A hybrid quantum algorithm to detect conical intersections

Emiel Koridon1,2, Joana Fraxanet3, Alexandre Dauphin3,4, Lucas Visscher2, Thomas E. O'Brien5,1, and Stefano Polla5,1

1Instituut-Lorentz, Universiteit Leiden, 2300RA Leiden, The Netherlands
2Theoretical Chemistry, Vrije Universiteit, 1081HV Amsterdam, The Netherlands
3ICFO - Institut de Ciències Fotòniques, 08860 Castelldefels (Barcelona), Spain
4PASQAL SAS, 2 av. Augustin Fresnel Palaiseau, 91120, France
5Google Research, Munich, 80636 Bavaria, Germany

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Abstract

Conical intersections are topologically protected crossings between the potential energy surfaces of a molecular Hamiltonian, known to play an important role in chemical processes such as photoisomerization and non-radiative relaxation. They are characterized by a non-zero Berry phase, which is a topological invariant defined on a closed path in atomic coordinate space, taking the value $\pi$ when the path encircles the intersection manifold. In this work, we show that for real molecular Hamiltonians, the Berry phase can be obtained by tracing a local optimum of a variational ansatz along the chosen path and estimating the overlap between the initial and final state with a control-free Hadamard test. Moreover, by discretizing the path into $N$ points, we can use $N$ single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or $\pi$), our procedure succeeds even for a cumulative error bounded by a constant; this allows us to bound the total sampling cost and to readily verify the success of the procedure. We demonstrate numerically the application of our algorithm on small toy models of the formaldimine molecule (${H_2C=NH}$).

In the last decade, variational quantum algorithms (VQAs) have been in the spotlight as a potential paradigm for tackling quantum simulation problems on noisy small-scale quantum computers. The typical requirement for high-precision results strongly hinders the application of these algorithms to computational chemistry. Achieving this high precision is extremely expensive due to the cost of sampling, made worse by the need for error mitigation and complex optimization. We identify a problem in quantum chemistry that can bypass the high precision requirement, we design an algorithm to solve it and benchmark it on a small molecular model.

In our work, we develop a VQA that detects the presence of a conical intersection by tracking the ground state around a loop in nuclear coordinate space. Conical intersections play a key role in photochemical reactions, for example in the process of vision. Identifying the presence of a conical intersection in a molecular model can be an important step in understanding or predicting the photochemical properties of a system.

The question we pose has a discrete answer (yes/no); this lifts the requirement of high precision. Furthermore, we simplify the optimization problem by using fixed-cost updates to track the ground state approximately, to the required level of precision. This allows to prove bounds on the cost of the algorithm, which is rare in the context of VQAs.

We perform numerical benchmarks of the algorithm, demonstrating its resilience to different levels of sampling noise. We release publicly the code we developed for this task, which includes a framework for orbital-optimized quantum circuit ansätze that supports automatic differentiation.

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► References

[1] A. K. Geim and K. S. Novoselov. The rise of graphene. Nature Materials, 6 (3): 183–191, March 2007. ISSN 1476-4660. 10.1038/​nmat1849.
https:/​/​doi.org/​10.1038/​nmat1849

[2] Michael Victor Berry. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392 (1802): 45–57, March 1984. 10.1098/​rspa.1984.0023.
https:/​/​doi.org/​10.1098/​rspa.1984.0023

[3] Wolfgang Domcke, David Yarkony, and Horst Köppel, editors. Conical Intersections: Theory, Computation and Experiment. Number v. 17 in Advanced Series in Physical Chemistry. World Scientific, Singapore ; Hackensack, NJ, 2011. ISBN 978-981-4313-44-5.

