{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T23:46:53Z","timestamp":1783036013069,"version":"3.54.6"},"reference-count":89,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T00:00:00Z","timestamp":1615766400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100014553","name":"Samsung Advanced Institute of Technology","doi-asserted-by":"crossref","award":["Global Research Partnership"],"award-info":[{"award-number":["Global Research Partnership"]}],"id":[{"id":"10.13039\/100014553","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000023","name":"Government of Canada","doi-asserted-by":"crossref","award":["Canada First Research Excellence Fund"],"award-info":[{"award-number":["Canada First Research Excellence Fund"]}],"id":[{"id":"10.13039\/501100000023","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Tensor networks represent the state-of-the-art in computational methods across many disciplines, including the classical simulation of quantum many-body systems and quantum circuits. Several applications of current interest give rise to tensor networks with irregular geometries. Finding the best possible contraction path for such networks is a central problem, with an exponential effect on computation time and memory footprint. In this work, we implement new randomized protocols that find very high quality contraction paths for arbitrary and large tensor networks. We test our methods on a variety of benchmarks, including the random quantum circuit instances recently implemented on Google quantum chips. We find that the paths obtained can be very close to optimal, and often many orders or magnitude better than the most established approaches. As different underlying geometries suit different methods, we also introduce a hyper-optimization approach, where both the method applied and its algorithmic parameters are tuned during the path finding. The increase in quality of contraction schemes found has significant practical implications for the simulation of quantum many-body systems and particularly for the benchmarking of new quantum chips. Concretely, we estimate a speed-up of over 10,000<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>\u00d7<\/mml:mo><\/mml:math>compared to the original expectation for the classical simulation of the Sycamore `supremacy' circuits.<\/jats:p>","DOI":"10.22331\/q-2021-03-15-410","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T16:38:12Z","timestamp":1615826292000},"page":"410","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":191,"title":["Hyper-optimized tensor network contraction"],"prefix":"10.22331","volume":"5","author":[{"given":"Johnnie","family":"Gray","sequence":"first","affiliation":[{"name":"Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom"},{"name":"Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefanos","family":"Kourtis","sequence":"additional","affiliation":[{"name":"Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom"},{"name":"Department of Physics, Boston University, Boston, MA, 02215, USA"},{"name":"Institut quantique & D\u00e9partement de physique, Universit\u00e9 de Sherbrooke, Qu\u00e9bec J1K 2R1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2021,3,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"F. Verstraete, V. Murg, and J. Cirac, Adv. Phys. 57, 143 (2008).","DOI":"10.1080\/14789940801912366"},{"key":"1","doi-asserted-by":"publisher","unstructured":"R. Or\u00fas, Ann. Phys. (N. Y). 349, 117 (2014).","DOI":"10.1016\/j.aop.2014.06.013"},{"key":"2","doi-asserted-by":"publisher","unstructured":"J. C. Bridgeman and C. T. Chubb, J. Phys. A Math. Theor. 50, 223001 (2017).","DOI":"10.1088\/1751-8121\/aa6dc3"},{"key":"3","unstructured":"J. D. Biamonte and V. Bergholm, arXiv:1708.00006 (2017), arXiv:1708.00006."