{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T16:30:20Z","timestamp":1774456220765,"version":"3.50.1"},"reference-count":110,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,4,9]],"date-time":"2024-04-09T00:00:00Z","timestamp":1712620800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004359","name":"Swedish Research Council","doi-asserted-by":"crossref","award":["016-06059"],"award-info":[{"award-number":["016-06059"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004359","name":"Swedish Research Council","doi-asserted-by":"crossref","award":["2018-05973"],"award-info":[{"award-number":["2018-05973"]}],"id":[{"id":"10.13039\/501100004359","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Marie Sk\u0142odowska-Curie","award":["01062864"],"award-info":[{"award-number":["01062864"]}]},{"name":"Marie Sk\u0142odowska-Curie","award":["101065117"],"award-info":[{"award-number":["101065117"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Variational quantum algorithms (VQAs) represent a promising approach to utilizing current quantum computing infrastructures. VQAs are based on a parameterized quantum circuit optimized in a closed loop via a classical algorithm. This hybrid approach reduces the quantum processing unit load but comes at the cost of a classical optimization that can feature a flat energy landscape. Existing optimization techniques, including either imaginary time-propagation, natural gradient, or momentum-based approaches, are promising candidates but place either a significant burden on the quantum device or suffer frequently from slow convergence. In this work, we propose the quantum Broyden adaptive natural gradient (qBang) approach, a novel optimizer that aims to distill the best aspects of existing approaches. By employing the Broyden approach to approximate updates in the Fisher information matrix and combining it with a momentum-based algorithm, qBang reduces quantum-resource requirements while performing better than more resource-demanding alternatives. Benchmarks for the barren plateau, quantum chemistry, and the max-cut problem demonstrate an overall stable performance with a clear improvement over existing techniques in the case of flat (but not exponentially flat) optimization landscapes. qBang introduces a new development strategy for gradient-based VQAs with a plethora of possible improvements.<\/jats:p>","DOI":"10.22331\/q-2024-04-09-1313","type":"journal-article","created":{"date-parts":[[2024,4,9]],"date-time":"2024-04-09T11:38:37Z","timestamp":1712662717000},"page":"1313","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":["Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4268-5485","authenticated-orcid":false,"given":"David","family":"Fitzek","sequence":"first","affiliation":[{"name":"Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, 412 96 Gothenburg, Sweden"},{"name":"Volvo Group Trucks Technology, 405 08 Gothenburg, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8235-3058","authenticated-orcid":false,"given":"Robert S.","family":"Jonsson","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, 412 96 Gothenburg, Sweden"},{"name":"Future Technologies, Saab Surveillance, 412 76 Gothenburg, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6479-1874","authenticated-orcid":false,"given":"Werner","family":"Dobrautz","sequence":"additional","affiliation":[{"name":"Department of Chemistry and Chemical Engineering, Chalmers University of Technology, 412 96 Gothenburg, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8557-733X","authenticated-orcid":false,"given":"Christian","family":"Sch\u00e4fer","sequence":"additional","affiliation":[{"name":"Department of Microtechnology and Nanoscience, MC2, Chalmers University of Technology, 412 96 Gothenburg, Sweden"},{"name":"Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden"}]}],"member":"9598","published-online":{"date-parts":[[2024,4,9]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles. ``Variational