[4] David R. Yarkony. Nonadiabatic Quantum Chemistry—Past, Present, and Future. Chemical Reviews, 112 (1): 481–498, January 2012. ISSN 0009-2665. 10.1021/​cr2001299.
https:/​/​doi.org/​10.1021/​cr2001299

[5] Dario Polli, Piero Altoè, Oliver Weingart, Katelyn M. Spillane, Cristian Manzoni, Daniele Brida, Gaia Tomasello, Giorgio Orlandi, Philipp Kukura, Richard A. Mathies, Marco Garavelli, and Giulio Cerullo. Conical intersection dynamics of the primary photoisomerization event in vision. Nature, 467 (7314): 440–443, September 2010. ISSN 1476-4687. 10.1038/​nature09346.
https:/​/​doi.org/​10.1038/​nature09346

[6] Gloria Olaso-González, Manuela Merchán, and Luis Serrano-Andrés. Ultrafast Electron Transfer in Photosynthesis: Reduced Pheophytin and Quinone Interaction Mediated by Conical Intersections. The Journal of Physical Chemistry B, 110 (48): 24734–24739, December 2006. ISSN 1520-6106, 1520-5207. 10.1021/​jp063915u.
https:/​/​doi.org/​10.1021/​jp063915u

[7] Howard E Zimmerman. Molecular Orbital Correlation Diagrams, Mobius Systems, and Factors Controlling Ground- and Excited-State Reactions. II. Journal of the American Chemical Society, 88 (7): 1566–1567, 1966. ISSN 0002-7863. 10.1021/​ja00959a053.
https:/​/​doi.org/​10.1021/​ja00959a053

[8] Fernando Bernardi, Massimo Olivucci, and Michael A. Robb. Potential energy surface crossings in organic photochemistry. Chemical Society Reviews, 25 (5): 321–328, 1996. ISSN 0306-0012. 10.1039/​cs9962500321.
https:/​/​doi.org/​10.1039/​cs9962500321

[9] Leticia González, Daniel Escudero, and Luis Serrano‐Andrés. Progress and Challenges in the Calculation of Electronic Excited States. ChemPhysChem, 13 (1): 28–51, 2012. ISSN 1439-4235. 10.1002/​cphc.201100200.
https:/​/​doi.org/​10.1002/​cphc.201100200

[10] Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21 (6-7): 467–488, June 1982. ISSN 0020-7748, 1572-9575. 10.1007/​BF02650179.
https:/​/​doi.org/​10.1007/​BF02650179

[11] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. Simulated Quantum Computation of Molecular Energies. Science, 309 (5741): 1704–1707, September 2005. 10.1126/​science.1113479.
https:/​/​doi.org/​10.1126/​science.1113479

[12] John Preskill. Quantum Computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/​q-2018-08-06-79.
https:/​/​doi.org/​10.22331/​q-2018-08-06-79

[13] Alberto Peruzzo, Jarrod R. 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 (1): 4213, September 2014. ISSN 2041-1723. 10.1038/​ncomms5213.
https:/​/​doi.org/​10.1038/​ncomms5213

[14] Jarrod R. McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, February 2016. ISSN 1367-2630. 10.1088/​1367-2630/​18/​2/​023023.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​2/​023023

[15] Dave Wecker, Matthew B Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Physical Review A, 92 (4): 042303, October 2015. ISSN 1050-2947. 10.1103/​PhysRevA.92.042303.
https:/​/​doi.org/​10.1103/​PhysRevA.92.042303

[16] 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, November 2018. ISSN 2041-1723. 10.1038/​s41467-018-07090-4.
https:/​/​doi.org/​10.1038/​s41467-018-07090-4

[17] Shiro Tamiya, Sho Koh, and Yuya O. Nakagawa. Calculating nonadiabatic couplings and berry's phase by variational quantum eigensolvers. Phys. Rev. Research, 3: 023244, Jun 2021. 10.1103/​PhysRevResearch.3.023244.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.023244

[18] Xiao Xiao, J. K. Freericks, and A. F. Kemper. Robust measurement of wave function topology on NISQ quantum computers, October 2022. URL https:/​/​doi.org/​10.22331/​q-2023-04-27-987.
https:/​/​doi.org/​10.22331/​q-2023-04-27-987

[19] Bruno Murta, G. Catarina, and J. Fernández-Rossier. Berry phase estimation in gate-based adiabatic quantum simulation. Phys. Rev. A, 101: 020302, Feb 2020. 10.1103/​PhysRevA.101.020302. URL https:/​/​doi.org/​10.1103/​PhysRevA.101.020302.
https:/​/​doi.org/​10.1103/​PhysRevA.101.020302

