Quantum Natural Gradient

James Stokes; Josh Izaac; Nathan Killoran; Giuseppe Carleo

doi:10.22331/q-2020-05-25-269

Quantum Natural Gradient

James Stokes¹, Josh Izaac², Nathan Killoran², and Giuseppe Carleo³

¹Center for Computational Quantum Physics and Center for Computational Mathematics, Flatiron Institute, New York, NY 10010 USA
²Xanadu, 777 Bay Street, Toronto, Canada
³Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USA

Published:	2020-05-25, volume 4, page 269
Eprint:	arXiv:1909.02108v3
Doi:	https://doi.org/10.22331/q-2020-05-25-269
Citation:	Quantum 4, 269 (2020).

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Abstract

A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.

► BibTeX data

@article{Stokes2020quantumnatural,
  doi = {10.22331/q-2020-05-25-269},
  url = {https://doi.org/10.22331/q-2020-05-25-269},
  title = {Quantum {N}atural {G}radient},
  author = {Stokes, James and Izaac, Josh and Killoran, Nathan and Carleo, Giuseppe},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {269},
  month = may,
  year = {2020}
}

► References

[1] Shun-Ichi Amari. Natural gradient works efficiently in learning. Neural Computation, 10 (2): 251–276, 1998. 10.1162/089976698300017746.
https://doi.org/10.1162/089976698300017746

[2] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, M. Sohaib Alam, Shahnawaz Ahmed, Juan Miguel Arrazola, Carsten Blank, Alain Delgado, Soran Jahangiri, Keri McKiernan, Johannes Jakob Meyer, Zeyue Niu, Antal Szàva, and Nathan Killoran. Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968, 2018.
arXiv:1811.04968

[3] Marin Bukov, Dries Sels, and Anatoli Polkovnikov. Geometric speed limit of accessible many-body state preparation. Physical Review X, 9 (1): 011034, 2019. 10.1103/PhysRevX.9.011034.
https://doi.org/10.1103/PhysRevX.9.011034

[4] Giuseppe Carleo, Federico Becca, Marco Schiró, and Michele Fabrizio. Localization and glassy dynamics of many-body quantum systems. Scientific reports, 2: 243, 2012. 10.1038/srep00243.
https://doi.org/10.1038/srep00243

[5] Giuseppe Carleo, Federico Becca, Laurent Sanchez-Palencia, Sandro Sorella, and Michele Fabrizio. Light-cone effect and supersonic correlations in one-and two-dimensional bosonic superfluids. Physical Review A, 89 (3): 031602, 2014. 10.1103/PhysRevA.89.031602.
https://doi.org/10.1103/PhysRevA.89.031602

[6] Ming-Cheng Chen, Ming Gong, Xiao-Si Xu, Xiao Yuan, Jian-Wen Wang, Can Wang, Chong Ying, Jin Lin, Yu Xu, Yulin Wu, et al. Demonstration of adiabatic variational quantum computing with a superconducting quantum coprocessor. arXiv preprint arXiv:1905.03150, 2019.
arXiv:1905.03150

[7] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. Efficient quantum measurement of pauli operators. arXiv preprint arXiv:1908.06942, 2019.
arXiv:1908.06942

[8] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao. The expressive power of parameterized quantum circuits. arXiv preprint arXiv:1810.11922, 2018.
arXiv:1810.11922

[9] Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors. arXiv preprint arXiv:1802.06002, 2018.
arXiv:1802.06002

[10] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014.
arXiv:1411.4028

[11] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong. Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families. arXiv preprint arXiv:1907.13623, 2019.
arXiv:1907.13623

[12] Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. arXiv preprint arXiv:1701.01450, 2017.
arXiv:1701.01450

[13] Aram Harrow and John Napp. Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms. arXiv preprint arXiv:1901.05374, 2019.
arXiv:1901.05374

[14] William James Huggins, Piyush Patil, Bradley Mitchell, K Birgitta Whaley, and Miles Stoudenmire. Towards quantum machine learning with tensor networks. Quantum Science and Technology, 4: 024001, 2018. 10.1088/2058-9565/aaea94.
https://doi.org/10.1088/2058-9565/aaea94

[15] Stanislaw Jastrzebski, Zachary Kenton, Devansh Arpit, Nicolas Ballas, Asja Fischer, Yoshua Bengio, and Amos Storkey. Three factors influencing minima in sgd. arXiv preprint arXiv:1711.04623, 2017.
arXiv:1711.04623

