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

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


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

► 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.

Cited by

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

[2] 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).

[3] 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).

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

[5] 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).

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

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

[8] 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).

[9] 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).

[10] 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).

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

[12] 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).

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

[14] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Trainability of Dissipative Perceptron-Based Quantum Neural Networks", arXiv:2005.12458, Physical Review Letters 128 18, 180505 (2022).

[15] 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.

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

[17] 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).

[18] 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).

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

[20] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", arXiv:2007.03638, SciPost Physics 10 2, 040 (2021).

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

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

[23] 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).

[24] 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).

[25] 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).

[26] 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).

[27] 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).

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

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

[30] 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).

[31] 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).

[32] 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).

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

[34] 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).

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

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

[37] 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).

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

[39] 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).

[40] 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).

[41] 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.

[42] 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.

[43] 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).

[44] 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).

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

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

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

[48] 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).

[49] 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).

[50] 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).

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

[52] 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).

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

[54] 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).

[55] 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).

[56] 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).

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

[58] 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).

[59] 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).

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

[61] 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).

[62] 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).

[63] 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).

[64] 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).

[65] 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).

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

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

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

[69] 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).

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

[71] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", arXiv:1909.09083, Quantum 4, 263 (2020).

[72] 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).

[73] 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).

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

[75] 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).

[76] 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).

[77] 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).

[78] 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).

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

[80] 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).

[81] 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).

[82] 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", arXiv:2111.05176.

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

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

[85] 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 Száva, and Nathan Killoran, "PennyLane: Automatic differentiation of hybrid quantum-classical computations", arXiv:1811.04968.

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

[87] Bálint Koczor and Simon C. Benjamin, "Quantum natural gradient generalised to non-unitary circuits", arXiv:1912.08660.

[88] Naoki Yamamoto, "On the natural gradient for variational quantum eigensolver", arXiv:1909.05074.

[89] Nikita A. Nemkov, Evgeniy O. Kiktenko, Ilia A. Luchnikov, and Aleksey K. Fedorov, "Efficient variational synthesis of quantum circuits with coherent multi-start optimization", arXiv:2205.01121.

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

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

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

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

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-17 06:33:57) and SAO/NASA ADS (last updated successfully 2022-05-17 06:33:58). The list may be incomplete as not all publishers provide suitable and complete citation data.