[20] Hugh Christopher Longuet-Higgins, U. Öpik, Maurice Henry Lecorney Pryce, and R. A. Sack. Studies of the Jahn-Teller effect .II. The dynamical problem. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 244 (1236): 1–16, February 1958. 10.1098/​rspa.1958.0022.
https:/​/​doi.org/​10.1098/​rspa.1958.0022

[21] C. Alden Mead and Donald G. Truhlar. On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei. The Journal of Chemical Physics, 70 (5): 2284–2296, March 1979. ISSN 0021-9606. 10.1063/​1.437734.
https:/​/​doi.org/​10.1063/​1.437734

[22] Ilya G. Ryabinkin, Loïc Joubert-Doriol, and Artur F. Izmaylov. Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections. Accounts of Chemical Research, 50 (7): 1785–1793, July 2017. ISSN 0001-4842. 10.1021/​acs.accounts.7b00220.
https:/​/​doi.org/​10.1021/​acs.accounts.7b00220

[23] Jacob Whitlow, Zhubing Jia, Ye Wang, Chao Fang, Jungsang Kim, and Kenneth R. Brown. Simulating conical intersections with trapped ions, February 2023. URL https:/​/​doi.org/​10.48550/​arXiv.2211.07319.
https:/​/​doi.org/​10.48550/​arXiv.2211.07319

[24] Christophe H. Valahu, Vanessa C. Olaya-Agudelo, Ryan J. MacDonell, Tomas Navickas, Arjun D. Rao, Maverick J. Millican, Juan B. Pérez-Sánchez, Joel Yuen-Zhou, Michael J. Biercuk, Cornelius Hempel, Ting Rei Tan, and Ivan Kassal. Direct observation of geometric phase in dynamics around a conical intersection. Nature Chemistry, 15 (11): 1503–1508, November 2023. ISSN 1755-4330, 1755-4349. 10.1038/​s41557-023-01300-3.
https:/​/​doi.org/​10.1038/​s41557-023-01300-3

[25] Christopher S. Wang, Nicholas E. Frattini, Benjamin J. Chapman, Shruti Puri, Steven M. Girvin, Michel H. Devoret, and Robert J. Schoelkopf. Observation of wave-packet branching through an engineered conical intersection. Physical Review X, 13 (1): 011008, January 2023. ISSN 2160-3308. 10.1103/​PhysRevX.13.011008.
https:/​/​doi.org/​10.1103/​PhysRevX.13.011008

[26] Emiel Koridon and Stefano Polla. auto_oo: an autodifferentiable framework for molecular orbital-optimized variational quantum algorithms. Zenodo, February 2024. URL https:/​/​doi.org/​10.5281/​zenodo.10639817.
https:/​/​doi.org/​10.5281/​zenodo.10639817

[27] E. Teller. The Crossing of Potential Surfaces. The Journal of Physical Chemistry, 41 (1): 109–116, January 1937. ISSN 0092-7325. 10.1021/​j150379a010.
https:/​/​doi.org/​10.1021/​j150379a010

[28] G. Herzberg and H. C. Longuet-Higgins. Intersection of potential energy surfaces in polyatomic molecules. Discussions of the Faraday Society, 35 (0): 77–82, January 1963. ISSN 0366-9033. 10.1039/​DF9633500077.
https:/​/​doi.org/​10.1039/​DF9633500077

[29] Trygve Helgaker, Poul Jørgensen, and Jeppe Olsen. Molecular Electronic-Structure Theory. Wiley, first edition, August 2000. ISBN 978-0-471-96755-2 978-1-119-01957-2. 10.1002/​9781119019572.
https:/​/​doi.org/​10.1002/​9781119019572

[30] R. Broer, L. Hozoi, and W. C. Nieuwpoort. Non-orthogonal approaches to the study of magnetic interactions. Molecular Physics, 101 (1-2): 233–240, January 2003. ISSN 0026-8976. 10.1080/​0026897021000035205.
https:/​/​doi.org/​10.1080/​0026897021000035205

[31] Valera Veryazov, Per Åke Malmqvist, and Björn O. Roos. How to select active space for multiconfigurational quantum chemistry? International Journal of Quantum Chemistry, 111 (13): 3329–3338, 2011. ISSN 1097-461X. 10.1002/​qua.23068.
https:/​/​doi.org/​10.1002/​qua.23068

