Hamiltonian variational ansatz without barren plateaus

Chae-Yeun Park and Nathan Killoran

Xanadu, Toronto, ON, M5G 2C8, Canada

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


Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed $\textit{barren plateaus}$, has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.

Variational quantum algorithms (VQAs) solve a target problem by optimizing the parameters of a quantum circuit. While VQAs are one of the most promising applications of a near-term quantum computer, the practical usefulness of VQAs is often questioned. One of the central issues is that quantum circuits with random parameters often have exponentially small gradients, limiting the trainability of the circuits. This problem, dubbed barren plateaus, has gained lots of interest recently, but a general solution is still unavailable. This work proposes a solution to the barren plateaus problem for the Hamiltonian variational ansatz, a type of quantum circuit ansatz widely studied for solving quantum many-body problems.

► BibTeX data

► References

[1] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019).

[2] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, et al. ``Quantum computational advantage using photons''. Science 370, 1460–1463 (2020).

[3] Lars S Madsen, Fabian Laudenbach, Mohsen Falamarzi Askarani, Fabien Rortais, Trevor Vincent, Jacob FF Bulmer, Filippo M Miatto, Leonhard Neuhaus, Lukas G Helt, Matthew J Collins, et al. ``Quantum computational advantage with a programmable photonic processor''. Nature 606, 75–81 (2022).

[4] John Preskill. ``Quantum computing in the NISQ era and beyond''. Quantum 2, 79 (2018).

[5] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. ``A quantum approximate optimization algorithm'' (2014). arXiv:1411.4028.

[6] 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''. Nat. Comm. 5, 1–7 (2014).

[7] Dave Wecker, Matthew B Hastings, and Matthias Troyer. ``Progress towards practical quantum variational algorithms''. Phys. Rev. A 92, 042303 (2015).

[8] 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, 242–246 (2017).

[9] Stuart Hadfield, Zhihui Wang, Bryan O’Gorman, Eleanor G Rieffel, Davide Venturelli, and Rupak Biswas. ``From the quantum approximate optimization algorithm to a quantum alternating operator ansatz''. Algorithms 12, 34 (2019).

[10] Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione. ``An introduction to quantum machine learning''. Contemporary Physics 56, 172–185 (2015).

[11] Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. ``Quantum machine learning''. Nature 549, 195–202 (2017).

[12] Maria Schuld and Nathan Killoran. ``Quantum machine learning in feature Hilbert spaces''. Phys. Rev. Lett. 122, 040504 (2019).

[13] Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme. ``A rigorous and robust quantum speed-up in supervised machine learning''. Nat. Phys. 17, 1013–1017 (2021).

[14] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. ``Variational quantum algorithms''. Nat. Rev. Phys. 3, 625–644 (2021).

[15] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes''. Nat. Comm. 9, 1–6 (2018).

[16] Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles. ``Cost function dependent barren plateaus in shallow parametrized quantum circuits''. Nat. Comm. 12, 1–12 (2021).

[17] Zoë Holmes, Kunal Sharma, Marco Cerezo, and Patrick J Coles. ``Connecting ansatz expressibility to gradient magnitudes and barren plateaus''. PRX Quantum 3, 010313 (2022).

[18] Sepp Hochreiter and Jürgen Schmidhuber. ``Long short-term memory''. Neural computation 9, 1735–1780 (1997).

[19] Xavier Glorot, Antoine Bordes, and Yoshua Bengio. ``Deep sparse rectifier neural networks''. In Proceedings of the fourteenth international conference on artificial intelligence and statistics. Pages 315–323. JMLR Workshop and Conference Proceedings (2011). url: https:/​/​proceedings.mlr.press/​v15/​glorot11a.html.

[20] Xavier Glorot and Yoshua Bengio. ``Understanding the difficulty of training deep feedforward neural networks''. In Proceedings of the thirteenth international conference on artificial intelligence and statistics. Pages 249–256. JMLR Workshop and Conference Proceedings (2010). url: https:/​/​proceedings.mlr.press/​v9/​glorot10a.html.

