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

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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.

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► References

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

[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.

[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.

[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.

[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.

[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.

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

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

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

[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.

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

[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.

[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.

[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.

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28] Dénes Petz. Information-geometry of quantum states. In Quantum Probability Communications: Volume X, pages 135–157. World Scientific, 1998. 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.

[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.

[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.

[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.

[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.

[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.

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

[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.

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[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] 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).

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

[4] 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).

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

[6] 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).

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

[8] 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).

[9] 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).

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

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

[12] Kaelan Donatella, Zakari Denis, Alexandre Le Boité, and Cristiano Ciuti, "Dynamics with autoregressive neural quantum states: Application to critical quench dynamics", Physical Review A 108 2, 022210 (2023).

[13] 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).

[14] 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).

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

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

[17] 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).

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

[19] 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).

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

[21] 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).

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

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

[24] 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).

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

[26] 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, and Astronomy 66 5, 250302 (2023).

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

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

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

[30] Christopher Roth, Attila Szabó, and Allan H. MacDonald, "High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks", Physical Review B 108 5, 054410 (2023).

[31] 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).

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

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

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

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

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

[37] 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).

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

[39] 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).

[40] 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).

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

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

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

[44] 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, 85 (2021).

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

[46] 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).

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

[48] Roeland Wiersema and Nathan Killoran, "Optimizing quantum circuits with Riemannian gradient flow", Physical Review A 107 6, 062421 (2023).

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

[50] 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).

[51] 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).

[52] 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).

[53] 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).

[54] 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).

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

[56] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini, "Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution", Quantum 7, 1131 (2023).

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

[58] 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).

[59] Daniel Claudino, Jerimiah Wright, Alexander J. McCaskey, and Travis S. Humble, "Benchmarking Adaptive Variational Quantum Eigensolvers", Frontiers in Chemistry 8, 1152 (2020).

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

[61] 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).

[62] 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).

[63] Bryce Fore, Jane M. Kim, Giuseppe Carleo, Morten Hjorth-Jensen, Alessandro Lovato, and Maria Piarulli, "Dilute neutron star matter from neural-network quantum states", Physical Review Research 5 3, 033062 (2023).

[64] Marc Illa, Caroline E. P. Robin, and Martin J. Savage, "Quantum simulations of SO(5) many-fermion systems using qudits", Physical Review C 108 6, 064306 (2023).

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

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

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

[68] 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).

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

[70] 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).

[71] 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).

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

[73] Di Luo, Jiayu Shen, Rumen Dangovski, and Marin Soljačić, "QuACK: Accelerating Gradient-Based Quantum Optimization with Koopman Operator Learning", arXiv:2211.01365, (2022).

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

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

[76] 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).

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

[78] 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).

[79] 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).

[80] 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).

[81] J. Gidi, B. Candia, A. D. Muñoz-Moller, A. Rojas, L. Pereira, M. Muñoz, L. Zambrano, and A. Delgado, "Stochastic optimization algorithms for quantum applications", Physical Review A 108 3, 032409 (2023).

[82] 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).

[83] Charles Moussa, Max Hunter Gordon, Michal Baczyk, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Resource frugal optimizer for quantum machine learning", Quantum Science and Technology 8 4, 045019 (2023).

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

[85] 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).

[86] Richard Meister, Cica Gustiani, and Simon C. Benjamin, "Exploring ab initio machine synthesis of quantum circuits", New Journal of Physics 25 7, 073018 (2023).

[87] 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).

[88] Xinglan Zhang and Feng Zhang, "Variational Quantum Computation Integer Factorization Algorithm", International Journal of Theoretical Physics 62 11, 245 (2023).

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

[90] 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).

[91] Matija Medvidović and Dries Sels, "Variational Quantum Dynamics of Two-Dimensional Rotor Models", PRX Quantum 4 4, 040302 (2023).

[92] Nikita Astrakhantsev, Guglielmo Mazzola, Ivano Tavernelli, and Giuseppe Carleo, "Phenomenological theory of variational quantum ground-state preparation", Physical Review Research 5 3, 033225 (2023).

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

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

[95] Cica Gustiani, Richard Meister, and Simon C. Benjamin, "Exploiting subspace constraints and ab initio variational methods for quantum chemistry", New Journal of Physics 25 7, 073019 (2023).

[96] Daniel Faílde, José Daniel Viqueira, Mariamo Mussa Juane, and Andrés Gómez, "Using Differential Evolution to avoid local minima in Variational Quantum Algorithms", Scientific Reports 13, 16230 (2023).

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

[98] Roeland Wiersema, Cunlu Zhou, Juan Felipe Carrasquilla, and Yong Baek Kim, "Measurement-induced entanglement phase transitions in variational quantum circuits", SciPost Physics 14 6, 147 (2023).

[99] 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).

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

[101] 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).

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

[103] Alistair W. R. Smith, A. J. Paige, and M. S. Kim, "Faster variational quantum algorithms with quantum kernel-based surrogate models", Quantum Science and Technology 8 4, 045016 (2023).

[104] 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).

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

[106] Ranyiliu Chen, Benchi Zhao, and Xin Wang, "Near-Term Efficient Quantum Algorithms for Entanglement Analysis", Physical Review Applied 20 2, 024071 (2023).

[107] Maximilian Amsler, Peter Deglmann, Matthias Degroote, Michael P. Kaicher, Matthew Kiser, Michael Kühn, Chandan Kumar, Andreas Maier, Georgy Samsonidze, Anna Schroeder, Michael Streif, Davide Vodola, Christopher Wever, and Qutac Material Science Working Group, "Classical and quantum trial wave functions in auxiliary-field quantum Monte Carlo applied to oxygen allotropes and a CuBr<SUB>2</SUB> model system", Journal of Chemical Physics 159 4, 044119 (2023).