[32] David R. Yarkony. Diabolical conical intersections. Reviews of Modern Physics, 68 (4): 985–1013, October 1996. 10.1103/​RevModPhys.68.985.
https:/​/​doi.org/​10.1103/​RevModPhys.68.985

[33] C. Alden Mead. The molecular Aharonov—Bohm effect in bound states. Chemical Physics, 49 (1): 23–32, June 1980. ISSN 0301-0104. 10.1016/​0301-0104(80)85035-X.
https:/​/​doi.org/​10.1016/​0301-0104(80)85035-X

[34] Stuart M. Harwood, Dimitar Trenev, Spencer T. Stober, Panagiotis Barkoutsos, Tanvi P. Gujarati, Sarah Mostame, and Donny Greenberg. Improving the Variational Quantum Eigensolver Using Variational Adiabatic Quantum Computing. ACM Transactions on Quantum Computing, 3 (1): 1:1–1:20, January 2022. ISSN 2643-6809. 10.1145/​3479197.
https:/​/​doi.org/​10.1145/​3479197

[35] C. Alden Mead. The ''noncrossing'' rule for electronic potential energy surfaces: The role of time-reversal invariance. The Journal of Chemical Physics, 70 (5): 2276–2283, March 1979. ISSN 0021-9606. 10.1063/​1.437733.
https:/​/​doi.org/​10.1063/​1.437733

[36] Rodney J. Bartlett, Stanislaw A. Kucharski, and Jozef Noga. Alternative coupled-cluster ansätze II. The unitary coupled-cluster method. Chemical Physics Letters, 155 (1): 133–140, February 1989. ISSN 0009-2614. 10.1016/​S0009-2614(89)87372-5.
https:/​/​doi.org/​10.1016/​S0009-2614(89)87372-5

[37] Jonathan Romero, Ryan Babbush, Jarrod R. McClean, Cornelius Hempel, Peter J. Love, and Alán Aspuru-Guzik. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Science and Technology, 4 (1): 014008, October 2018. ISSN 2058-9565. 10.1088/​2058-9565/​aad3e4.
https:/​/​doi.org/​10.1088/​2058-9565/​aad3e4

[38] Gian-Luca R. Anselmetti, David Wierichs, Christian Gogolin, and Robert M. Parrish. Local, expressive, quantum-number-preserving vqe ansatze for fermionic systems. New Journal of Physics, 23, 4 2021. 10.1088/​1367-2630/​ac2cb3.
https:/​/​doi.org/​10.1088/​1367-2630/​ac2cb3

[39] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99 (3): 032331, March 2019. ISSN 2469-9926, 2469-9934. 10.1103/​PhysRevA.99.032331.
https:/​/​doi.org/​10.1103/​PhysRevA.99.032331

[40] Hans Jorgen Aa. Jensen and Poul Jorgensen. A direct approach to second-order MCSCF calculations using a norm extended optimization scheme. The Journal of Chemical Physics, 80 (3): 1204–1214, February 1984. ISSN 0021-9606. 10.1063/​1.446797.
https:/​/​doi.org/​10.1063/​1.446797

[41] Benjamin Helmich-Paris. A trust-region augmented Hessian implementation for restricted and unrestricted Hartree–Fock and Kohn–Sham methods. The Journal of Chemical Physics, 154 (16): 164104, April 2021. ISSN 0021-9606. 10.1063/​5.0040798.
https:/​/​doi.org/​10.1063/​5.0040798

[42] Thomas E. O'Brien, Stefano Polla, Nicholas C. Rubin, William J. Huggins, Sam McArdle, Sergio Boixo, Jarrod R. McClean, and Ryan Babbush. Error Mitigation via Verified Phase Estimation. PRX Quantum, 2 (2), oct 2021. 10.1103/​prxquantum.2.020317.
https:/​/​doi.org/​10.1103/​prxquantum.2.020317

[43] Stefano Polla, Gian-Luca R. Anselmetti, and Thomas E. O'Brien. Optimizing the information extracted by a single qubit measurement. Physical Review A, 108 (1): 012403, July 2023. 10.1103/​PhysRevA.108.012403.
https:/​/​doi.org/​10.1103/​PhysRevA.108.012403

[44] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer Series in Operations Research. Springer, New York, 2nd ed edition, 2006. ISBN 978-0-387-30303-1.