quantum algorithms&apos;&apos;. Nature Reviews Physics 3, 625\u2013644 (2021).","DOI":"10.1038\/s42254-021-00348-9"},{"key":"1","doi-asserted-by":"publisher","unstructured":"K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik. ``Noisy intermediate-scale quantum algorithms&apos;&apos;. Reviews of Modern Physics 94, 015004 (2022).","DOI":"10.1103\/RevModPhys.94.015004"},{"key":"2","doi-asserted-by":"publisher","unstructured":"J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson. ``The Variational Quantum Eigensolver: A review of methods and best practices&apos;&apos;. Physics Reports 986, 1\u2013128 (2022).","DOI":"10.1016\/j.physrep.2022.08.003"},{"key":"3","doi-asserted-by":"publisher","unstructured":"F. Arute et al. ``Quantum supremacy using a programmable superconducting processor.&apos;&apos;. Nature 574, 505\u2013510 (2019).","DOI":"10.1038\/s41586-019-1666-5"},{"key":"4","doi-asserted-by":"publisher","unstructured":"C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage. ``Trapped-ion quantum computing: Progress and challenges&apos;&apos;. Applied Physics Reviews 6, 021314 (2019).","DOI":"10.1063\/1.5088164"},{"key":"5","doi-asserted-by":"publisher","unstructured":"A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller. ``Practical quantum advantage in quantum simulation&apos;&apos;. Nature 607, 667\u2013676 (2022).","DOI":"10.1038\/s41586-022-04940-6"},{"key":"6","doi-asserted-by":"publisher","unstructured":"S. Bravyi, O. Dial, J. M. Gambetta, D. Gil, and Z. Nazario. ``The future of quantum computing with superconducting qubits&apos;&apos;. Journal of Applied Physics 132, 160902 (2022).","DOI":"10.1063\/5.0082975"},{"key":"7","doi-asserted-by":"publisher","unstructured":"J. Preskill. ``Quantum computing in the NISQ era and beyond&apos;&apos;. Quantum 2, 79 (2018).","DOI":"10.22331\/q-2018-08-06-79"},{"key":"8","doi-asserted-by":"publisher","unstructured":"A. Peruzzo, J. McClean, P. Shadbolt, M. H. Yung, X. Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O&apos;Brien. ``A variational eigenvalue solver on a photonic quantum processor&apos;&apos;. Nature Communications 5 (2014).","DOI":"10.1038\/ncomms5213"},{"key":"9","doi-asserted-by":"publisher","unstructured":"D. Wecker, M. B. Hastings, and M. Troyer. ``Progress towards practical quantum variational algorithms&apos;&apos;. Phys. Rev. A 92, 042303 (2015).","DOI":"10.1103\/PhysRevA.92.042303"},{"key":"10","doi-asserted-by":"publisher","unstructured":"J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik. ``The theory of variational hybrid quantum-classical algorithms&apos;&apos;. New Journal of Physics 18, 023023 (2016).","DOI":"10.1088\/1367-2630\/18\/2\/023023"},{"key":"11","doi-asserted-by":"publisher","unstructured":"S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan. ``Hybrid quantum-classical algorithms and quantum error mitigation&apos;&apos;. Journal of the Physical Society of Japan 90, 032001 (2021).","DOI":"10.7566\/jpsj.90.032001"},{"key":"12","unstructured":"D. P. Kingma and J. Ba. ``Adam: A method for stochastic optimization&apos;&apos; (2017). arXiv:1412.6980."},{"key":"13","doi-asserted-by":"publisher","unstructured":"K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii. ``Quantum circuit learning&apos;&apos;. Physical Review A 98, 032309 (2018).","DOI":"10.1103\/PhysRevA.98.032309"},{"key":"14","doi-asserted-by":"publisher","unstructured":"L. Banchi and G. E. Crooks. ``Measuring analytic gradients of general quantum evolution with the stochastic parameter shift rule&apos;&apos;. Quantum 5, 386 (2021).","DOI":"10.22331\/q-2021-01-25-386"},{"key":"15","doi-asserted-by":"publisher","unstructured":"M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran. ``Evaluating analytic gradients on quantum hardware&apos;&apos;. Physical Review A 99, 032331 (2019).","DOI":"10.1103\/PhysRevA.99.032331"},{"key":"16","doi-asserted-by":"publisher","unstructured":"L. D&apos;Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol. ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics&apos;&apos;. Advances in Physics 65, 239\u2013362 (2016).","DOI":"10.1080\/00018732.2016.1198134"},{"key":"17","doi-asserted-by":"publisher","unstructured":"J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven. ``Barren plateaus in quantum neural network training landscapes&apos;&apos;. Nature Communications 9, 4812 (2018).","DOI":"10.1038\/s41467-018-07090-4"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles. ``Connecting ansatz expressibility to gradient magnitudes and barren plateaus&apos;&apos;. PRX Quantum 3, 010313 (2022).","DOI":"10.1103\/PRXQuantum.3.010313"},{"key":"19","doi-asserted-by":"publisher","unstructured":"M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles. ``Cost function dependent barren plateaus in shallow parametrized quantum circuits&apos;&apos;. Nature Communications 12, 1791 (2021).","DOI":"10.1038\/s41467-021-21728-w"},{"key":"20","doi-asserted-by":"publisher","unstructured":"S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles. ``Noise-induced barren plateaus in variational quantum algorithms&apos;&apos;. Nature Communications 12 (2021).","DOI":"10.1038\/s41467-021-27045-6"},{"key":"21","doi-asserted-by":"publisher","unstructured":"J. Stokes, J. Izaac, N. Killoran, and G. Carleo. ``Quantum Natural Gradient&apos;&apos;. Quantum 4, 269 (2020).","DOI":"10.22331\/q-2020-05-25-269"},{"key":"22","doi-asserted-by":"publisher","unstructured":"J. Gacon, C. Zoufal, G. Carleo, and S. Woerner. ``Simultaneous perturbation stochastic approximation of the quantum Fisher information&apos;&apos;. Quantum 5, 567 (2021).","DOI":"10.22331\/q-2021-10-20-567"},{"key":"23","doi-asserted-by":"publisher","unstructured":"J. Liu, H. Yuan, X.-M. Lu, and X. Wang. ``Quantum Fisher information matrix and multiparater estimation&apos;&apos;. Journal of Physics A: Mathematical and Theoretical 53, 023001 (2020).","DOI":"10.1088\/1751-8121\/ab5d4d"},{"key":"24","doi-asserted-by":"publisher","unstructured":"D. Wierichs, C. Gogolin, and M. Kastoryano. ``Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer&apos;&apos;. Physical Review Research 2, 043246 (2020).","DOI":"10.1103\/PhysRevResearch.2.043246"},{"key":"25","doi-asserted-by":"publisher","unstructured":"B. Koczor and S. C. Benjamin. ``Quantum natural gradient generalized to noisy and nonunitary circuits&apos;&apos;. Phys. Rev. A 106, 062416 (2022).","DOI":"10.1103\/PhysRevA.106.062416"},{"key":"26","doi-asserted-by":"publisher","unstructured":"J. L. Beckey, M. Cerezo, A. Sone, and P. J. Coles. ``Variational quantum algorithm for estimating the quantum Fisher information&apos;&apos;. Physical Review Research 4, 013083 (2022).","DOI":"10.1103\/PhysRevResearch.4.013083"},{"key":"27","doi-asserted-by":"publisher","unstructured":"J. Gacon, J. Nys, R. Rossi, S. Woerner, and G. Carleo. ``Variational quantum time evolution without the quantum geometric tensor&apos;&apos;. Phys. Rev. Res. 6, 013143 (2024).","DOI":"10.1103\/PhysRevResearch.6.013143"},{"key":"28","doi-asserted-by":"publisher","unstructured":"C. G. Broyden. ``The convergence of a class of double-rank minimization algorithms 1. General considerations&apos;&apos;. IMA Journal of Applied Mathematics 6, 76\u201390 (1970).","DOI":"10.1093\/imamat\/6.1.76"},{"key":"29","doi-asserted-by":"publisher","unstructured":"M. Motta, C. Sun, A. T. K. Tan, M. J. O. Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brandao, and G. K.-L. Chan. ``Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution&apos;&apos;. Nature Physics 16, 205\u2013210 (2020).","DOI":"10.1038\/s41567-019-0704-4"},{"key":"30","doi-asserted-by":"publisher","unstructured":"S. McArdle, T. Jones, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan. ``Variational ansatz-based quantum simulation of imaginary time evolution&apos;&apos;. npj Quantum Information 5, 75 (2019).","DOI":"10.1038\/s41534-019-0187-2"},{"key":"31","doi-asserted-by":"publisher","unstructured":"X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. Benjamin. ``Theory of variational quantum simulation&apos;&apos;. Quantum 3, 191 (2019).","DOI":"10.22331\/q-2019-10-07-191"},{"key":"32","doi-asserted-by":"publisher","unstructured":"C. Cao, Z. An, S.