},{"key":"4","doi-asserted-by":"publisher","unstructured":"M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007).","DOI":"10.1103\/PhysRevLett.99.120601"},{"key":"5","doi-asserted-by":"publisher","unstructured":"G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015).","DOI":"10.1103\/PhysRevLett.115.180405"},{"key":"6","doi-asserted-by":"publisher","unstructured":"G. Evenbly, Phys. Rev. B 95, 045117 (2017).","DOI":"10.1103\/PhysRevB.95.045117"},{"key":"7","doi-asserted-by":"publisher","unstructured":"A. Cichocki, N. Lee, I. Oseledets, A.-H. Phan, Q. Zhao, and D. P. Mandic, Found. Trends Mach. Learn. 9, 249 (2016).","DOI":"10.1561\/2200000059"},{"key":"8","doi-asserted-by":"publisher","unstructured":"A. Cichocki, N. Lee, I. Oseledets, A.-H. Phan, Q. Zhao, M. Sugiyama, and D. P. Mandic, Found. Trends Mach. Learn. 9, 431 (2017).","DOI":"10.1561\/2200000067"},{"key":"9","doi-asserted-by":"publisher","unstructured":"L. Due\u00f1as-Osorio, M. Y. Vardi, and J. Rojo, Struct. Saf. 75, 110 (2018).","DOI":"10.1016\/j.strusafe.2018.05.004"},{"key":"10","doi-asserted-by":"publisher","unstructured":"I. L. Markov and Y. Shi, SIAM J. Comput. 38, 963 (2008).","DOI":"10.1137\/050644756"},{"key":"11","unstructured":"E. Stoudenmire and D. J. Schwab, in Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (Curran Associates, Inc., 2016) pp. 4799\u20134807."},{"key":"12","doi-asserted-by":"publisher","unstructured":"E. M. Stoudenmire, Quantum Sci. Technol. 3, 034003 (2018).","DOI":"10.1088\/2058-9565\/aaba1a"},{"key":"13","unstructured":"C. Roberts, A. Milsted, M. Ganahl, A. Zalcman, B. Fontaine, Y. Zou, J. Hidary, G. Vidal, and S. Leichenauer, arXiv:1905.01330 (2019), arXiv:1905.01330."},{"key":"14","doi-asserted-by":"publisher","unstructured":"H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008).","DOI":"10.1103\/PhysRevLett.101.090603"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Z.-C. Gu and X.-G. Wen, Phys. Rev. B 80, 155131 (2009).","DOI":"10.1103\/PhysRevB.80.155131"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Phys. Rev. B 86, 045139 (2012).","DOI":"10.1103\/PhysRevB.86.045139"},{"key":"17","doi-asserted-by":"publisher","unstructured":"H.-H. Zhao, Z. Y. Xie, T. Xiang, and M. Imada, Phys. Rev. B 93, 125115 (2016).","DOI":"10.1103\/PhysRevB.93.125115"},{"key":"18","doi-asserted-by":"publisher","unstructured":"M. Bal, M. Mari\u00ebn, J. Haegeman, and F. Verstraete, Phys. Rev. Lett. 118, 250602 (2017).","DOI":"10.1103\/PhysRevLett.118.250602"},{"key":"19","doi-asserted-by":"publisher","unstructured":"S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017).","DOI":"10.1103\/PhysRevLett.118.110504"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Y.-Y. Shi, L.-M. Duan, and G. Vidal, Phys. Rev. A 74, 022320 (2006).","DOI":"10.1103\/PhysRevA.74.022320"},{"key":"21","doi-asserted-by":"publisher","unstructured":"L. G. Valiant, SIAM J. Comput. 37, 1565 (2008).","DOI":"10.1137\/070682575"},{"key":"22","doi-asserted-by":"publisher","unstructured":"S. Bravyi, Contemp. Math. 482, 179 (2008), arXiv:0801.2989.","DOI":"10.1090\/conm\/482\/09419"},{"key":"23","doi-asserted-by":"publisher","unstructured":"M. Aguado and G. Vidal, Phys. Rev. Lett. 100, 070404 (2008).","DOI":"10.1103\/PhysRevLett.100.070404"},{"key":"24","doi-asserted-by":"publisher","unstructured":"R. K\u00f6nig, B. W. Reichardt, and G. Vidal, Phys. Rev. B 79, 195123 (2009).","DOI":"10.1103\/PhysRevB.79.195123"},{"key":"25","doi-asserted-by":"publisher","unstructured":"S. J. Denny, J. D. Biamonte, D. Jaksch, and S. R. Clark, J. Phys. A Math. Theor. 45, 015309 (2012).","DOI":"10.1088\/1751-8113\/45\/1\/015309"},{"key":"26","doi-asserted-by":"publisher","unstructured":"L. G. Valiant, Theor. Comput. Sci. 8, 189 (1979).","DOI":"10.1016\/0304-3975(79)90044-6"},{"key":"27","doi-asserted-by":"publisher","unstructured":"C. Damm, M. Holzer, and P. McKenzie, Comput. Complex. 