[45] Eugene P. Wigner. Characteristic Vectors of Bordered Matrices With Infinite Dimensions. Annals of Mathematics, 62 (3): 548–564, 1955. ISSN 0003-486X. 10.2307/​1970079.
https:/​/​doi.org/​10.2307/​1970079

[46] Saad Yalouz, Bruno Senjean, Jakob Günther, Francesco Buda, Thomas E O'Brien, and Lucas Visscher. A state-averaged orbital-optimized hybrid quantum–classical algorithm for a democratic description of ground and excited states. Quantum Science and Technology, 6 (2): 024004, jan 2021. ISSN 2058-9565. 10.1088/​2058-9565/​abd334.
https:/​/​doi.org/​10.1088/​2058-9565/​abd334

[47] Saad Yalouz, Emiel Koridon, Bruno Senjean, Benjamin Lasorne, Francesco Buda, and Lucas Visscher. Analytical nonadiabatic couplings and gradients within the state-averaged orbital-optimized variational quantum eigensolver. Journal of Chemical Theory and Computation, 18 (2): 776–794, 2022. 10.1021/​acs.jctc.1c00995. PMID: 35029988.
https:/​/​doi.org/​10.1021/​acs.jctc.1c00995

[48] Per‐Olov Löwdin. On the non‐orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals. The Journal of Chemical Physics, 18 (3): 365–375, 1950. 10.1063/​1.1747632.
https:/​/​doi.org/​10.1063/​1.1747632

[49] Xavier Bonet-Monroig, Ryan Babbush, and Thomas E. O'Brien. Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States. Physical Review X, 10 (3): 031064, September 2020. 10.1103/​PhysRevX.10.031064.
https:/​/​doi.org/​10.1103/​PhysRevX.10.031064

[50] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. Quantum computing enhanced computational catalysis. Physical Review Research, 3 (3): 033055, July 2021. ISSN 2643-1564. 10.1103/​PhysRevResearch.3.033055.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033055

[51] Jeffrey Cohn, Mario Motta, and Robert M. Parrish. Quantum Filter Diagonalization with Compressed Double-Factorized Hamiltonians. PRX Quantum, 2 (4): 040352, December 2021. 10.1103/​PRXQuantum.2.040352.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.040352

[52] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J Huggins, Lev Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E. O'Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z. Jamie Yao, Ping Yeh, and Adam Zalcman. Hartree-Fock on a superconducting qubit quantum computer. Science, 369 (6507): 1084–1089, August 2020. ISSN 0036-8075. 10.1126/​science.abb9811.
https:/​/​doi.org/​10.1126/​science.abb9811

[53] Patrick Huembeli and Alexandre Dauphin. Characterizing the loss landscape of variational quantum circuits. Quantum Science and Technology, 6 (2): 025011, February 2021. ISSN 2058-9565. 10.1088/​2058-9565/​abdbc9.
https:/​/​doi.org/​10.1088/​2058-9565/​abdbc9

[54] Hirotoshi Hirai. Excited-state molecular dynamics simulation based on variational quantum algorithms, November 2022. URL https:/​/​doi.org/​10.48550/​arXiv.2211.02302.
https:/​/​doi.org/​10.48550/​arXiv.2211.02302

[55] Vlasta Bonačić-Koutecký and Josef Michl. Photochemicalsyn-anti isomerization of a Schiff base: A two-dimensional description of a conical intersection in formaldimine. Theoretica chimica acta, 68 (1): 45–55, July 1985. ISSN 1432-2234. 10.1007/​BF00698750.
https:/​/​doi.org/​10.1007/​BF00698750

[56] Robert R. Birge. Nature of the primary photochemical events in rhodopsin and bacteriorhodopsin. Biochimica et Biophysica Acta (BBA) - Bioenergetics, 1016 (3): 293–327, April 1990. ISSN 0005-2728. 10.1016/​0005-2728(90)90163-X.
https:/​/​doi.org/​10.1016/​0005-2728(90)90163-X