-Y. Hou, D. L. Zhou, and B. Zeng. ``Quantum imaginary time evolution steered by reinforcement learning&apos;&apos;. Communications Physics 5, 57 (2022).","DOI":"10.1038\/s42005-022-00837-y"},{"key":"33","doi-asserted-by":"publisher","unstructured":"V. Havl\u00ed\u010dek, A. D. C\u00f3rcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta. ``Supervised learning with quantum-enhanced feature spaces&apos;&apos;. Nature 567, 209\u2013212 (2019).","DOI":"10.1038\/s41586-019-0980-2"},{"key":"34","doi-asserted-by":"publisher","unstructured":"A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta. ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets&apos;&apos;. Nature 549, 242\u2013246 (2017).","DOI":"10.1038\/nature23879"},{"key":"35","unstructured":"E. Farhi, J. Goldstone, and S. Gutmann. ``A Quantum Approximate Optimization Algorithm&apos;&apos; (2014). arXiv:1411.4028."},{"key":"36","doi-asserted-by":"publisher","unstructured":"S. Sim, P. D. Johnson, and A. Aspuru-Guzik. ``Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms&apos;&apos;. Advanced Quantum Technologies 2, 1900070 (2019).","DOI":"10.1002\/qute.201900070"},{"key":"37","doi-asserted-by":"publisher","unstructured":"D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin. ``General parameter-shift rules for quantum gradients&apos;&apos;. Quantum 6, 677 (2022).","DOI":"10.22331\/q-2022-03-30-677"},{"key":"38","doi-asserted-by":"publisher","unstructured":"A. Lucas. ``Ising formulations of many NP problems&apos;&apos;. Frontiers in Physics 2, 1\u201314 (2014).","DOI":"10.3389\/fphy.2014.00005"},{"key":"39","doi-asserted-by":"publisher","unstructured":"S. Hadfield, Z. Wang, B. O&apos;Gorman, E. G. Rieffel, D. Venturelli, and R. Biswas. ``From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz&apos;&apos;. Algorithms 12, 34 (2019).","DOI":"10.3390\/a12020034"},{"key":"40","unstructured":"M. Svensson, M. Andersson, M. Gr\u00f6nkvist, P. Vikst\u00e5l, D. Dubhashi, G. Ferrini, and G. Johansson. ``A Heuristic Method to solve large-scale Integer Linear Programs by combining Branch-and-Price with a Quantum Algorithm&apos;&apos; (2021). arXiv:2103.15433."},{"key":"41","doi-asserted-by":"publisher","unstructured":"W. Lavrijsen, A. Tudor, J. M\u00fcller, C. Iancu, and W. de Jong. ``Classical optimizers for noisy intermediate-scale quantum devices&apos;&apos;. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE). Pages 267\u2013277. (2020).","DOI":"10.1109\/QCE49297.2020.00041"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. Johnson, M. Kieferov\u00e1, I. D. Kivlichan, T. Menke, B. Peropadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik. ``Quantum chemistry in the age of quantum computing&apos;&apos;. Chemical Reviews 119, 10856\u201310915 (2019).","DOI":"10.1021\/acs.chemrev.8b00803"},{"key":"43","doi-asserted-by":"publisher","unstructured":"V. Lordi and J. M. Nichol. ``Advances and opportunities in materials science for scalable quantum computing&apos;&apos;. MRS Bulletin 46, 589\u2013595 (2021).","DOI":"10.1557\/s43577-021-00133-0"},{"key":"44","unstructured":"G. E. Crooks. ``Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition&apos;&apos; (2019). quant-ph:1905.13311."},{"key":"45","unstructured":"J. Martens. ``New insights and perspectives on the natural gradient method&apos;&apos;. Journal of Machine Learning Research 21, 1\u201376 (2020). url: https:\/\/www.jmlr.org\/papers\/v21\/17-678.html."