[16] Tyson Jones and Simon C Benjamin. Quantum compilation and circuit optimisation via energy dissipation. arXiv preprint arXiv:1811.03147, 2018.
arXiv:1811.03147

[17] Tyson Jones, Suguru Endo, Sam McArdle, Xiao Yuan, and Simon C Benjamin. Variational quantum algorithms for discovering hamiltonian spectra. Physical Review A, 99 (6): 062304, 2019. 10.1103/PhysRevA.99.062304.
https://doi.org/10.1103/PhysRevA.99.062304

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

[19] Michael Kolodrubetz, Dries Sels, Pankaj Mehta, and Anatoli Polkovnikov. Geometry and non-adiabatic response in quantum and classical systems. Physics Reports, 697: 1–87, 2017. 10.1016/j.physrep.2017.07.001.
https://doi.org/10.1016/j.physrep.2017.07.001

[20] PH Kramer and Marcos Saraceno. Geometry of the time-dependent variational principle in quantum mechanics. Springer, 1981. 10.1007/3-540-10271-X_317.
https://doi.org/10.1007/3-540-10271-X_317

[21] Ying Li and Simon C Benjamin. Efficient variational quantum simulator incorporating active error minimization. Physical Review X, 7 (2): 021050, 2017. 10.1103/PhysRevX.7.021050.
https://doi.org/10.1103/PhysRevX.7.021050

[22] Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, and James Stokes. Fisher-rao metric, geometry, and complexity of neural networks. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 888–896, 2019. arXiv preprint arXiv:1711.01530.
arXiv:1711.01530

[23] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1): 1–6, 2019. 10.1038/s41534-019-0187-2.
https://doi.org/10.1038/s41534-019-0187-2

[24] 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. 10.1038/s41467-018-07090-4.
https://doi.org/10.1038/s41467-018-07090-4

[25] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning. Physical Review A, 98 (3): 032309, 2018. 10.1103/PhysRevA.98.032309.
https://doi.org/10.1103/PhysRevA.98.032309

[26] Behnam Neyshabur, Ruslan R Salakhutdinov, and Nati Srebro. Path-SGD: Path-normalized optimization in deep neural networks. In Advances in Neural Information Processing Systems, pages 2422–2430, 2015. arXiv preprint arXiv:1506.02617.
arXiv:1506.02617

[27] 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. 10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213

[28] Dénes Petz. Information-geometry of quantum states. In Quantum Probability Communications: Volume X, pages 135–157. World Scientific, 1998. 10.1142/9789812816054_0006.
https://doi.org/10.1142/9789812816054_0006

[29] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, 2018. 10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[30] Maria Schuld, Alex Bocharov, Krysta Svore, and Nathan Wiebe. Circuit-centric quantum classifiers. arXiv preprint arXiv:1804.00633, 2018. 10.1103/PhysRevA.101.032308.
https://doi.org/10.1103/PhysRevA.101.032308
arXiv:1804.00633

[31] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99 (3): 032331, 2019. 10.1103/PhysRevA.99.032331.
https://doi.org/10.1103/PhysRevA.99.032331

[32] Sandro Sorella, Michele Casula, and Dario Rocca. Weak binding between two aromatic rings: Feeling the van der waals attraction by quantum monte carlo methods. The Journal of Chemical Physics, 127 (1): 014105, 2007. 10.1063/1.2746035.
https://doi.org/10.1063/1.2746035

[33] James C Spall et al. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control, 37 (3): 332–341, 1992. 10.1109/9.119632.
https://doi.org/10.1109/9.119632

[34] F Wilczek and A Shapere. Geometric phases in physics. Geometric Phases In Physics. Series: Advanced Series in Mathematical Physics, ISBN: 978-9971-5-0621-6. WORLD SCIENTIFIC, Edited by F Wilczek and A Shapere, vol. 5, 5, 1989. 10.1142/0613.
https://doi.org/10.1142/0613

[35] Xanadu Quantum Technologies. PennyLane source code. https://github.com/XanaduAI/pennylane, 2019. [Online; accessed 3-Mar-2020].
https://github.com/XanaduAI/pennylane

[36] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, 2019. 10.22331/q-2019-10-07-191.
https://doi.org/10.22331/q-2019-10-07-191

Cited by

[1] Kazuhiro Seki and Seiji Yunoki, "Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer", Physical Review A 105 3, 032419 (2022).