[57] M Chahre. Trigger and Amplification Mechanisms in Visual Phototransduction. Annual Review of Biophysics and Biophysical Chemistry, 14 (1): 331–360, 1985. 10.1146/​annurev.bb.14.060185.001555.
https:/​/​doi.org/​10.1146/​annurev.bb.14.060185.001555

[58] 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, July 2022. URL https:/​/​doi.org/​10.48550/​arXiv.1811.04968.
https:/​/​doi.org/​10.48550/​arXiv.1811.04968

[59] Qiming Sun, Xing Zhang, Samragni Banerjee, Peng Bao, Marc Barbry, Nick S. Blunt, Nikolay A. Bogdanov, George H. Booth, Jia Chen, Zhi-Hao Cui, Janus J. Eriksen, Yang Gao, Sheng Guo, Jan Hermann, Matthew R. Hermes, Kevin Koh, Peter Koval, Susi Lehtola, Zhendong Li, Junzi Liu, Narbe Mardirossian, James D. McClain, Mario Motta, Bastien Mussard, Hung Q. Pham, Artem Pulkin, Wirawan Purwanto, Paul J. Robinson, Enrico Ronca, Elvira R. Sayfutyarova, Maximilian Scheurer, Henry F. Schurkus, James E. T. Smith, Chong Sun, Shi-Ning Sun, Shiv Upadhyay, Lucas K. Wagner, Xiao Wang, Alec White, James Daniel Whitfield, Mark J. Williamson, Sebastian Wouters, Jun Yang, Jason M. Yu, Tianyu Zhu, Timothy C. Berkelbach, Sandeep Sharma, Alexander Yu. Sokolov, and Garnet Kin-Lic Chan. Recent developments in the PySCF program package. The Journal of Chemical Physics, 153 (2): 024109, July 2020. ISSN 0021-9606. 10.1063/​5.0006074.
https:/​/​doi.org/​10.1063/​5.0006074

[60] William J. Huggins, Jarrod R. McClean, Nicholas C. Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers. npj Quantum Information, 7 (1): 1–9, February 2021. ISSN 2056-6387. 10.1038/​s41534-020-00341-7.
https:/​/​doi.org/​10.1038/​s41534-020-00341-7

[61] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake. Fermionic partial tomography via classical shadows. Physical Review Letters, 127 (11): 110504, September 2021. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.127.110504.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.110504

[62] Seonghoon Choi, Tzu-Ching Yen, and Artur F. Izmaylov. Improving quantum measurements by introducing "ghost" Pauli products. Journal of Chemical Theory and Computation, 18 (12): 7394–7402, December 2022. ISSN 1549-9618, 1549-9626. 10.1021/​acs.jctc.2c00837.
https:/​/​doi.org/​10.1021/​acs.jctc.2c00837

[63] Alexander Gresch and Martin Kliesch. Guaranteed efficient energy estimation of quantum many-body Hamiltonians using ShadowGrouping, September 2023. URL https:/​/​doi.org/​10.48550/​arXiv.2301.03385.
https:/​/​doi.org/​10.48550/​arXiv.2301.03385

[64] Emiel Koridon, Saad Yalouz, Bruno Senjean, Francesco Buda, Thomas E. O'Brien, and Lucas Visscher. Orbital transformations to reduce the 1-norm of the electronic structure hamiltonian for quantum computing applications. Phys. Rev. Res., 3: 033127, Aug 2021. 10.1103/​PhysRevResearch.3.033127.
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.033127

[65] Edward G. Hohenstein, Oumarou Oumarou, Rachael Al-Saadon, Gian-Luca R. Anselmetti, Maximilian Scheurer, Christian Gogolin, and Robert M. Parrish. Efficient Quantum Analytic Nuclear Gradients with Double Factorization, July 2022. URL https:/​/​doi.org/​10.48550/​arXiv.2207.13144.
https:/​/​doi.org/​10.48550/​arXiv.2207.13144

[66] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin. General parameter-shift rules for quantum gradients. Quantum, 6: 677, March 2022. ISSN 2521-327X. 10.22331/​q-2022-03-30-677. URL https:/​/​doi.org/​10.22331/​q-2022-03-30-677.
https:/​/​doi.org/​10.22331/​q-2022-03-30-677