},{"key":"46","doi-asserted-by":"publisher","unstructured":"J. Martens and I. Sutskever. ``Training deep and recurrent networks with Hessian-free optimization&apos;&apos;. Pages 479\u2013535. Springer Berlin Heidelberg. (2012).","DOI":"10.1007\/978-3-642-35289-8_27"},{"key":"47","doi-asserted-by":"publisher","unstructured":"D. F. Shanno. ``Conditioning of quasi-Newton methods for function minimization&apos;&apos;. Mathematics of Computation 24, 647\u2013656 (1970).","DOI":"10.1090\/s0025-5718-1970-0274029-x"},{"key":"48","doi-asserted-by":"publisher","unstructured":"R. Fletcher. ``A new approach to variable metric algorithms&apos;&apos;. The Computer Journal 13, 317\u2013322 (1970).","DOI":"10.1093\/comjnl\/13.3.317"},{"key":"49","doi-asserted-by":"publisher","unstructured":"D. Goldfarb. ``A family of variable-metric methods derived by variational means&apos;&apos;. Mathematics of Computation 24, 23\u201326 (1970).","DOI":"10.1090\/s0025-5718-1970-0258249-6"},{"key":"50","unstructured":"S. Ruder. ``An overview of gradient descent optimization algorithms&apos;&apos; (2016). arXiv:1609.04747."},{"key":"51","doi-asserted-by":"publisher","unstructured":"G. C. Wick. ``Properties of Bethe-Salpeter wave functions&apos;&apos;. Phys. Rev. 96, 1124\u20131134 (1954).","DOI":"10.1103\/PhysRev.96.1124"},{"key":"52","doi-asserted-by":"publisher","unstructured":"T. Tsuchimochi, Y. Ryo, S. L. Ten-no, and K. Sasasako. ``Improved algorithms of quantum imaginary time evolution for ground and excited states of molecular systems&apos;&apos;. Journal of Chemical Theory and Computation (2023).","DOI":"10.1021\/acs.jctc.2c00906"},{"key":"53","doi-asserted-by":"publisher","unstructured":"W. von der Linden. ``A quantum Monte Carlo approach to many-body physics&apos;&apos;. Physics Reports 220, 53\u2013162 (1992).","DOI":"10.1016\/0370-1573(92)90029-y"},{"key":"54","doi-asserted-by":"publisher","unstructured":"D. M. Ceperley. ``Path integrals in the theory of condensed helium&apos;&apos;. Rev. Mod. Phys. 67, 279\u2013355 (1995).","DOI":"10.1103\/RevModPhys.67.279"},{"key":"55","doi-asserted-by":"publisher","unstructured":"N. Trivedi and D. M. Ceperley. ``Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study&apos;&apos;. Phys. Rev. B 41, 4552\u20134569 (1990).","DOI":"10.1103\/PhysRevB.41.4552"},{"key":"56","doi-asserted-by":"publisher","unstructured":"K. Guther, R. J. Anderson, N. S. Blunt, N. A. Bogdanov, D. Cleland, N. Dattani, W. Dobrautz, K. Ghanem, P. Jeszenszki, N. Liebermann, et al. ``NECI: N-Electron Configuration Interaction with an emphasis on state-of-the-art stochastic methods&apos;&apos;. The Journal of Chemical Physics 153, 034107 (2020).","DOI":"10.1063\/5.0005754"},{"key":"57","doi-asserted-by":"publisher","unstructured":"A. McLachlan. ``A variational solution of the time-dependent Schrodinger equation&apos;&apos;. Molecular Physics 8, 39\u201344 (1964).","DOI":"10.1080\/00268976400100041"},{"key":"58","doi-asserted-by":"publisher","unstructured":"C. Zoufal, D. Sutter, and S. Woerner. ``Error bounds for variational quantum time evolution&apos;&apos;. Phys. Rev. Appl. 20, 044059 (2023).","DOI":"10.1103\/PhysRevApplied.20.044059"},{"key":"59","doi-asserted-by":"publisher","unstructured":"G. Fubini. ``Sulla teoria delle funzioni automorfe e delle loro trasformazioni&apos;&apos;. Annali di Matematica Pura ed Applicata 14, 33\u201367 (1908).","DOI":"10.1007\/bf02420184"},{"key":"60","doi-asserted-by":"publisher","unstructured":"E. Study. ``K\u00fcrzeste wege im komplexen gebiet&apos;&apos;. Mathematische Annalen 60, 321\u2013378 (1905).","DOI":"10.1007\/bf01457616"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Y. Yao, P. Cussenot, R. A. Wolf, and F. Miatto. ``Complex natural gradient optimization for optical quantum circuit design&apos;&apos;. Phys. Rev. A 105, 052402 (2022).","DOI":"10.1103\/PhysRevA.105.052402"},{"key":"62","doi-asserted-by":"publisher","unstructured":"F. Wilczek and A. Shapere. ``Geometric phases in physics&apos;&apos;. World Scientific Publishing. (1989).","DOI":"10.1142\/0613"},{"key":"63","doi-asserted-by":"publisher","unstructured":"L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and J. I. Cirac. ``Geometry of variational methods: dynamics of closed quantum systems&apos;&apos;. SciPost Phys. 9, 048 (2020).","DOI":"10.21468\/SciPostPhys.9.4.048"},{"key":"64","unstructured":"S. Zhou and L. Jiang. ``An exact correspondence between the quantum Fisher information and the Bures metric&apos;&apos; (2019). arXiv:1910.08473."