[2] 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).

[3] 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).

[4] Y. S. Teo, "Optimized numerical gradient and Hessian estimation for variational quantum algorithms", Physical Review A 107 4, 042421 (2023).

[5] Ljubomir Budinski, "Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method", Quantum Information Processing 20 2, 57 (2021).

[6] Johannes Jakob Meyer, "Fisher Information in Noisy Intermediate-Scale Quantum Applications", Quantum 5, 539 (2021).

[7] Maria Schuld and Nathan Killoran, "Is Quantum Advantage the Right Goal for Quantum Machine Learning?", PRX Quantum 3 3, 030101 (2022).

[8] 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).

[9] J. Cortés-Vega, J. F. Barra, L. Pereira, and A. Delgado, "Detecting entanglement of unknown states by violating the Clauser–Horne–Shimony–Holt inequality", Quantum Information Processing 22 5, 203 (2023).

[10] Guang Yang, Su-Ya Chao, Min Nie, Yuan-Hua Liu, and Mei-Ling Zhang, "Construction method of hybrid quantum long-short term memory neural network for image classification", Acta Physica Sinica 72 5, 058901 (2023).

[11] Seunghyeok Oh, Jaeho Choi, Jong-Kook Kim, and Joongheon Kim, 2021 International Conference on Information Networking (ICOIN) 50 (2021) ISBN:978-1-7281-9101-0.

[12] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, "Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information", Quantum 5, 567 (2021).

[13] R. R. Ferguson, L. Dellantonio, A. Al Balushi, K. Jansen, W. Dür, and C. A. Muschik, "Measurement-Based Variational Quantum Eigensolver", Physical Review Letters 126 22, 220501 (2021).

[14] Gabriel Matos, Chris N. Self, Zlatko Papić, Konstantinos Meichanetzidis, and Henrik Dreyer, "Characterization of variational quantum algorithms using free fermions", Quantum 7, 966 (2023).

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

[16] André Sequeira, Luis Paulo Santos, and Luis Soares Barbosa, "Policy gradients using variational quantum circuits", Quantum Machine Intelligence 5 1, 18 (2023).

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

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

[19] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, "Quantum Algorithms for Quantum Chemistry and Quantum Materials Science", Chemical Reviews 120 22, 12685 (2020).

[20] Korbinian Kottmann, Friederike Metz, Joana Fraxanet, and Niccolò Baldelli, "Variational quantum anomaly detection: Unsupervised mapping of phase diagrams on a physical quantum computer", Physical Review Research 3 4, 043184 (2021).

[21] Jakob S Kottmann, Sumner Alperin-Lea, Teresa Tamayo-Mendoza, Alba Cervera-Lierta, Cyrille Lavigne, Tzu-Ching Yen, Vladyslav Verteletskyi, Philipp Schleich, Abhinav Anand, Matthias Degroote, Skylar Chaney, Maha Kesibi, Naomi Grace Curnow, Brandon Solo, Georgios Tsilimigkounakis, Claudia Zendejas-Morales, Artur F Izmaylov, and Alán Aspuru-Guzik, "TEQUILA: a platform for rapid development of quantum algorithms", Quantum Science and Technology 6 2, 024009 (2021).

[22] 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).

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

[24] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics & Astronomy 66 5, 250302 (2023).

[25] Jack Y. Araz, Sebastian Schenk, and Michael Spannowsky, "Toward a quantum simulation of nonlinear sigma models with a topological term", Physical Review A 107 3, 032619 (2023).

[26] 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).

[27] Kathleen E. Hamilton, Emily Lynn, and Raphael C. Pooser, " Mode connectivity in the loss landscape of parameterized quantum circuits", Quantum Machine Intelligence 4 1, 10 (2022).

[28] Kevin J Sung, Jiahao Yao, Matthew P Harrigan, Nicholas C Rubin, Zhang Jiang, Lin Lin, Ryan Babbush, and Jarrod R McClean, "Using models to improve optimizers for variational quantum algorithms", Quantum Science and Technology 5 4, 044008 (2020).

[29] Massimiliano Incudini, Fabio Tarocco, Riccardo Mengoni, Alessandra Di Pierro, and Antonio Mandarino, "Computing graph edit distance on quantum devices", Quantum Machine Intelligence 4 2, 24 (2022).

