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

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

► 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] 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).

[2] 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).

[3] 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).

[4] 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).

[5] 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).

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

[7] Max Wilson, Rachel Stromswold, Filip Wudarski, Stuart Hadfield, Norm M. Tubman, and Eleanor G. Rieffel, "Optimizing quantum heuristics with meta-learning", arXiv:1908.03185, Quantum Machine Intelligence 3 1, 13 (2021).

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

[9] 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).

[10] 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).

[11] 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).

[12] 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).

[13] 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).

[14] 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).

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

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

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

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

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

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

[21] 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).

[22] 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).

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

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

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

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

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

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

[29] 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).

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

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

[32] 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).

[33] 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).

[34] 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).

[35] 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).

[36] 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).

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

[38] 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 2100114 (2021).

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

[40] 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).

[41] 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).

[42] 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).

[43] 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).

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

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

[46] 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).

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

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

[49] 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.

[50] 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).

[51] 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).

[52] 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).

[53] 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).

[54] Ryan LaRose, Arkin Tikku, Étude O’Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational quantum state diagonalization", arXiv:1810.10506, npj Quantum Information 5 1, 57 (2019).

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

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

[57] 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).

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

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

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

[61] 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).

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

[63] 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).

[64] 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).

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

[66] 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).

[67] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, Reviews of Modern Physics 92 1, 015003 (2020).

[68] 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.

[69] 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.

[70] Johannes Bausch, "Recurrent Quantum Neural Networks", arXiv:2006.14619.

[71] 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.

[72] 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.

The above citations are from Crossref's cited-by service (last updated successfully 2021-12-07 19:54:17) and SAO/NASA ADS (last updated successfully 2021-12-07 19:54:18). The list may be incomplete as not all publishers provide suitable and complete citation data.