{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T17:40:17Z","timestamp":1774633217766,"version":"3.50.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T00:00:00Z","timestamp":1658275200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T00:00:00Z","timestamp":1658275200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004489","name":"Mitacs","doi-asserted-by":"publisher","award":["FR48717"],"award-info":[{"award-number":["FR48717"]}],"id":[{"id":"10.13039\/501100004489","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004489","name":"Mitacs","doi-asserted-by":"publisher","award":["FR46305"],"award-info":[{"award-number":["FR46305"]}],"id":[{"id":"10.13039\/501100004489","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004489","name":"Mitacs","doi-asserted-by":"publisher","award":["FR48717"],"award-info":[{"award-number":["FR48717"]}],"id":[{"id":"10.13039\/501100004489","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011332","name":"Innovation, Science and Economic Development Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100011332","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nat Mach Intell"],"DOI":"10.1038\/s42256-022-00509-0","type":"journal-article","created":{"date-parts":[[2022,7,20]],"date-time":"2022-07-20T16:03:18Z","timestamp":1658332998000},"page":"618-624","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["Neural Error Mitigation of Near-Term Quantum Simulations"],"prefix":"10.1038","volume":"4","author":[{"given":"Elizabeth R.","family":"Bennewitz","sequence":"first","affiliation":[]},{"given":"Florian","family":"Hopfmueller","sequence":"additional","affiliation":[]},{"given":"Bohdan","family":"Kulchytskyy","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7263-3462","authenticated-orcid":false,"given":"Juan","family":"Carrasquilla","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9591-9727","authenticated-orcid":false,"given":"Pooya","family":"Ronagh","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,20]]},"reference":[{"key":"509_CR1","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1007\/BF02650179","volume":"21","author":"RP Feynman","year":"1982","unstructured":"Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21, 467\u2013488 (1982).","journal-title":"Int. J. Theor. Phys."},{"key":"509_CR2","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1147\/rd.176.0525","volume":"17","author":"CH Bennett","year":"1973","unstructured":"Bennett, C. H. Logical reversibility of computation. IBM J. Res. Dev. 17, 525\u2013532 (1973).","journal-title":"IBM J. Res. Dev."},{"key":"509_CR3","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1007\/BF01011339","volume":"22","author":"P Benioff","year":"1980","unstructured":"Benioff, P. The computer as a physical system: a microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22, 563\u2013591 (1980).","journal-title":"J. Stat. Phys."},{"key":"509_CR4","volume-title":"Computable and Noncomputable","author":"Y Manin","year":"1980","unstructured":"Manin, Y. Computable and Noncomputable (in Russian) (Sovetskoye Radio, 1980)."},{"key":"509_CR5","doi-asserted-by":"crossref","unstructured":"Preskill, J. Simulating quantum field theory with a quantum computer. In Proc. 36th Annual International Symposium on Lattice Field Theory (LATTICE2018) 334 (East Lansing, MI, USA, 2018).","DOI":"10.22323\/1.334.0024"},{"key":"509_CR6","doi-asserted-by":"publisher","first-page":"10856","DOI":"10.1021\/acs.chemrev.8b00803","volume":"119","author":"Y Cao","year":"2019","unstructured":"Cao, Y. et al. Quantum chemistry in the age of quantum computing. Chem. Rev. 119, 10856\u201310915 (2019).","journal-title":"Chem. Rev."},{"key":"509_CR7","doi-asserted-by":"publisher","first-page":"015003","DOI":"10.1103\/RevModPhys.92.015003","volume":"92","author":"S McArdle","year":"2020","unstructured":"McArdle, S., Endo, S., Aspuru-Guzik, A., Benjamin, S. C. & Yuan, X. Quantum computational chemistry. Rev. Mod. Phys. 92, 015003 (2020).","journal-title":"Rev. Mod. Phys."},{"key":"509_CR8","doi-asserted-by":"publisher","first-page":"12685","DOI":"10.1021\/acs.chemrev.9b00829","volume":"120","author":"B Bauer","year":"2020","unstructured":"Bauer, B., Bravyi, S., Motta, M. & Kin-Lic Chan, G. Quantum algorithms for quantum chemistry and quantum materials science. Chem. Rev. 120, 12685\u201312717 (2020).","journal-title":"Chem. Rev."