[21] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. ``Delving deep into rectifiers: Surpassing human-level performance on imagenet classification''. In Proceedings of the IEEE international conference on computer vision. Pages 1026–1034. (2015).

[22] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, and Dacheng Tao. ``Toward trainability of quantum neural networks'' (2020). arXiv:2011.06258.

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

[24] Arthur Pesah, Marco Cerezo, Samson Wang, Tyler Volkoff, Andrew T Sornborger, and Patrick J Coles. ``Absence of barren plateaus in quantum convolutional neural networks''. Phys. Rev. X 11, 041011 (2021).

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

[26] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. ``An initialization strategy for addressing barren plateaus in parametrized quantum circuits''. Quantum 3, 214 (2019).

[27] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar. ``Graph neural network initialisation of quantum approximate optimisation''. Quantum 6, 861 (2022).

[28] Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, and Dacheng Tao. ``Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits''. In Advances in Neural Information Processing Systems. Volume 35, pages 18612–18627. (2022). url: https:/​/​doi.org/​10.48550/​arXiv.2203.09376.

[29] 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''. Phys. Rev. A 106, L060401 (2022).

[30] Manuel S Rudolph, Jacob Miller, Danial Motlagh, Jing Chen, Atithi Acharya, and Alejandro Perdomo-Ortiz. ``Synergistic pretraining of parametrized quantum circuits via tensor networks''. Nature Communications 14, 8367 (2023).

[31] 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, 020319 (2020).

[32] 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).

[33] Ying Li and Simon C Benjamin. ``Efficient variational quantum simulator incorporating active error minimization''. Phys. Rev. X 7, 021050 (2017).

[34] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin. ``Theory of variational quantum simulation''. Quantum 3, 191 (2019).

[35] Cristina Cirstoiu, Zoe 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, 1–10 (2020).

[36] Sheng-Hsuan Lin, Rohit Dilip, Andrew G Green, Adam Smith, and Frank Pollmann. ``Real-and imaginary-time evolution with compressed quantum circuits''. PRX Quantum 2, 010342 (2021).

[37] Conor Mc Keever and Michael Lubasch. ``Classically optimized hamiltonian simulation''. Phys. Rev. Res. 5, 023146 (2023).

[38] Josh M Deutsch. ``Quantum statistical mechanics in a closed system''. Phys. Rev. A 43, 2046 (1991).

[39] Mark Srednicki. ``Chaos and quantum thermalization''. Phys. Rev. E 50, 888 (1994).

[40] Marcos Rigol, Vanja Dunjko, and Maxim Olshanii. ``Thermalization and its mechanism for generic isolated quantum systems''. Nature 452, 854–858 (2008).

[41] Peter Reimann. ``Foundation of statistical mechanics under experimentally realistic conditions''. Phys. Rev. Lett. 101, 190403 (2008).

[42] Noah Linden, Sandu Popescu, Anthony J Short, and Andreas Winter. ``Quantum mechanical evolution towards thermal equilibrium''. Phys. Rev. E 79, 061103 (2009).

[43] Anthony J Short. ``Equilibration of quantum systems and subsystems''. New Journal of Physics 13, 053009 (2011).

[44] Christian Gogolin and Jens Eisert. ``Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems''. Reports on Progress in Physics 79, 056001 (2016).

[45] Yichen Huang, Fernando GSL Brandão, Yong-Liang Zhang, et al. ``Finite-size scaling of out-of-time-ordered correlators at late times''. Phys. Rev. Lett. 123, 010601 (2019).

[46] Daniel A Roberts and Beni Yoshida. ``Chaos and complexity by design''. Journal of High Energy Physics 2017, 1–64 (2017).

[47] Hyungwon Kim, Tatsuhiko N Ikeda, and David A Huse. ``Testing whether all eigenstates obey the eigenstate thermalization hypothesis''. Phys. Rev. E 90, 052105 (2014).