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

[31] Jack Y. Araz and Michael Spannowsky, "Classical versus quantum: Comparing tensor-network-based quantum circuits on Large Hadron Collider data", Physical Review A 106 6, 062423 (2022).

[32] Tobias Haug and M.S. Kim, "Scalable Measures of Magic Resource for Quantum Computers", PRX Quantum 4 1, 010301 (2023).

[33] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and opportunities in quantum machine learning", Nature Computational Science 2 9, 567 (2022).

[34] 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).

[35] 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).

[36] Mauro Rigo, Benjamin Hall, Morten Hjorth-Jensen, Alessandro Lovato, and Francesco Pederiva, "Solving the nuclear pairing model with neural network quantum states", Physical Review E 107 2, 025310 (2023).

[37] 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.

[38] 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).

[39] 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).

[40] Gregory Boyd and Bálint Koczor, "Training Variational Quantum Circuits with CoVaR: Covariance Root Finding with Classical Shadows", Physical Review X 12 4, 041022 (2022).

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

[42] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[43] Nikita A. Nemkov, Evgeniy O. Kiktenko, Ilia A. Luchnikov, and Aleksey K. Fedorov, "Efficient variational synthesis of quantum circuits with coherent multi-start optimization", Quantum 7, 993 (2023).

[44] Ilia Luchnikov, Alexander Ryzhov, Sergey Filippov, and Henni Ouerdane, "QGOpt: Riemannian optimization for quantum technologies", SciPost Physics 10 3, 079 (2021).

[45] David Wierichs, Christian Gogolin, and Michael Kastoryano, "Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer", Physical Review Research 2 4, 043246 (2020).

[46] Juneseo Lee, Alicia B. Magann, Herschel A. Rabitz, and Christian Arenz, "Progress toward favorable landscapes in quantum combinatorial optimization", Physical Review A 104 3, 032401 (2021).

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

[48] 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).

[49] V. Armaos, Dimitrios A. Badounas, Paraskevas Deligiannis, Konstantinos Lianos, and Yordan S. Yordanov, "Efficient Parabolic Optimisation Algorithm for Adaptive VQE Implementations", SN Computer Science 3 6, 443 (2022).

[50] Donghwa Lee, Jinil Lee, Seongjin Hong, Hyang-Tag Lim, Young-Wook Cho, Sang-Wook Han, Hyundong Shin, Junaid ur Rehman, and Yong-Su Kim, "Error-mitigated photonic variational quantum eigensolver using a single-photon ququart", Optica 9 1, 88 (2022).

[51] Andrew Blance and Michael Spannowsky, "Quantum machine learning for particle physics using a variational quantum classifier", Journal of High Energy Physics 2021 2, 212 (2021).

[52] Aram W. Harrow and John C. Napp, "Low-Depth Gradient Measurements Can Improve Convergence in Variational Hybrid Quantum-Classical Algorithms", Physical Review Letters 126 14, 140502 (2021).

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

[54] Matthew T. Scoggins and Armin Rahmani, "Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: Application to the XXZ model on the square lattice", Physical Review Research 3 4, 043165 (2021).

[55] 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).

[56] Galan Moody, Volker J Sorger, Daniel J Blumenthal, Paul W Juodawlkis, William Loh, Cheryl Sorace-Agaskar, Alex E Jones, Krishna C Balram, Jonathan C F Matthews, Anthony Laing, Marcelo Davanco, Lin Chang, John E Bowers, Niels Quack, Christophe Galland, Igor Aharonovich, Martin A Wolff, Carsten Schuck, Neil Sinclair, Marko Lončar, Tin Komljenovic, David Weld, Shayan Mookherjea, Sonia Buckley, Marina Radulaski, Stephan Reitzenstein, Benjamin Pingault, Bartholomeus Machielse, Debsuvra Mukhopadhyay, Alexey Akimov, Aleksei Zheltikov, Girish S Agarwal, Kartik Srinivasan, Juanjuan Lu, Hong X Tang, Wentao Jiang, Timothy P McKenna, Amir H Safavi-Naeini, Stephan Steinhauer, Ali W Elshaari, Val Zwiller, Paul S Davids, Nicholas Martinez, Michael Gehl, John Chiaverini, Karan K Mehta, Jacquiline Romero, Navin B Lingaraju, Andrew M Weiner, Daniel Peace, Robert Cernansky, Mirko Lobino, Eleni Diamanti, Luis Trigo Vidarte, and Ryan M Camacho, "2022 Roadmap on integrated quantum photonics", Journal of Physics: Photonics 4 1, 012501 (2022).