},{"key":"509_CR9","doi-asserted-by":"publisher","first-page":"1704","DOI":"10.1126\/science.1113479","volume":"309","author":"A Aspuru-Guzik","year":"2005","unstructured":"Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704\u20131707 (2005).","journal-title":"Science"},{"key":"509_CR10","doi-asserted-by":"publisher","first-page":"015004","DOI":"10.1103\/RevModPhys.94.015004","volume":"94","author":"K Bharti","year":"2022","unstructured":"Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 94, 015004 (2022).","journal-title":"Rev. Mod. Phys."},{"key":"509_CR11","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1038\/s42254-021-00348-9","volume":"3","author":"M Cerezo","year":"2021","unstructured":"Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625\u2013644 (2021).","journal-title":"Nat. Rev. Phys."},{"key":"509_CR12","doi-asserted-by":"publisher","first-page":"4213","DOI":"10.1038\/ncomms5213","volume":"5","author":"A Peruzzo","year":"2014","unstructured":"Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).","journal-title":"Nat. Commun."},{"key":"509_CR13","doi-asserted-by":"publisher","first-page":"032001","DOI":"10.7566\/JPSJ.90.032001","volume":"90","author":"S Endo","year":"2021","unstructured":"Endo, S., Cai, Z., Benjamin, S. C. & Yuan, X. Hybrid quantum-classical algorithms and quantum error mitigation. J. Phys. Soc. Jpn 90, 032001 (2021).","journal-title":"J. Phys. Soc. Jpn"},{"key":"509_CR14","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1080\/00107514.2019.1667078","volume":"60","author":"J Roffe","year":"2019","unstructured":"Roffe, J. Quantum error correction: an introductory guide. Contemp. Phys. 60, 226\u2013245 (2019).","journal-title":"Contemp. Phys."},{"key":"509_CR15","doi-asserted-by":"publisher","first-page":"074001","DOI":"10.1088\/1361-6633\/aab406","volume":"81","author":"V Dunjko","year":"2018","unstructured":"Dunjko, V. & Briegel, H. J. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Rep. Progr. Phys. 81, 074001 (2018).","journal-title":"Rep. Progr. Phys."},{"key":"509_CR16","first-page":"1797528","volume":"5","author":"J Carrasquilla","year":"2020","unstructured":"Carrasquilla, J. Machine learning for quantum matter. Adv. Phys. X 5, 1797528 (2020).","journal-title":"Adv. Phys. X"},{"key":"509_CR17","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1038\/s41567-018-0048-5","volume":"14","author":"G Torlai","year":"2018","unstructured":"Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447\u2013450 (2018).","journal-title":"Nat. Phys."},{"key":"509_CR18","doi-asserted-by":"crossref","unstructured":"Becca, F. & Sorella, S. Quantum Monte Carlo Approaches for Correlated Systems (Cambridge Univ. Press, 2017).","DOI":"10.1017\/9781316417041"},{"key":"509_CR19","unstructured":"Vaswani, A. et al. Attention is all you need. In Proc. 31st Conference on Neural Information Processing Systems (NIPS 2017) (Long Beach, CA, USA, 2017)."},{"key":"509_CR20","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1038\/s41534-021-00404-3","volume":"7","author":"Z Cai","year":"2021","unstructured":"Cai, Z. Multi-exponential error extrapolation and combining error mitigation techniques for NISQ applications. npj Quantum Inf. 7, 80 (2021).","journal-title":"npj Quantum Inf."},{"key":"509_CR21","doi-asserted-by":"publisher","first-page":"230504","DOI":"10.1103\/PhysRevLett.123.230504","volume":"123","author":"G Torlai","year":"2019","unstructured":"Torlai, G. et al. Integrating neural networks with a quantum simulator for state reconstruction. Phys. Rev. Lett. 123, 230504 (2019).","journal-title":"Phys. Rev. Lett."},{"key":"509_CR22","doi-asserted-by":"publisher","first-page":"eaaw5686","DOI":"10.1126\/sciadv.aaw5686","volume":"5","author":"C Song","year":"2019","unstructured":"Song, C. et al. Quantum computation with universal error mitigation on a superconducting quantum processor. Sci. Adv. 5, eaaw5686 (2019).","journal-title":"Sci. Adv."},{"key":"509_CR23","doi-asserted-by":"publisher","first-page":"034026","DOI":"10.1103\/PhysRevApplied.15.034026","volume":"15","author":"J Sun","year":"2021","unstructured":"Sun, J. et al. Mitigating realistic noise in practical noisy intermediate-scale quantum devices. Phys. Rev. Appl. 15, 034026 (2021).","journal-title":"Phys. Rev. Appl."},{"key":"509_CR24","doi-asserted-by":"publisher","first-page":"022060","DOI":"10.1103\/PhysRevResearch.2.022060","volume":"2","author":"G Torlai","year":"2020","unstructured":"Torlai, G., Mazzola, G., Carleo, G. & Mezzacapo, A. Precise measurement of quantum observables with neural-network estimators. Phys. Rev. Res. 2, 022060 (2020).","journal-title":"Phys. Rev. Res."