[48] Tomotaka Kuwahara, Takashi Mori, and Keiji Saito. ``Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems''. Annals of Physics 367, 96–124 (2016).

[49] David Wierichs, Christian Gogolin, and Michael Kastoryano. ``Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer''. Phys. Rev. Research 2, 043246 (2020).

[50] Chae-Yeun Park. ``Efficient ground state preparation in variational quantum eigensolver with symmetry breaking layers'' (2021). arXiv:2106.02509.

[51] Jan Lukas Bosse and Ashley Montanaro. ``Probing ground-state properties of the kagome antiferromagnetic heisenberg model using the variational quantum eigensolver''. Phys. Rev. B 105, 094409 (2022).

[52] Joris Kattemölle and Jasper van Wezel. ``Variational quantum eigensolver for the heisenberg antiferromagnet on the kagome lattice''. Phys. Rev. B 106, 214429 (2022).

[53] Diederik P. Kingma and Jimmy Ba. ``Adam: A method for stochastic optimization''. In 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings. (2015). url: https:/​/​doi.org/​10.48550/​arXiv.1412.6980.

[54] Tyson Jones and Julien Gacon. ``Efficient calculation of gradients in classical simulations of variational quantum algorithms'' (2020). arXiv:2009.02823.

[55] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M. Sohaib Alam, Guillermo Alonso-Linaje, et al. ``Pennylane: Automatic differentiation of hybrid quantum-classical computations'' (2018). arXiv:1811.04968.

[56] Lodewyk FA Wessels and Etienne Barnard. ``Avoiding false local minima by proper initialization of connections''. IEEE Transactions on Neural Networks 3, 899–905 (1992).

[57] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. ``Quantum circuit learning''. Phys. Rev. A 98, 032309 (2018).

[58] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. ``Evaluating analytic gradients on quantum hardware''. Phys. Rev. A 99, 032331 (2019).

[59] Masuo Suzuki. ``General theory of fractal path integrals with applications to many-body theories and statistical physics''. Journal of Mathematical Physics 32, 400–407 (1991).

[60] Michael A. Nielsen. ``A geometric approach to quantum circuit lower bounds'' (2005). arXiv:quant-ph/​0502070.

[61] Michael A Nielsen, Mark R Dowling, Mile Gu, and Andrew C Doherty. ``Quantum computation as geometry''. Science 311, 1133–1135 (2006).

[62] Douglas Stanford and Leonard Susskind. ``Complexity and shock wave geometries''. Phys. Rev. D 90, 126007 (2014).

[63] Jonas Haferkamp, Philippe Faist, Naga BT Kothakonda, Jens Eisert, and Nicole Yunger Halpern. ``Linear growth of quantum circuit complexity''. Nat. Phys. 18, 528–532 (2022).

[64] Adam R Brown, Leonard Susskind, and Ying Zhao. ``Quantum complexity and negative curvature''. Phys. Rev. D 95, 045010 (2017).

[65] Adam R Brown and Leonard Susskind. ``Second law of quantum complexity''. Phys. Rev. D 97, 086015 (2018).

[66] Yu Chen. ``Universal logarithmic scrambling in many body localization'' (2016). arXiv:1608.02765.

[67] Ruihua Fan, Pengfei Zhang, Huitao Shen, and Hui Zhai. ``Out-of-time-order correlation for many-body localization''. Science Bulletin 62, 707–711 (2017).

[68] Juhee Lee, Dongkyu Kim, and Dong-Hee Kim. ``Typical growth behavior of the out-of-time-ordered commutator in many-body localized systems''. Phys. Rev. B 99, 184202 (2019).

[69] Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles. ``Noise-induced barren plateaus in variational quantum algorithms''. Nat. Comm. 12, 6961 (2021).

[70] ``PennyLane–Lightning plugin https:/​/​github.com/​PennyLaneAI/​pennylane-lightning'' (2023).

[71] ``PennyLane–Lightning-GPU plugin https:/​/​github.com/​PennyLaneAI/​pennylane-lightning-gpu'' (2023).