[57] Thomas Hubregtsen, Frederik Wilde, Shozab Qasim, and Jens Eisert, "Single-component gradient rules for variational quantum algorithms", Quantum Science and Technology 7 3, 035008 (2022).

[58] 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).

[59] James Stokes, Brian Chen, and Shravan Veerapaneni, "Numerical and geometrical aspects of flow-based variational quantum Monte Carlo", Machine Learning: Science and Technology 4 2, 021001 (2023).

[60] Weikang Li, Zhi-de Lu, and Dong-Ling Deng, "Quantum Neural Network Classifiers: A Tutorial", SciPost Physics Lecture Notes 61 (2022).

[61] Lennart Bittel and Martin Kliesch, "Training Variational Quantum Algorithms Is NP-Hard", Physical Review Letters 127 12, 120502 (2021).

[62] Yuan Yao, Pierre Cussenot, Richard A. Wolf, and Filippo Miatto, "Complex natural gradient optimization for optical quantum circuit design", Physical Review A 105 5, 052402 (2022).

[63] 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).

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

[65] Shi-Xin Zhang, Jonathan Allcock, Zhou-Quan Wan, Shuo Liu, Jiace Sun, Hao Yu, Xing-Han Yang, Jiezhong Qiu, Zhaofeng Ye, Yu-Qin Chen, Chee-Kong Lee, Yi-Cong Zheng, Shao-Kai Jian, Hong Yao, Chang-Yu Hsieh, and Shengyu Zhang, "TensorCircuit: a Quantum Software Framework for the NISQ Era", Quantum 7, 912 (2023).

[66] Chang Yu Hsieh, Qiming Sun, Shengyu Zhang, and Chee Kong Lee, "Unitary-coupled restricted Boltzmann machine ansatz for quantum simulations", npj Quantum Information 7 1, 19 (2021).

[67] 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).

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

[69] James Stokes, Javier Robledo Moreno, Eftychios A. Pnevmatikakis, and Giuseppe Carleo, "Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states", Physical Review B 102 20, 205122 (2020).

[70] Ibrahim Gad, Aboul Ella Hassanien, Ashraf Darwish, and Mincong Tang, Lecture Notes in Operations Research 693 (2022) ISBN:978-981-16-8655-9.

[71] Robert J. Webber and Michael Lindsey, "Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo", Physical Review Research 4 3, 033099 (2022).

[72] 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).

[73] Vu Tuan Hai and Le Bin Ho, "Universal compilation for quantum state tomography", Scientific Reports 13 1, 3750 (2023).

[74] Bo Peng and Karol Kowalski, "Variational quantum solver employing the PDS energy functional", Quantum 5, 473 (2021).

[75] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm, "Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe", EPJ Quantum Technology 9 1, 19 (2022).

[76] Junyu Liu, Zimu Li, Han Zheng, Xiao Yuan, and Jinzhao Sun, "Towards a variational Jordan–Lee–Preskill quantum algorithm", Machine Learning: Science and Technology 3 4, 045030 (2022).

[77] Stefano Markidis, "Programming Quantum Neural Networks on NISQ Systems: An Overview of Technologies and Methodologies", Entropy 25 4, 694 (2023).

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

[79] 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).

[80] 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).

[81] Tomonori Shirakawa, Kazuhiro Seki, and Seiji Yunoki, "Discretized quantum adiabatic process for free fermions and comparison with the imaginary-time evolution", Physical Review Research 3 1, 013004 (2021).

[82] 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).

[83] 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).

[84] Anthony N. Ciavarella and Ivan A. Chernyshev, "Preparation of the SU(3) lattice Yang-Mills vacuum with variational quantum methods", Physical Review D 105 7, 074504 (2022).

[85] Matija Medvidović and Giuseppe Carleo, "Classical variational simulation of the Quantum Approximate Optimization Algorithm", npj Quantum Information 7 1, 101 (2021).

[86] 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).

[87] Hirofumi Nishi, Taichi Kosugi, and Yu-ichiro Matsushita, "Implementation of quantum imaginary-time evolution method on NISQ devices by introducing nonlocal approximation", npj Quantum Information 7 1, 85 (2021).