},{"key":"65","doi-asserted-by":"publisher","unstructured":"V. Giovannetti, S. Lloyd, and L. Maccone. ``Advances in quantum metrology&apos;&apos;. Nature Photonics 5, 222\u2013229 (2011).","DOI":"10.1038\/nphoton.2011.35"},{"key":"66","doi-asserted-by":"publisher","unstructured":"D. Petz and C. Sud\u00e1r. ``Geometries of quantum states&apos;&apos;. Journal of Mathematical Physics 37, 2662\u20132673 (1996).","DOI":"10.1063\/1.531535"},{"key":"67","doi-asserted-by":"publisher","unstructured":"J. P. Provost and G. Vallee. ``Riemannian structure on manifolds of quantum states&apos;&apos;. Communications in Mathematical Physics 76, 289\u2013301 (1980).","DOI":"10.1007\/bf02193559"},{"key":"68","doi-asserted-by":"publisher","unstructured":"C.-Y. Park and M. J. Kastoryano. ``Geometry of learning neural quantum states&apos;&apos;. Physical Review Research 2, 023232 (2020).","DOI":"10.1103\/PhysRevResearch.2.023232"},{"key":"69","doi-asserted-by":"publisher","unstructured":"S. L. Braunstein and C. M. Caves. ``Statistical distance and the geometry of quantum states&apos;&apos;. Phys. Rev. Lett. 72, 3439\u20133443 (1994).","DOI":"10.1103\/PhysRevLett.72.3439"},{"key":"70","doi-asserted-by":"publisher","unstructured":"P. Facchi, R. Kulkarni, V. Man&apos;ko, G. Marmo, E. Sudarshan, and F. Ventriglia. ``Classical and quantum Fisher information in the geometrical formulation of quantum mechanics&apos;&apos;. Physics Letters A 374, 4801\u20134803 (2010).","DOI":"10.1016\/j.physleta.2010.10.005"},{"key":"71","doi-asserted-by":"publisher","unstructured":"S.-I. Amari. ``Neural learning in structured parameter spaces: natural Riemannian gradient&apos;&apos;. In Proceedings of the 9th International Conference on Neural Information Processing Systems. Pages 127\u2013\u2013133. NIPS&apos;96. MIT Press (1996).","DOI":"10.5555\/2998981.2998999"},{"key":"72","doi-asserted-by":"publisher","unstructured":"S.-i. Amari. ``Natural gradient works efficiently in learning&apos;&apos;. Neural Computation 10, 251\u2013276 (1998).","DOI":"10.1162\/089976698300017746"},{"key":"73","doi-asserted-by":"publisher","unstructured":"S.-i. Amari and S. Douglas. ``Why natural gradient?&apos;&apos;. In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP &apos;98 (Cat. No.98CH36181). Volume 2, pages 1213\u20131216. (1998).","DOI":"10.1109\/ICASSP.1998.675489"},{"key":"74","doi-asserted-by":"publisher","unstructured":"S.-i. Amari, H. Park, and K. Fukumizu. ``Adaptive method of realizing natural gradient learning for multilayer perceptrons&apos;&apos;. Neural Computation 12, 1399\u20131409 (2000).","DOI":"10.1162\/089976600300015420"},{"key":"75","doi-asserted-by":"publisher","unstructured":"J. J. Meyer. ``Fisher information in noisy intermediate-scale quantum applications&apos;&apos;. Quantum 5, 539 (2021).","DOI":"10.22331\/q-2021-09-09-539"},{"key":"76","doi-asserted-by":"publisher","unstructured":"P. Huembeli and A. Dauphin. ``Characterizing the loss landscape of variational quantum circuits&apos;&apos;. Quantum Science and Technology 6, 025011 (2021).","DOI":"10.1088\/2058-9565\/abdbc9"},{"key":"77","doi-asserted-by":"publisher","unstructured":"E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti. ``An initialization strategy for addressing barren plateaus in parametrized quantum circuits&apos;&apos;. Quantum 3, 214 (2019).","DOI":"10.22331\/q-2019-12-09-214"},{"key":"78","doi-asserted-by":"publisher","unstructured":"I. O. Sokolov, W. Dobrautz, H. Luo, A. Alavi, and I. Tavernelli. ``Orders of magnitude increased accuracy for quantum many-body problems on quantum computers via an exact transcorrelated method&apos;&apos;. Phys. Rev. Res. 5, 023174 (2023).","DOI":"10.1103\/PhysRevResearch.5.023174"},{"key":"79","unstructured":"W. Dobrautz, I. O. Sokolov, K. Liao, P. L. R\u00edos, M. Rahm, A. Alavi, and I. Tavernelli. ``Ab initio transcorrelated method enabling accurate quantum chemistry on near-term quantum hardware&apos;&apos; (2023). arXiv:2303.02007."