11, 54 (2002).","DOI":"10.1007\/s00037-000-0170-4"},{"key":"28","doi-asserted-by":"crossref","unstructured":"B. M. Terhal and D. P. DiVincenzo, Quant. Inf. Comp. 4, 134 (2004), arXiv:0205133 [quant-ph].","DOI":"10.26421\/QIC4.2-5"},{"key":"29","doi-asserted-by":"publisher","unstructured":"M. J. Bremner, R. Jozsa, and D. J. Shepherd, Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 459 (2010).","DOI":"10.1098\/rspa.2010.0301"},{"key":"30","doi-asserted-by":"publisher","unstructured":"S. Aaronson and A. Arkhipov, Theory Comput. 9, 143 (2013).","DOI":"10.4086\/toc.2013.v009a004"},{"key":"31","doi-asserted-by":"publisher","unstructured":"R. Jozsa and M. V. den Nest, arXiv:1305.6190 (2013), https:\/\/doi.org\/10.26421\/qic14.7-8-7, arXiv:1305.6190.","DOI":"10.26421\/qic14.7-8-7"},{"key":"32","doi-asserted-by":"publisher","unstructured":"T. Morimae, K. Fujii, and J. F. Fitzsimons, Phys. Rev. Lett. 112, 130502 (2014).","DOI":"10.1103\/PhysRevLett.112.130502"},{"key":"33","doi-asserted-by":"publisher","unstructured":"J. Carolan, C. Harrold, C. Sparrow, E. Martin-Lopez, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O'Brien, and A. Laing, Science 349, 711 (2015).","DOI":"10.1126\/science.aab3642"},{"key":"34","unstructured":"E. Farhi and A. W. Harrow, arXiv:1602.07674 (2016), arXiv:1602.07674."},{"key":"35","doi-asserted-by":"publisher","unstructured":"S. Aaronson, A. Bouland, G. Kuperberg, and S. Mehraban, in Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (2017) pp. 317\u2013327.","DOI":"10.1145\/3055399.3055453"},{"key":"36","unstructured":"S. Aaronson and L. Chen, arXiv:1612.05903 (2016), arXiv:1612.05903."},{"key":"37","doi-asserted-by":"publisher","unstructured":"S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Nat. Phys. 14, 595 (2018).","DOI":"10.1038\/s41567-018-0124-x"},{"key":"38","doi-asserted-by":"publisher","unstructured":"A. Bouland, B. Fefferman, C. Nirkhe, and U. Vazirani, Nat. Phys. 15, 159 (2019).","DOI":"10.1038\/s41567-018-0318-2"},{"key":"39","doi-asserted-by":"publisher","unstructured":"E. S. Fried, N. P. D. Sawaya, Y. Cao, I. D. Kivlichan, J. Romero, and A. Aspuru-Guzik, PLoS One 13, e0208510 (2018).","DOI":"10.1371\/journal.pone.0208510"},{"key":"40","unstructured":"J. Chen, F. Zhang, C. Huang, M. Newman, and Y. Shi, arXiv:1805.01450 (2018), arXiv:1805.01450."},{"key":"41","doi-asserted-by":"publisher","unstructured":"E. F. Dumitrescu, A. L. Fisher, T. D. Goodrich, T. S. Humble, B. D. Sullivan, and A. L. Wright, PLoS One 13, e0207827 (2018).","DOI":"10.1371\/journal.pone.0207827"},{"key":"42","doi-asserted-by":"publisher","unstructured":"S. Kourtis, C. Chamon, E. R. Mucciolo, and A. E. Ruckenstein, SciPost Phys. 7, 60 (2019).","DOI":"10.21468\/SciPostPhys.7.5.060"},{"key":"43","unstructured":"J. M. Dudek, L. Due\u00f1as-Osorio, and M. Y. Vardi, arXiv:1908.04381 (2019), arXiv:1908.04381."},{"key":"44","doi-asserted-by":"publisher","unstructured":"F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandr\u00e0, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Nature 574, 505 (2019).","DOI":"10.1038\/s41586-019-1666-5"},{"key":"45","doi-asserted-by":"publisher","unstructured":"D. Bienstock, Journal of Combinatorial Theory, Series B 49, 103 (1990).","DOI":"10.1016\/0095-8956(90)90066-9"},{"key":"46","doi-asserted-by":"publisher","unstructured":"B. O'Gorman, in 14th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2019), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 135, edited by W. van Dam and L. Mancinska (Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2019) pp. 10:1\u201310:19.","DOI":"10.4230\/LIPIcs.TQC.2019.10"},{"key":"47","doi-asserted-by":"publisher","unstructured":"R. N. C. Pfeifer, J. Haegeman, and F. Verstraete, Phys. Rev. E 90, 033315 (2014).","DOI":"10.1103\/PhysRevE.90.033315"},{"key":"48","doi-asserted-by":"publisher","unstructured":"D. G. A. Smith and J. Gray, J. Open Source Softw. 