[88] Zhaoqi Leng, Pranav Mundada, Saeed Ghadimi, and Andrew Houck, "Efficient Algorithms for High-Dimensional Quantum Optimal Control of a Transmon Qubit", Physical Review Applied 19 4, 044034 (2023).

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

[90] Kentaro Yamamoto, David Zsolt Manrique, Irfan T. Khan, Hideaki Sawada, and David Muñoz Ramo, "Quantum hardware calculations of periodic systems with partition-measurement symmetry verification: Simplified models of hydrogen chain and iron crystals", Physical Review Research 4 3, 033110 (2022).

[91] 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).

[92] Maiyuren Srikumar, Charles D Hill, and Lloyd C L Hollenberg, "Clustering and enhanced classification using a hybrid quantum autoencoder", Quantum Science and Technology 7 1, 015020 (2022).

[93] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits", Physical Review A 103 5, 052416 (2021).

[94] Yiyou Chen, Hideyuki Miyahara, Louis-S. Bouchard, and Vwani Roychowdhury, "Quantum approximation of normalized Schatten norms and applications to learning", Physical Review A 106 5, 052409 (2022).

[95] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, "Estimating the gradient and higher-order derivatives on quantum hardware", Physical Review A 103 1, 012405 (2021).

[96] Amara Katabarwa, Sukin Sim, Dax Enshan Koh, and Pierre-Luc Dallaire-Demers, "Connecting geometry and performance of two-qubit parameterized quantum circuits", Quantum 6, 782 (2022).

[97] Kosuke Ito, Wataru Mizukami, and Keisuke Fujii, "Universal noise-precision relations in variational quantum algorithms", Physical Review Research 5 2, 023025 (2023).

[98] Ilya Piatrenka and Marian Rusek, Lecture Notes in Computer Science 13353, 247 (2022) ISBN:978-3-031-08759-2.

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

[100] Damian Hofmann, Giammarco Fabiani, Johan Mentink, Giuseppe Carleo, and Michael Sentef, "Role of stochastic noise and generalization error in the time propagation of neural-network quantum states", SciPost Physics 12 5, 165 (2022).

[101] David Amaro, Matthias Rosenkranz, Nathan Fitzpatrick, Koji Hirano, and Mattia Fiorentini, "A case study of variational quantum algorithms for a job shop scheduling problem", EPJ Quantum Technology 9 1, 5 (2022).

[102] 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).

[103] Fumiyoshi Kobayashi, Kosuke Mitarai, and Keisuke Fujii, "Parent Hamiltonian as a benchmark problem for variational quantum eigensolvers", Physical Review A 105 5, 052415 (2022).

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

[105] Guglielmo Mazzola, "Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers", Physical Review A 104 2, 022431 (2021).

[106] Danny Paulson, Luca Dellantonio, Jan F. Haase, Alessio Celi, Angus Kan, Andrew Jena, Christian Kokail, Rick van Bijnen, Karl Jansen, Peter Zoller, and Christine A. Muschik, "Simulating 2D Effects in Lattice Gauge Theories on a Quantum Computer", PRX Quantum 2 3, 030334 (2021).

[107] Daniel Claudino, Jerimiah Wright, Alexander J. McCaskey, and Travis S. Humble, "Benchmarking Adaptive Variational Quantum Eigensolvers", Frontiers in Chemistry 8, 606863 (2020).

[108] Yifan Zhou and Peng Zhang, "Noise-Resilient Quantum Machine Learning for Stability Assessment of Power Systems", IEEE Transactions on Power Systems 38 1, 475 (2023).

[109] Shouvanik Chakrabarti, Rajiv Krishnakumar, Guglielmo Mazzola, Nikitas Stamatopoulos, Stefan Woerner, and William J. Zeng, "A Threshold for Quantum Advantage in Derivative Pricing", Quantum 5, 463 (2021).

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

[111] Seunghyeok Oh, Jaeho Choi, and Joongheon Kim, 2020 International Conference on Information and Communication Technology Convergence (ICTC) 236 (2020) ISBN:978-1-7281-6758-9.

[112] Shraddha Mishra and Chi-Yi Tsai, "QSurfNet: a hybrid quantum convolutional neural network for surface defect recognition", Quantum Information Processing 22 5, 179 (2023).

[113] Raphael César de Souza Pimenta and Anibal Thiago Bezerra, "Revisiting semiconductor bulk hamiltonians using quantum computers", Physica Scripta 98 4, 045804 (2023).