[72] ``GitHub repository https:/​/​github.com/​XanaduAI/​hva-without-barren-plateaus'' (2023).

[73] Wilhelm Magnus. ``On the exponential solution of differential equations for a linear operator''. Commun. Pure. Appl. Math. 7, 649–673 (1954).

[74] Dmitry Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers. ``A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems''. Commun. Math. Phys. 354, 809–827 (2017).

Cited by

[1] Yaswitha Gujju, Atsushi Matsuo, and Rudy Raymond, "Quantum machine learning on near-term quantum devices: Current state of supervised and unsupervised techniques for real-world applications", Physical Review Applied 21 6, 067001 (2024).

[2] Ricard Puig-i-Valls, Marc Drudis, Supanut Thanasilp, and Zoë Holmes, "Variational quantum simulation: a case study for understanding warm starts", arXiv:2404.10044, (2024).

[3] M. Cerezo, Martin Larocca, Diego García-Martín, N. L. Diaz, Paolo Braccia, Enrico Fontana, Manuel S. Rudolph, Pablo Bermejo, Aroosa Ijaz, Supanut Thanasilp, Eric R. Anschuetz, and Zoë Holmes, "Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing", arXiv:2312.09121, (2023).

[4] Richard D. P. East, Guillermo Alonso-Linaje, and Chae-Yeun Park, "All you need is spin: SU(2) equivariant variational quantum circuits based on spin networks", arXiv:2309.07250, (2023).

[5] Han Qi, Lei Wang, Hongsheng Zhu, Abdullah Gani, and Changqing Gong, "The barren plateaus of quantum neural networks: review, taxonomy and trends", Quantum Information Processing 22 12, 435 (2023).

[6] Antonio Anna Mele, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck França, and Yihui Quek, "Noise-induced shallow circuits and absence of barren plateaus", arXiv:2403.13927, (2024).

[7] Chukwudubem Umeano, Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "What can we learn from quantum convolutional neural networks?", arXiv:2308.16664, (2023).

[8] Lukas Broers and Ludwig Mathey, "Mitigated barren plateaus in the time-nonlocal optimization of analog quantum-algorithm protocols", Physical Review Research 6 1, 013076 (2024).

[9] Jiaqi Miao, Chang-Yu Hsieh, and Shi-Xin Zhang, "Neural-network-encoded variational quantum algorithms", Physical Review Applied 21 1, 014053 (2024).

[10] Julien Gacon, Jannes Nys, Riccardo Rossi, Stefan Woerner, and Giuseppe Carleo, "Variational quantum time evolution without the quantum geometric tensor", Physical Review Research 6 1, 013143 (2024).

[11] Chandan Sarma, Olivia Di Matteo, Abhishek Abhishek, and Praveen C. Srivastava, "Prediction of the neutron drip line in oxygen isotopes using quantum computation", Physical Review C 108 6, 064305 (2023).

[12] Jesús Cobos, David F. Locher, Alejandro Bermudez, Markus Müller, and Enrique Rico, "Noise-aware variational eigensolvers: a dissipative route for lattice gauge theories", arXiv:2308.03618, (2023).

[13] Yanqi Song, Yusen Wu, Sujuan Qin, Qiaoyan Wen, Jingbo B. Wang, and Fei Gao, "Trainability Analysis of Quantum Optimization Algorithms from a Bayesian Lens", arXiv:2310.06270, (2023).

[14] Julien Gacon, "Scalable Quantum Algorithms for Noisy Quantum Computers", arXiv:2403.00940, (2024).

[15] Zheng Qin, Xiufan Li, Yang Zhou, Shikun Zhang, Rui Li, Chunxiao Du, and Zhisong Xiao, "Applicability of Measurement-based Quantum Computation towards Physically-driven Variational Quantum Eigensolver", arXiv:2307.10324, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 05:49:56) and SAO/NASA ADS (last updated successfully 2024-06-22 05:49:57). The list may be incomplete as not all publishers provide suitable and complete citation data.