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.

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

► 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] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402.

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

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

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

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

[6] Laura Gentini, Alessandro Cuccoli, Stefano Pirandola, Paola Verrucchi, and Leonardo Banchi, "Noise-Assisted Variational Hybrid Quantum-Classical Optimization", arXiv:1912.06744.

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

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

[9] Suguru Endo, Iori Kurata, and Yuya O. Nakagawa, "Calculation of the Green's function on near-term quantum computers", arXiv:1909.12250.

[10] Sirui Lu, Lu-Ming Duan, and Dong-Ling Deng, "Quantum Adversarial Machine Learning", arXiv:2001.00030.

[11] David Wierichs, Christian Gogolin, and Michael Kastoryano, "Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer", arXiv:2004.14666.

[12] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Trainability of Dissipative Perceptron-Based Quantum Neural Networks", arXiv:2005.12458.

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

[14] Tianchen Zhao, Giuseppe Carleo, James Stokes, and Shravan Veerapaneni, "Natural evolution strategies and quantum approximate optimization", arXiv:2005.04447.

[15] Leonardo Banchi and Gavin E. Crooks, "Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule", arXiv:2005.10299.

[16] Barnaby van Straaten and Bálint Koczor, "Measurement cost of metric-aware variational quantum algorithms", arXiv:2005.05172.

[17] Hirofumi Nishi, Taichi Kosugi, and Yu-ichiro Matsushita, "Implementation of quantum imaginary-time evolution method on NISQ devices: Nonlocal approximation", arXiv:2005.12715.

[18] Kouhei Nakaji and Naoki Yamamoto, "Expressibility of the alternating layered ansatz for quantum computation", arXiv:2005.12537.

The above citations are from SAO/NASA ADS (last updated successfully 2020-07-14 07:08:34). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-07-14 07:08:33).