[114] 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).

[115] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin, "General parameter-shift rules for quantum gradients", Quantum 6, 677 (2022).

[116] Dieter Jaksch, Peyman Givi, Andrew J. Daley, and Thomas Rung, "Variational Quantum Algorithms for Computational Fluid Dynamics", AIAA Journal 61 5, 1885 (2023).

[117] Tim Weaving, Alexis Ralli, William M. Kirby, Andrew Tranter, Peter J. Love, and Peter V. Coveney, "A Stabilizer Framework for the Contextual Subspace Variational Quantum Eigensolver and the Noncontextual Projection Ansatz", Journal of Chemical Theory and Computation 19 3, 808 (2023).

[118] Jeffmin Lin, Gil Goldshlager, and Lin Lin, "Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation", Journal of Computational Physics 474, 111765 (2023).

[119] 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).

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

[121] Enrico Fontana, M. Cerezo, Andrew Arrasmith, Ivan Rungger, and Patrick J. Coles, "Non-trivial symmetries in quantum landscapes and their resilience to quantum noise", Quantum 6, 804 (2022).

[122] J. J. Postema, P. Bonizzi, G. Koekoek, R. L. Westra, and S. J. J. M. F. Kokkelmans, "Hybrid quantum singular spectrum decomposition for time series analysis", AVS Quantum Science 5 2, 023803 (2023).

[123] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer, "Effect of data encoding on the expressive power of variational quantum-machine-learning models", Physical Review A 103 3, 032430 (2021).

[124] Bálint Koczor and Simon C. Benjamin, "Quantum analytic descent", Physical Review Research 4 2, 023017 (2022).

[125] Leonardo Alchieri, Davide Badalotti, Pietro Bonardi, and Simone Bianco, "An introduction to quantum machine learning: from quantum logic to quantum deep learning", Quantum Machine Intelligence 3 2, 28 (2021).

[126] Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, and Dacheng Tao, "Efficient Measure for the Expressivity of Variational Quantum Algorithms", Physical Review Letters 128 8, 080506 (2022).

[127] Sirui Lu, Lu-Ming Duan, and Dong-Ling Deng, "Quantum adversarial machine learning", Physical Review Research 2 3, 033212 (2020).

[128] 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).

[129] Quoc Chuong Nguyen, Le Bin Ho, Lan Nguyen Tran, and Hung Q Nguyen, "Qsun: an open-source platform towards practical quantum machine learning applications", Machine Learning: Science and Technology 3 1, 015034 (2022).

[130] Saahil Patel, Benjamin Collis, William Duong, Daniel Koch, Massimiliano Cutugno, Laura Wessing, and Paul Alsing, "Information loss and run time from practical application of quantum data compression", Physica Scripta 98 4, 045111 (2023).

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

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

[133] Giorgio Tosti Balducci, Boyang Chen, Matthias Möller, Marc Gerritsma, and Roeland De Breuker, "Review and perspectives in quantum computing for partial differential equations in structural mechanics", Frontiers in Mechanical Engineering 8, 914241 (2022).

[134] Shui-Yuan Huang, Wan-Jia An, De-Shun Zhang, and Nan-Run Zhou, "Image classification and adversarial robustness analysis based on hybrid quantum–classical convolutional neural network", Optics Communications 533, 129287 (2023).

[135] Takashi Tsuchimochi, Yoohee Ryo, Seiichiro L. Ten-no, and Kazuki Sasasako, "Improved Algorithms of Quantum Imaginary Time Evolution for Ground and Excited States of Molecular Systems", Journal of Chemical Theory and Computation 19 2, 503 (2023).

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

[137] Jun Qi, Xiao-Lei Zhang, and Javier Tejedor, ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 1 (2023) ISBN:978-1-7281-6327-7.

[138] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", Quantum 4, 263 (2020).

[139] Bojia Duan and Chang-Yu Hsieh, "Hamiltonian-based data loading with shallow quantum circuits", Physical Review A 106 5, 052422 (2022).

[140] 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).

[141] Alessandro Carbone, Davide Emilio Galli, Mario Motta, and Barbara Jones, "Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers", Symmetry 14 3, 624 (2022).

[142] Vishal S. Ngairangbam, Michael Spannowsky, and Michihisa Takeuchi, "Anomaly detection in high-energy physics using a quantum autoencoder", Physical Review D 105 9, 095004 (2022).

