, 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 197 (2022) ISBN:978-1-6654-9113-6.

[136] Yun-Fei Niu, Shuo Zhang, and Wan-Su Bao, "Warm Starting Variational Quantum Algorithms with Near Clifford Circuits", Electronics 12 2, 347 (2023).

[137] Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, and Guo-Ping Guo, "Variational Quantum Algorithms for the Steady States of Open Quantum Systems ", Chinese Physics Letters 38 8, 080301 (2021).

[138] Arthur Pesah, M. Cerezo, Samson Wang, Tyler Volkoff, Andrew T. Sornborger, and Patrick J. Coles, "Absence of Barren Plateaus in Quantum Convolutional Neural Networks", Physical Review X 11 4, 041011 (2021).

[139] Mohammad Pirhooshyaran and Tamás Terlaky, "Quantum circuit design search", Quantum Machine Intelligence 3 2, 25 (2021).

[140] Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, and Henry Yuen, "Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz", PRX Quantum 1 2, 020319 (2020).

[141] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn, "Avoiding Barren Plateaus Using Classical Shadows", PRX Quantum 3 2, 020365 (2022).

[142] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", Reviews of Modern Physics 92 1, 015003 (2020).

[143] Arthur G. Rattew, Shaohan Hu, Marco Pistoia, Richard Chen, and Steve Wood, "A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver", arXiv:1910.09694, (2019).

[144] Guillaume Verdon, Michael Broughton, Jarrod R. McClean, Kevin J. Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, and Masoud Mohseni, "Learning to learn with quantum neural networks via classical neural networks", arXiv:1907.05415, (2019).

[145] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, (2018).

[146] Xia Liu, Geng Liu, Jiaxin Huang, Hao-Kai Zhang, and Xin Wang, "Mitigating barren plateaus of variational quantum eigensolvers", arXiv:2205.13539, (2022).

[147] Sukin Sim, Peter D. Johnson, and Alan Aspuru-Guzik, "Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms", arXiv:1905.10876, (2019).

[148] Noah L. Wach, Manuel S. Rudolph, Fred Jendrzejewski, and Sebastian Schmitt, "Data re-uploading with a single qudit", arXiv:2302.13932, (2023).

[149] Max Wilson, Sam Stromswold, Filip Wudarski, Stuart Hadfield, Norm M. Tubman, and Eleanor Rieffel, "Optimizing quantum heuristics with meta-learning", arXiv:1908.03185, (2019).

[150] Johannes Bausch, "Recurrent Quantum Neural Networks", arXiv:2006.14619, (2020).

[151] Hao-Kai Zhang, Chengkai Zhu, Geng Liu, and Xin Wang, "Fundamental limitations on optimization in variational quantum algorithms", arXiv:2205.05056, (2022).

[152] Cenk Tüysüz, Kristiane Novotny, Carla Rieger, Federico Carminati, Bilge Demirköz, Daniel Dobos, Fabio Fracas, Karolos Potamianos, Sofia Vallecorsa, and Jean-Roch Vlimant, "Performance of Particle Tracking Using a Quantum Graph Neural Network", arXiv:2012.01379, (2020).

[153] Ryan LaRose, Arkin Tikku, Étude O'Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum State Diagonalization", arXiv:1810.10506, (2018).

[154] Marco Ballarin, Stefano Mangini, Simone Montangero, Chiara Macchiavello, and Riccardo Mengoni, "Entanglement entropy production in Quantum Neural Networks", arXiv:2206.02474, (2022).

[155] Alexey Uvarov, "Variational quantum algorithms for local Hamiltonian problems", arXiv:2208.11220, (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2023-03-22 00:28:58) and SAO/NASA ADS (last updated successfully 2023-03-22 00:28:59). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.