},{"key":"509_CR25","doi-asserted-by":"publisher","first-page":"4682","DOI":"10.1103\/PhysRevLett.83.4682","volume":"83","author":"R Assaraf","year":"1999","unstructured":"Assaraf, R. & Caffarel, M. Zero-variance principle for Monte Carlo algorithms. Phys. Rev. Lett. 83, 4682 (1999).","journal-title":"Phys. Rev. Lett."},{"key":"509_CR26","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1038\/nature23879","volume":"549","author":"A Kandala","year":"2017","unstructured":"Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242\u2013246 (2017).","journal-title":"Nature"},{"key":"509_CR27","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1038\/s41586-019-1177-4","volume":"569","author":"C Kokail","year":"2019","unstructured":"Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355\u2013360 (2019).","journal-title":"Nature"},{"key":"509_CR28","doi-asserted-by":"publisher","first-page":"032331","DOI":"10.1103\/PhysRevA.98.032331","volume":"98","author":"N Klco","year":"2018","unstructured":"Klco, N. et al. Quantum-classical computation of Schwinger model dynamics using quantum computers. Phys. Rev. A 98, 032331 (2018).","journal-title":"Phys. Rev. A"},{"key":"509_CR29","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1038\/nature18318","volume":"534","author":"EA Martinez","year":"2016","unstructured":"Martinez, E. A. et al. Real-time dynamics of lattice gauge theories with a few-qubit quantum computer. Nature 534, 516\u2013519 (2016).","journal-title":"Nature"},{"key":"509_CR30","doi-asserted-by":"publisher","first-page":"144002","DOI":"10.1063\/5.0043322","volume":"118","author":"OV Borzenkova","year":"2021","unstructured":"Borzenkova, O. V. et al. Variational simulation of Schwinger's Hamiltonian with polarization qubits. Appl. Phys. Lett. 118, 144002 (2021).","journal-title":"Appl. Phys. Lett."},{"key":"509_CR31","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1126\/science.aau4963","volume":"364","author":"T Brydges","year":"2019","unstructured":"Brydges, T. et al. Probing Renyi entanglement entropy via randomized measurements. Science 364, 260\u2013263 (2019).","journal-title":"Science"},{"key":"509_CR32","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1038\/nature15750","volume":"528","author":"R Islam","year":"2015","unstructured":"Islam, R. et al. Measuring entanglement entropy in a quantum many-body system. Nature 528, 77\u201383 (2015).","journal-title":"Nature"},{"key":"509_CR33","doi-asserted-by":"publisher","first-page":"025011","DOI":"10.1088\/2058-9565\/abdbc9","volume":"6","author":"P Huembeli","year":"2021","unstructured":"Huembeli, P. & Dauphin, A. Characterizing the loss landscape of variational quantum circuits. Quantum Sci. Technol. 6, 025011 (2021).","journal-title":"Quantum Sci. Technol."},{"key":"509_CR34","doi-asserted-by":"publisher","first-page":"023232","DOI":"10.1103\/PhysRevResearch.2.023232","volume":"2","author":"C-Y Park","year":"2020","unstructured":"Park, C.-Y. & Kastoryano, M. J. Geometry of learning neural quantum states. Phys. Rev. Res. 2, 023232 (2020).","journal-title":"Phys. Rev. Res."},{"key":"509_CR35","doi-asserted-by":"publisher","first-page":"147","DOI":"10.21468\/SciPostPhys.10.6.147","volume":"10","author":"M Bukov","year":"2021","unstructured":"Bukov, M., Schmitt, M. & Dupont, M. Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape. SciPost Phys. 10, 147 (2021).","journal-title":"SciPost Phys."},{"key":"509_CR36","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","volume":"9","author":"S Hochreiter","year":"1997","unstructured":"Hochreiter, S. & Schmidhuber, J. Long short-term memory. Neur. Comput. 9, 1735\u20131780 (1997).","journal-title":"Neur. Comput."},{"key":"509_CR37","doi-asserted-by":"publisher","first-page":"023358","DOI":"10.1103\/PhysRevResearch.2.023358","volume":"2","author":"M Hibat-Allah","year":"2020","unstructured":"Hibat-Allah, M., Ganahl, M., Hayward, L. E., Melko, R. G. & Carrasquilla, J. Recurrent neural-network wavefunctions. Phys. Rev. Res. 2, 023358 (2020).","journal-title":"Phys. Rev. Res."},{"key":"509_CR38","doi-asserted-by":"publisher","first-page":"020503","DOI":"10.1103\/PhysRevLett.124.020503","volume":"124","author":"O Sharir","year":"2020","unstructured":"Sharir, O., Levine, Y., Wies, N., Carleo, G. & Shashua, A. Deep autoregressive models for the efficient variational simulation of many-body quantum systems. Phys. Rev. Lett. 124, 020503 (2020).","journal-title":"Phys. Rev. Lett."