[143] 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).

[144] Daniel Claudino, Alexander J. McCaskey, and Dmitry I. Lyakh, " A Backend-agnostic, Quantum-classical Framework for Simulations of Chemistry in C ++ ", ACM Transactions on Quantum Computing 4 1, 1 (2023).

[145] Phillip C. Lotshaw, Travis S. Humble, Rebekah Herrman, James Ostrowski, and George Siopsis, "Empirical performance bounds for quantum approximate optimization", Quantum Information Processing 20 12, 403 (2021).

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

[147] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M. Sohaib Alam, Guillermo Alonso-Linaje, B. AkashNarayanan, Ali Asadi, Juan Miguel Arrazola, Utkarsh Azad, Sam Banning, Carsten Blank, Thomas R Bromley, Benjamin A. Cordier, Jack Ceroni, Alain Delgado, Olivia Di Matteo, Amintor Dusko, Tanya Garg, Diego Guala, Anthony Hayes, Ryan Hill, Aroosa Ijaz, Theodor Isacsson, David Ittah, Soran Jahangiri, Prateek Jain, Edward Jiang, Ankit Khandelwal, Korbinian Kottmann, Robert A. Lang, Christina Lee, Thomas Loke, Angus Lowe, Keri McKiernan, Johannes Jakob Meyer, J. A. Montañez-Barrera, Romain Moyard, Zeyue Niu, Lee James O'Riordan, Steven Oud, Ashish Panigrahi, Chae-Yeun Park, Daniel Polatajko, Nicolás Quesada, Chase Roberts, Nahum Sá, Isidor Schoch, Borun Shi, Shuli Shu, Sukin Sim, Arshpreet Singh, Ingrid Strandberg, Jay Soni, Antal Száva, Slimane Thabet, Rodrigo A. Vargas-Hernández, Trevor Vincent, Nicola Vitucci, Maurice Weber, David Wierichs, Roeland Wiersema, Moritz Willmann, Vincent Wong, Shaoming Zhang, and Nathan Killoran, "PennyLane: Automatic differentiation of hybrid quantum-classical computations", arXiv:1811.04968, (2018).

[148] Nguyen Tan Viet, Nguyen Thi Chuong, Vu Thi Ngoc Huyen, and Le Bin Ho, "tqix.pis: A toolbox for quantum dynamics simulation of spin ensembles in Dicke basis", arXiv:2209.01168, (2022).

[149] Kazuhiro Seki, Tomonori Shirakawa, and Seiji Yunoki, "Symmetry-adapted variational quantum eigensolver", Physical Review A 101 5, 052340 (2020).

[150] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252, (2020).

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

[152] Naoki Yamamoto, "On the natural gradient for variational quantum eigensolver", arXiv:1909.05074, (2019).

[153] Marc Illa, Caroline E. P. Robin, and Martin J. Savage, "Quantum Simulations of SO(5) Many-Fermion Systems using Qudits", arXiv:2305.11941, (2023).

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

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

[156] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", arXiv:2007.03638, (2020).

[157] Weiyuan Gong and Dong-Ling Deng, "Universal Adversarial Examples and Perturbations for Quantum Classifiers", arXiv:2102.07788, (2021).

[158] Lennart Bittel, Jens Watty, and Martin Kliesch, "Fast gradient estimation for variational quantum algorithms", arXiv:2210.06484, (2022).

[159] Kazuki Osawa, Satoki Ishikawa, Rio Yokota, Shigang Li, and Torsten Hoefler, "ASDL: A Unified Interface for Gradient Preconditioning in PyTorch", arXiv:2305.04684, (2023).

[160] Tianchen Zhao, Giuseppe Carleo, James Stokes, and Shravan Veerapaneni, "Natural evolution strategies and variational Monte Carlo", arXiv:2005.04447, (2020).

[161] Le Bin Ho, "Stochastic approach for quantum metrology with generic Hamiltonians", arXiv:2204.01055, (2022).

[162] Nguyen Tan Viet, Nguyen Thi Chuong, Vu Thi Ngoc Huyen, and Le Bin Ho, "tqix.pis: A toolbox for quantum dynamics simulation of spin ensembles in Dicke basis", Computer Physics Communications 286, 108686 (2023).

[163] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini, "Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution", arXiv:2305.14294, (2023).

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Quantum Natural Gradient

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