3, 753 (2018).","DOI":"10.21105\/joss.00753"},{"key":"49","unstructured":"V. Gogate and R. Dechter, in Proceedings of the 20th conference on Uncertainty in artificial intelligence (AUAI Press, 2004) pp. 201\u2013208, arXiv:1207.4109."},{"key":"50","doi-asserted-by":"publisher","unstructured":"M. Hamann and B. Strasser, J. Exp. Algorithmics 23, 1.2 (2018).","DOI":"10.1145\/3173045"},{"key":"51","unstructured":"B. Strasser, arXiv:1709.08949 (2017), arXiv:1709.08949."},{"key":"52","unstructured":"M. A. Porter, J.-P. Onnela, and P. J. Mucha, Not. Am. Math. Soc. 56, 1082 (2009), arXiv:0902.3788."},{"key":"53","doi-asserted-by":"publisher","unstructured":"S. Fortunato, Phys. Rep. 486, 75 (2010).","DOI":"10.1016\/j.physrep.2009.11.002"},{"key":"54","doi-asserted-by":"publisher","unstructured":"M. Girvan and M. E. J. Newman, Proc. Natl. Acad. Sci. 99, 7821 (2002).","DOI":"10.1073\/pnas.122653799"},{"key":"55","doi-asserted-by":"publisher","unstructured":"S. Schlag, V. Henne, T. Heuer, H. Meyerhenke, P. Sanders, and C. Schulz, in 2016 Proc. Eighteenth Work. Algorithm Eng. Exp. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2016) pp. 53\u201367.","DOI":"10.1137\/1.9781611974317.5"},{"key":"56","doi-asserted-by":"publisher","unstructured":"Y. Akhremtsev, T. Heuer, P. Sanders, and S. Schlag, in 2017 Proc. Ninteenth Work. Algorithm Eng. Exp. (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017) pp. 28\u201342.","DOI":"10.1137\/1.9781611974768.3"},{"key":"57","unstructured":"D. A. Papa and I. L. Markov, in Handbook of Approximation Algorithms and Metaheuristics (2007)."},{"key":"58","doi-asserted-by":"publisher","unstructured":"B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and N. de Freitas, Proceedings of the IEEE 104, 148 (2016).","DOI":"10.1109\/JPROC.2015.2494218"},{"key":"59","unstructured":"L. Gustafson, Bayesian Tuning and Bandits: An Extensible, Open Source Library for AutoML, M. eng thesis, Massachusetts Institute of Technology, Cambridge, MA (2018)."},{"key":"60","doi-asserted-by":"publisher","unstructured":"C. K. Williams and C. E. Rasmussen, Gaussian processes for machine learning, Vol. 2 (MIT press Cambridge, MA, 2006).","DOI":"10.7551\/mitpress\/3206.001.0001"},{"key":"61","unstructured":"S. Boixo, S. V. Isakov, V. N. Smelyanskiy, and H. Neven, arXiv:1712.05384 (2017), arXiv:1712.05384."},{"key":"62","doi-asserted-by":"publisher","unstructured":"J. Gray, Journal of Open Source Software 3, 819 (2018).","DOI":"10.21105\/joss.00819"},{"key":"63","unstructured":"J. Gray, ``cotengra,'' https:\/\/github.com\/jcmgray\/cotengra (2020)."},{"key":"64","doi-asserted-by":"publisher","unstructured":"B. Villalonga, D. Lyakh, S. Boixo, H. Neven, T. S. Humble, R. Biswas, E. G. Rieffel, A. Ho, and S. Mandr\u00e0, Quantum Science and Technology 5, 034003 (2020).","DOI":"10.1088\/2058-9565\/ab7eeb"},{"key":"65","doi-asserted-by":"publisher","unstructured":"I. L. Markov and Y. Shi, Algorithmica 59 (2009), https:\/\/doi.org\/10.1007\/s00453-009-9312-5.","DOI":"10.1007\/s00453-009-9312-5"},{"key":"66","doi-asserted-by":"publisher","unstructured":"F. Viger and M. Latapy, in COCOON'05 Proc. 11th Annu. Int. Conf. Comput. Comb. (2005) pp. 440\u2013449.","DOI":"10.1007\/11533719_45"},{"key":"67","doi-asserted-by":"publisher","unstructured":"K. Meichanetzidis and S. Kourtis, Phys. Rev. E 100, 033303 (2019).","DOI":"10.1103\/PhysRevE.100.033303"},{"key":"68","doi-asserted-by":"publisher","unstructured":"\u00c9. Fusy, Random Struct. Algorithms 35, 464 (2009).","DOI":"10.1002\/rsa.20275"},{"key":"69","doi-asserted-by":"publisher","unstructured":"J.