},{"key":"509_CR39","doi-asserted-by":"publisher","first-page":"032610","DOI":"10.1103\/PhysRevA.104.032610","volume":"104","author":"J Carrasquilla","year":"2021","unstructured":"Carrasquilla, J. et al. Probabilistic simulation of quantum circuits using a deep-learning architecture. Phys. Rev. A 104, 032610 (2021).","journal-title":"Phys. Rev. A"},{"key":"509_CR40","unstructured":"Kingma, D. P. & Ba, J. Adam: a method for stochastic optimization. Preprint at https:\/\/arxiv.org\/abs\/1412.6980 (2014)."},{"key":"509_CR41","doi-asserted-by":"publisher","first-page":"602","DOI":"10.1126\/science.aag2302","volume":"355","author":"G Carleo","year":"2017","unstructured":"Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602\u2013606 (2017).","journal-title":"Science"},{"key":"509_CR42","doi-asserted-by":"publisher","first-page":"2368","DOI":"10.1038\/s41467-020-15724-9","volume":"11","author":"K Choo","year":"2020","unstructured":"Choo, K., Mezzacapo, A. & Carleo, G. Fermionic neural-network states for ab-initio electronic structure. Nat. Commun. 11, 2368 (2020).","journal-title":"Nat. Commun."},{"key":"509_CR43","doi-asserted-by":"publisher","first-page":"14491","DOI":"10.1039\/C9CP02269D","volume":"21","author":"L Otis","year":"2019","unstructured":"Otis, L. & Neuscamman, E. Complementary first and second derivative methods for ansatz optimization in variational Monte Carlo. Phys. Chem. Chem. Phys. 21, 14491\u201314510 (2019).","journal-title":"Phys. Chem. Chem. Phys."},{"key":"509_CR44","doi-asserted-by":"publisher","first-page":"210","DOI":"10.1006\/aphy.2002.6254","volume":"298","author":"SB Bravyi","year":"2002","unstructured":"Bravyi, S. B. & Kitaev, A. Y. Fermionic quantum computation. Ann. Phys. 298, 210\u2013226 (2002).","journal-title":"Ann. Phys."},{"key":"509_CR45","doi-asserted-by":"publisher","first-page":"224109","DOI":"10.1063\/1.4768229","volume":"137","author":"JT Seeley","year":"2012","unstructured":"Seeley, J. T., Richard, M. J. & Love, P. J. The Bravyi\u2013Kitaev transformation for quantum computation of electronic structure. J. Chem. Phys. 137, 224109 (2012).","journal-title":"J. Chem. Phys."},{"key":"509_CR46","unstructured":"Bravyi, S., Gambetta, J. M., Mezzacapo, A. & Temme, K. Tapering off qubits to simulate fermionic Hamiltonians. Preprint at https:\/\/arxiv.org\/abs\/1701.08213 (2017)."},{"key":"509_CR47","unstructured":"Qiskit Development Team. Qiskit: An Open Source Framework for Quantum Computation, ver. 0.23.0. https:\/\/qiskit.org (IBM, 2019)."},{"key":"509_CR48","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1109\/7.705889","volume":"34","author":"JC Spall","year":"1998","unstructured":"Spall, J. C. Implementation of the simultaneous perturbation algorithm for stochastic optimization. IEEE Trans. Aerosp. Electron. Syst. 34, 817\u2013823 (1998).","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"509_CR49","doi-asserted-by":"publisher","first-page":"021318","DOI":"10.1063\/1.5089550","volume":"6","author":"P Krantz","year":"2019","unstructured":"Krantz, P. et al. A quantum engineer\u2019s guide to superconducting qubits. Appl. Phys. Rev. 6, 021318 (2019).","journal-title":"Appl. Phys. Rev."},{"key":"509_CR50","doi-asserted-by":"publisher","unstructured":"Bennewitz, E. R., Hopfmueller, F., Kulchytskyy, B., Carrasquilla, J. & Ronagh, P. 1QB-Information-Technologies\/NEM: 1QB-Information-Technologies\/Neural Error Mitigation. Zenodo https:\/\/doi.org\/10.5281\/zenodo.6466405 (2022).","DOI":"10.5281\/zenodo.6466405"}],"container-title":["Nature Machine Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s42256-022-00509-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s42256-022-00509-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s42256-022-00509-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T09:37:46Z","timestamp":1669369066000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s42256-022-00509-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,20]]},"references-count":50,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["509"],"URL":"https:\/\/doi.org\/10.1038\/s42256-022-00509-0","relation":{},"ISSN":["2522-5839"],"issn-type":[{"value":"2522-5839","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,20]]},"assertion":[{"value":"23 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 June 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}