[108] 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).

[109] 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).

[110] 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).

[111] 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).

[112] 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).

[113] 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).

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

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

[116] 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).

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

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

[119] 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).

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

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

[122] 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).

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

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

[125] Maximilian Balthasar Mansky, Jonas Nüßlein, David Bucher, Daniëlle Schuman, Sebastian Zielinski, and Claudia Linnhoff-Popien, "Sampling Problems on a Quantum Computer", arXiv:2402.16341, (2024).

[126] Eimantas Ledinauskas and Egidijus Anisimovas, "Scalable imaginary time evolution with neural network quantum states", SciPost Physics 15 6, 229 (2023).

[127] 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).

[128] Xin Wang, Bo Qi, Yabo Wang, and Daoyi Dong, "Entanglement-variational hardware-efficient ansatz for eigensolvers", Physical Review Applied 21 3, 034059 (2024).

[129] 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).

[130] Benedikt Fauseweh, "Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges", Nature Communications 15, 2123 (2024).

[131] 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).

[132] Stefano Mangini, "Variational quantum algorithms for machine learning: theory and applications", arXiv:2306.09984, (2023).

[133] Yagnik Chatterjee, Eric Bourreau, and Marko J. Rančić, "Solving various NP-hard problems using exponentially fewer qubits on a quantum computer", Physical Review A 109 5, 052441 (2024).

[134] Jonas Beck, Jonathan Bodky, Johannes Motruk, Tobias Müller, Ronny Thomale, and Pratyay Ghosh, "Phase diagram of the J -J<SUB>d</SUB> Heisenberg model on the maple leaf lattice: Neural networks and density matrix renormalization group", Physical Review B 109 18, 184422 (2024).

[135] Yusuke Nomura, "Boltzmann machines and quantum many-body problems", Journal of Physics Condensed Matter 36 7, 073001 (2024).

[136] João C. Getelina, Niladri Gomes, Thomas Iadecola, Peter P. Orth, and Yong-Xin Yao, "Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations", SciPost Physics 15 3, 102 (2023).

[137] Eimantas Ledinauskas and Egidijus Anisimovas, "Universal Performance Gap of Neural Quantum States Applied to the Hofstadter-Bose-Hubbard Model", arXiv:2405.01981, (2024).

[138] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).

[139] Xinglan Zhang, Feng Zhang, Yankun Guo, and Fei Chen, "Variational quantum multidimensional scaling algorithm", Quantum Information Processing 23 3, 77 (2024).

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

[141] Jing-Kai Fang, Yue-Feng Lin, Jun-Han Huang, Yibo Chen, Gao-Ming Fan, Yuhui Sun, Guanru Feng, Cong Guo, Tiejun Meng, Yong Zhang, Xun Xu, Jingen Xiang, and Yuxiang Li, "Divide-and-Conquer Quantum Algorithm for Hybrid d e n o v o Genome Assembly of Short and Long Reads", PRX Life 2 2, 023006 (2024).

[142] Emiel Koridon, Joana Fraxanet, Alexandre Dauphin, Lucas Visscher, Thomas E. O'Brien, and Stefano Polla, "A hybrid quantum algorithm to detect conical intersections", Quantum 8, 1259 (2024).

[143] Qi-Ming Ding, Yi-Ming Huang, and Xiao Yuan, "Molecular docking via quantum approximate optimization algorithm", Physical Review Applied 21 3, 034036 (2024).

[144] Pablo Bermejo, Borja Aizpurua, and Román Orús, "Improving gradient methods via coordinate transformations: Applications to quantum machine learning", Physical Review Research 6 2, 023069 (2024).

[145] David Fitzek, Robert S. Jonsson, Werner Dobrautz, and Christian Schäfer, "Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes", Quantum 8, 1313 (2024).

[146] João C. Getelina, Cai-Zhuang Wang, Thomas Iadecola, Yong-Xin Yao, and Peter P. Orth, "Adaptive variational ground state preparation for spin-1 models on qubit-based architectures", Physical Review B 109 8, 085128 (2024).

[147] Maniraman Periyasamy, Axel Plinge, Christopher Mutschler, Daniel D. Scherer, and Wolfgang Mauerer, "Guided-SPSA: Simultaneous Perturbation Stochastic Approximation assisted by the Parameter Shift Rule", arXiv:2404.15751, (2024).

[148] Han Qi, Sihui Xiao, Zhuo Liu, Changqing Gong, and Abdullah Gani, "Variational quantum algorithms: fundamental concepts, applications and challenges", Quantum Information Processing 23 6, 224 (2024).

[149] Jane Kim, Gabriel Pescia, Bryce Fore, Jannes Nys, Giuseppe Carleo, Stefano Gandolfi, Morten Hjorth-Jensen, and Alessandro Lovato, "Neural-network quantum states for ultra-cold Fermi gases", Communications Physics 7 1, 148 (2024).

[150] Vu Tuan Hai, Nguyen Tan Viet, and Le Bin Ho, "«qo|op»: A quantum object optimizer", SoftwareX 26, 101726 (2024).

[151] Tao Cheng, Run-Sheng Zhao, Shuang Wang, Rui Wang, and Hong-Yang Ma, "Analysis of learnability of a novel hybrid quantum–classical convolutional neural network in image classification", Chinese Physics B 33 4, 040303 (2024).

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