[67] Nicholas C Rubin, Ryan Babbush, and Jarrod McClean. Application of fermionic marginal constraints to hybrid quantum algorithms. New Journal of Physics, 20 (5): 053020, may 2018. 10.1088/​1367-2630/​aab919. URL https:/​/​dx.doi.org/​10.1088/​1367-2630/​aab919.
https:/​/​doi.org/​10.1088/​1367-2630/​aab919

[68] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum Natural Gradient. Quantum, 4: 269, May 2020. ISSN 2521-327X. 10.22331/​q-2020-05-25-269. URL https:/​/​doi.org/​10.22331/​q-2020-05-25-269.
https:/​/​doi.org/​10.22331/​q-2020-05-25-269

[69] Johannes Jakob Meyer. Fisher Information in Noisy Intermediate-Scale Quantum Applications. Quantum, 5: 539, September 2021. ISSN 2521-327X. 10.22331/​q-2021-09-09-539.
https:/​/​doi.org/​10.22331/​q-2021-09-09-539

[70] Shun-ichi Amari. Natural Gradient Works Efficiently in Learning. Neural Computation, 10 (2): 251–276, 02 1998. ISSN 0899-7667. 10.1162/​089976698300017746.
https:/​/​doi.org/​10.1162/​089976698300017746

[71] Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, and James Stokes. Fisher-Rao Metric, Geometry, and Complexity of Neural Networks, February 2019. URL https:/​/​doi.org/​10.48550/​arXiv.1711.01530.
https:/​/​doi.org/​10.48550/​arXiv.1711.01530

[72] János K. Asóth, László Oroszlány, and András Pályi. A short course on topological insulators: band structure and edge states in one and two dimensions. Springer, 2016. ISBN 9783319256078 9783319256054.

[73] J. Zak. Berry's phase for energy bands in solids. Phys. Rev. Lett., 62: 2747–2750, Jun 1989. 10.1103/​PhysRevLett.62.2747.
https:/​/​doi.org/​10.1103/​PhysRevLett.62.2747

[74] Yasuhiro Hatsugai. Quantized berry phases as a local order parameter of a quantum liquid. Journal of the Physical Society of Japan, 75 (12): 123601, 2006. 10.1143/​JPSJ.75.123601.
https:/​/​doi.org/​10.1143/​JPSJ.75.123601

[75] Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki. Chern numbers in discretized brillouin zone: Efficient method of computing (spin) hall conductances. Journal of the Physical Society of Japan, 74 (6): 1674–1677, 2005. 10.1143/​JPSJ.74.1674.
https:/​/​doi.org/​10.1143/​JPSJ.74.1674

[76] Shiing-shen Chern. Characteristic Classes of Hermitian Manifolds. Annals of Mathematics, 47 (1): 85–121, 1946. ISSN 0003-486X. 10.2307/​1969037.
https:/​/​doi.org/​10.2307/​1969037

[77] Roberta Citro and Monika Aidelsburger. Thouless pumping and topology. Nature Reviews Physics, 5 (2): 87–101, January 2023. ISSN 2522-5820. 10.1038/​s42254-022-00545-0.
https:/​/​doi.org/​10.1038/​s42254-022-00545-0

[78] D. J. Thouless. Stability conditions and nuclear rotations in the Hartree-Fock theory. Nuclear Physics, 21: 225–232, November 1960. ISSN 0029-5582. 10.1016/​0029-5582(60)90048-1.
https:/​/​doi.org/​10.1016/​0029-5582(60)90048-1

Cited by

[1] Yuchen Wang and David A. Mazziotti, "Quantum simulation of conical intersections", Physical Chemistry Chemical Physics (2024).

[2] Kumar J. B. Ghosh and Sumit Ghosh, "Exploring exotic configurations with anomalous features with deep learning: Application of classical and quantum-classical hybrid anomaly detection", Physical Review B 108 16, 165408 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-12 04:19:30) and SAO/NASA ADS (last updated successfully 2024-04-12 04:19:32). The list may be incomplete as not all publishers provide suitable and complete citation data.