},{"key":"80","doi-asserted-by":"publisher","unstructured":"T. R. Bromley, J. M. Arrazola, S. Jahangiri, J. Izaac, N. Quesada, A. D. Gran, M. Schuld, J. Swinarton, Z. Zabaneh, and N. Killoran. ``Applications of near-term photonic quantum computers: software and algorithms&apos;&apos;. Quantum Science and Technology 5, 034010 (2020).","DOI":"10.1088\/2058-9565\/ab8504"},{"key":"81","doi-asserted-by":"publisher","unstructured":"H. Park, S.-i. Amari, and K. Fukumizu. ``Adaptive natural gradient learning algorithms for various stochastic models&apos;&apos;. Neural Networks 13, 755\u2013\u2013764 (2000).","DOI":"10.1016\/S0893-6080(00)00051-4"},{"key":"82","doi-asserted-by":"publisher","unstructured":"S.-i. Amari. ``Information geometry and its applications&apos;&apos;. Springer. (2016).","DOI":"10.1007\/978-4-431-55978-8"},{"key":"83","doi-asserted-by":"crossref","unstructured":"S. Dash, F. Vicentini, M. Ferrero, and A. Georges. ``Efficiency of neural quantum states in light of the quantum geometric tensor&apos;&apos; (2024). arXiv:2402.01565.","DOI":"10.21203\/rs.3.rs-3964607\/v1"},{"key":"84","unstructured":"D. Fitzek, R. S. Jonsson, W. Dobrautz, and C. Sch\u00e4fer (2023). code: davidfitzek\/qflow."},{"key":"85","doi-asserted-by":"publisher","unstructured":"B. van Straaten and B. Koczor. ``Measurement cost of metric-aware variational quantum algorithms&apos;&apos;. PRX Quantum 2, 030324 (2021).","DOI":"10.1103\/PRXQuantum.2.030324"},{"key":"86","doi-asserted-by":"publisher","unstructured":"A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola. ``Numerical methods for the solution of ill-posed problems&apos;&apos;. Springer Dordrecht. (1995).","DOI":"10.1007\/978-94-015-8480-7"},{"key":"87","unstructured":"V. Bergholm, J. Izaac, M. Schuld, et al. ``PennyLane: Automatic differentiation of hybrid quantum-classical computations&apos;&apos; (2018). arXiv:1811.04968."},{"key":"88","doi-asserted-by":"publisher","unstructured":"T. Helgaker, P. J\u00f8rgensen, and J. Olsen. ``Molecular electronic-structure theory&apos;&apos;. John Wiley & Sons. (2000).","DOI":"10.1002\/9781119019572"},{"key":"89","doi-asserted-by":"publisher","unstructured":"Q. Sun, X. Zhang, S. Banerjee, P. Bao, et al. ``Recent developments in the PySCF program package&apos;&apos;. The Journal of Chemical Physics 153, 024109 (2020).","DOI":"10.1063\/5.0006074"},{"key":"90","doi-asserted-by":"publisher","unstructured":"J. Nocedal and S. J. Wright. ``Numerical optimization&apos;&apos;. Springer Science+Business Media. (2006).","DOI":"10.1007\/978-0-387-40065-5"},{"key":"91","doi-asserted-by":"publisher","unstructured":"J. M. K\u00fcbler, A. Arrasmith, L. Cincio, and P. J. Coles. ``An Adaptive Optimizer for Measurement-Frugal Variational Algorithms&apos;&apos;. Quantum 4, 263 (2020).","DOI":"10.22331\/q-2020-05-11-263"},{"key":"92","unstructured":"D. Fitzek, R. S. Jonsson, W. Dobrautz, and C. Sch\u00e4fer (2023). code: davidfitzek\/qbang."},{"key":"93","unstructured":"M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. O. Marrero, M. Larocca, and M. Cerezo. ``A unified theory of barren plateaus for deep parametrized quantum circuits&apos;&apos; (2023). arXiv:2309.09342."},{"key":"94","unstructured":"E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia. ``The adjoint is all you need: Characterizing barren plateaus in quantum ans\u00e4tze&apos;&apos; (2023). arXiv:2309.07902."},{"key":"95","doi-asserted-by":"publisher","unstructured":"M. Larocca, N. Ju, D. Garc\u00eda-Mart\u00edn, P. J. Coles, and M. Cerezo. ``Theory of overparametrization in quantum neural networks&apos;&apos;. Nature Computational Science 3, 542\u2013551 (2023).","DOI":"10.1038\/s43588-023-00467-6"},{"key":"96","doi-asserted-by":"publisher","unstructured":"Y. Du, M.