-Y. Cai and V. Choudhary, Theoretical Computer Science 384, 22 (2007), theory and Applications of Models of Computation.","DOI":"10.1016\/j.tcs.2007.05.015"},{"key":"70","doi-asserted-by":"publisher","unstructured":"F. Bacchus, S. Dalmao, and T. Pitassi, in 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings. (2003) pp. 340\u2013351.","DOI":"10.1109\/SFCS.2003.1238208"},{"key":"71","doi-asserted-by":"publisher","unstructured":"C. Domshlak and J. Hoffmann, J. Artif. Int. Res. 30, 565\u2013620 (2007).","DOI":"10.1613\/jair.2289"},{"key":"72","unstructured":"C. P. Gomes, A. Sabharwal, and B. Selman, in Handbook of Satisfiability (2009)."},{"key":"73","unstructured":"``1st international competition on model counting (mc 2020),''."},{"key":"74","doi-asserted-by":"publisher","unstructured":"J. Dudek, V. Phan, and M. Vardi, ``Addmc: Weighted model counting with algebraic decision diagrams,'' (2020).","DOI":"10.1609\/aaai.v34i02.5505"},{"key":"75","unstructured":"E. Farhi, J. Goldstone, and S. Gutmann, (2014), arXiv:1411.4028 [quant-ph]."},{"key":"76","doi-asserted-by":"crossref","unstructured":"C. Huang, M. Szegedy, F. Zhang, X. Gao, J. Chen, and Y. Shi, (2019), arXiv:1909.02559 [quant-ph].","DOI":"10.1155\/2019\/1847130"},{"key":"77","unstructured":"J. Preskill, arXiv:1203.5813 (2012), arXiv:1203.5813."},{"key":"78","unstructured":"E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, T. Magerlein, E. Solomonik, E. W. Draeger, E. T. Holland, and R. Wisnieff, arXiv:1710.05867 (2017), arXiv:1710.05867."},{"key":"79","unstructured":"F. Zhang, C. Huang, M. Newman, J. Cai, H. Yu, Z. Tian, B. Yuan, H. Xu, J. Wu, X. Gao, J. Chen, M. Szegedy, and Y. Shi, arXiv:1907.11217 (2019), arXiv:1907.11217."},{"key":"80","doi-asserted-by":"publisher","unstructured":"B. Villalonga, S. Boixo, B. Nelson, C. Henze, E. Rieffel, R. Biswas, and S. Mandr\u00e0, npj Quantum Inf. 5, 86 (2019).","DOI":"10.1038\/s41534-019-0196-1"},{"key":"81","doi-asserted-by":"publisher","unstructured":"C. Guo, Y. Liu, M. Xiong, S. Xue, X. Fu, A. Huang, X. Qiang, P. Xu, J. Liu, S. Zheng, H.-L. Huang, M. Deng, D. Poletti, W.-S. Bao, and J. Wu, Phys. Rev. Lett. 123, 190501 (2019).","DOI":"10.1103\/PhysRevLett.123.190501"},{"key":"82","unstructured":"I. L. Markov, A. Fatima, S. V. Isakov, and S. Boixo, arXiv preprint arXiv:1807.10749 (2018)."},{"key":"83","unstructured":"https:\/\/github.com\/sboixo\/GRCS."},{"key":"84","doi-asserted-by":"publisher","unstructured":"I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Phys. Rev. Lett. 120, 110501 (2018).","DOI":"10.1103\/PhysRevLett.120.110501"},{"key":"85","unstructured":"J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, and S. Wanderman-Milne, ``JAX: composable transformations of Python+NumPy programs,'' (2018)."},{"key":"86","doi-asserted-by":"publisher","unstructured":"S. Bravyi, M. Suchara, and A. Vargo, Phys. Rev. A 90, 032326 (2014).","DOI":"10.1103\/PhysRevA.90.032326"},{"key":"87","doi-asserted-by":"publisher","unstructured":"A. J. Ferris and D. Poulin, Phys. Rev. Lett. 113, 030501 (2014).","DOI":"10.1103\/PhysRevLett.113.030501"},{"key":"88","unstructured":"C. T. Chubb and S. T. Flammia, arXiv:1809.10704 (2018)."}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2021-03-15-410\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,12,21]],"date-time":"2022-12-21T15:05:24Z","timestamp":1671635124000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2021-03-15-410\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,15]]},"references-count":89,"URL":"https:\/\/doi.org\/10.22331\/q-2021-03-15-410","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,15]]},"article-number":"410"}}