-H. Hsieh, T. Liu, and D. Tao. ``Expressive power of parametrized quantum circuits&apos;&apos;. Phys. Rev. Res. 2, 033125 (2020).","DOI":"10.1103\/PhysRevResearch.2.033125"},{"key":"97","doi-asserted-by":"publisher","unstructured":"L. Funcke, T. Hartung, K. Jansen, S. K\u00fchn, and P. Stornati. ``Dimensional expressivity analysis of parametric quantum circuits&apos;&apos;. Quantum 5, 422 (2021).","DOI":"10.22331\/q-2021-03-29-422"},{"key":"98","doi-asserted-by":"publisher","unstructured":"Y. Du, Z. Tu, X. Yuan, and D. Tao. ``Efficient measure for the expressivity of variational quantum algorithms&apos;&apos;. Phys. Rev. Lett. 128, 080506 (2022).","DOI":"10.1103\/PhysRevLett.128.080506"},{"key":"99","doi-asserted-by":"publisher","unstructured":"R. D&apos;Cunha, T. D. Crawford, M. Motta, and J. E. Rice. ``Challenges in the use of quantum computing hardware-efficient ans\u00e4tze in electronic structure theory&apos;&apos;. The Journal of Physical Chemistry A (2023).","DOI":"10.1021\/acs.jpca.2c08430"},{"key":"100","doi-asserted-by":"publisher","unstructured":"H. Shima. ``The geometry of Hessian structures&apos;&apos;. World Scientific. (2007).","DOI":"10.1007\/978-3-642-40020-9_4"},{"key":"101","doi-asserted-by":"publisher","unstructured":"L. Campos Venuti and P. Zanardi. ``Quantum critical scaling of the geometric tensors&apos;&apos;. Phys. Rev. Lett. 99, 095701 (2007).","DOI":"10.1103\/PhysRevLett.99.095701"},{"key":"102","doi-asserted-by":"publisher","unstructured":"M. Bukov, D. Sels, and A. Polkovnikov. ``Geometric speed limit of accessible many-body state preparation&apos;&apos;. Phys. Rev. X 9, 011034 (2019).","DOI":"10.1103\/PhysRevX.9.011034"},{"key":"103","doi-asserted-by":"publisher","unstructured":"M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov. ``Geometry and non-adiabatic response in quantum and classical systems&apos;&apos;. Physics Reports 697, 1\u201387 (2017).","DOI":"10.1016\/j.physrep.2017.07.001"},{"key":"104","doi-asserted-by":"publisher","unstructured":"S. Pancharatnam. ``Generalized theory of interference, and its applications&apos;&apos;. Proceedings of the Indian Academy of Sciences - Section A 44, 247\u2013262 (1956).","DOI":"10.1007\/bf03046050"},{"key":"105","doi-asserted-by":"publisher","unstructured":"M. V. Berry. ``Quantal phase factors accompanying adiabatic changes&apos;&apos;. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45\u201357 (1984).","DOI":"10.1098\/rspa.1984.0023"},{"key":"106","doi-asserted-by":"publisher","unstructured":"J. Broeckhove, L. Lathouwers, E. Kesteloot, and P. V. Leuven. ``On the equivalence of time-dependent variational principles&apos;&apos;. Chemical Physics Letters 149, 547\u2013550 (1988).","DOI":"10.1016\/0009-2614(88)80380-4"},{"key":"107","doi-asserted-by":"publisher","unstructured":"S. Sorella. ``Green function Monte Carlo with stochastic reconfiguration&apos;&apos;. Phys. Rev. Lett. 80, 4558\u20134561 (1998).","DOI":"10.1103\/PhysRevLett.80.4558"},{"key":"108","doi-asserted-by":"publisher","unstructured":"S. Sorella and L. Capriotti. ``Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem&apos;&apos;. Phys. Rev. B 61, 2599\u20132612 (2000).","DOI":"10.1103\/PhysRevB.61.2599"},{"key":"109","doi-asserted-by":"publisher","unstructured":"G. Mazzola, A. Zen, and S. Sorella. ``Finite-temperature electronic simulations without the Born-Oppenheimer constraint&apos;&apos;. The Journal of Chemical Physics 137, 134112 (2012).","DOI":"10.1063\/1.4755992"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-04-09-1313\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,4,9]],"date-time":"2024-04-09T11:39:25Z","timestamp":1712662765000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-04-09-1313\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,9]]},"references-count":110,"URL":"https:\/\/doi.org\/10.22331\/q-2024-04-09-1313","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,9]]},"article-number":"1313"}}