{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T00:57:23Z","timestamp":1776214643841,"version":"3.50.1"},"reference-count":114,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T00:00:00Z","timestamp":1765497600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Variational quantum algorithms (VQAs) represent a promising pathway toward achieving practical quantum advantage on near-term hardware. Despite this promise, for generic, expressive ans\u00e4tze, their scalability is critically hindered by barren plateaus\u2014regimes of exponentially vanishing gradients. We demonstrate that initializing a hardware-efficient, Floquet-structured ansatz within the many-body localized (MBL) phase mitigates barren plateaus and enhances algorithmic trainability. Through analysis of the inverse participation ratio, entanglement entropy, and a novel low-weight stabilizer R\u00e9nyi entropy, we characterize a distinct MBL--thermalization transition. Below a critical kick strength, the circuit avoids forming a unitary 2-design, exhibits robust area-law entanglement, and maintains non-vanishing gradients. Leveraging this MBL regime facilitates the efficient variational preparation of ground states for several model Hamiltonians with significantly reduced computational resources. Crucially, experiments on a 127-qubit superconducting processor provide evidence for the preservation of trainable gradients in the MBL phase for a kicked Heisenberg chain, validating our approach on contemporary noisy hardware. Our findings position MBL-based initialization as a viable strategy for developing scalable VQAs and motivate broader integration of localization into quantum algorithm design.<\/jats:p>","DOI":"10.22331\/q-2025-12-12-1942","type":"journal-article","created":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T15:17:26Z","timestamp":1765552646000},"page":"1942","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Exploiting many-body localization for scalable variational quantum simulation"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5589-7503","authenticated-orcid":false,"given":"Chenfeng","family":"Cao","sequence":"first","affiliation":[{"name":"HK Institute of Quantum Science $\\&$ Technology, The University of Hong Kong, Hong Kong, China"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, 14195 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yeqing","family":"Zhou","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Swamit","family":"Tannu","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Wisconsin-Madison, Madison, WI 53706, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nic","family":"Shannon","sequence":"additional","affiliation":[{"name":"Theory of Quantum Matter Unit, Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0412, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Joynt","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA"},{"name":"Theoretical Sciences Visiting Program (TSVP), Okinawa Institute of Science and Technology Graduate University, Onna, 904-0495, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,12,12]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Al\u00e1n Aspuru-Guzik, and Jeremy L. O&apos;Brien. ``A variational eigenvalue solver on a photonic quantum processor&apos;&apos;. Nature Communications 5, 4213 (2014).","DOI":"10.1038\/ncomms5213"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Dave Wecker, Matthew B. Hastings, and Matthias Troyer. ``Progress towards practical quantum variational algorithms&apos;&apos;. Phys. Rev. A 92, 042303 (2015).","DOI":"10.1103\/PhysRevA.92.042303"},{"key":"2","unstructured":"Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. ``A quantum approximate optimization algorithm&apos;&apos; (2014). arXiv:1411.4028."},{"key":"3","doi-asserted-by":"publisher","unstructured":"Leo Zhou, Sheng-Tao Wang, Soonwon Choi, Hannes Pichler, and Mikhail D. Lukin. ``Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices&apos;&apos;. Phys. Rev. X 10, 021067 (2020).","DOI":"10.1103\/PhysRevX.10.021067"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven. ``Barren plateaus in quantum neural network training landscapes&apos;&apos;. Nature Communications 9, 4812 (2018).","DOI":"10.1038\/s41467-018-07090-4"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Carlos Ortiz Marrero, M\u00e1ria Kieferov\u00e1, and Nathan Wiebe. ``Entanglement-induced barren plateaus&apos;&apos;. PRX Quantum 2, 040316 (2021).","DOI":"10.1103\/PRXQuantum.2.040316"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo. ``Diagnosing Barren Plateaus with Tools from Quantum Optimal Control&apos;&apos;. Quantum 6, 824 (2022).","DOI":"10.22331\/q-2022-09-29-824"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Enrique Cervero Mart\u00edn, Kirill Plekhanov, and Michael Lubasch. ``Barren plateaus in quantum tensor network optimization&apos;&apos;. Quantum 7, 974 (2023).","DOI":"10.22331\/q-2023-04-13-974"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Mart\u00edn Larocca, Supanut Thanasilp, Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J. Coles, Lukasz Cincio, Jarrod R. McClean, Zo\u00eb Holmes, and M. Cerezo. ``Barren plateaus in variational quantum computing&apos;&apos;. Nature Reviews Physics 7, 174\u2013189 (2025).","DOI":"10.1038\/s42254-025-00813-9"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Michael Ragone, Bojko N. Bakalov, Fr\u00e9d\u00e9ric Sauvage, Alexander F. Kemper, Carlos Ortiz Marrero, Mart\u00edn Larocca, and M. Cerezo. ``A lie algebraic theory of barren plateaus for deep parameterized quantum circuits&apos;&apos;. Nature Communications 15, 7172 (2024).","DOI":"10.1038\/s41467-024-49909-3"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Enrico Fontana, Dylan Herman, Shouvanik Chakrabarti, Niraj Kumar, Romina Yalovetzky, Jamie Heredge, Shree Hari Sureshbabu, and Marco Pistoia. ``Characterizing barren plateaus in quantum ans\u00e4tze with the adjoint representation&apos;&apos;. Nature Communications 15, 7171 (2024).","DOI":"10.1038\/s41467-024-49910-w"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Eric R. Anschuetz and Bobak T. Kiani. ``Quantum variational algorithms are swamped with traps&apos;&apos;. Nature Communications 13, 7760 (2022).","DOI":"10.1038\/s41467-022-35364-5"},{"key":"12","unstructured":"Eric R. Anschuetz. ``A unified theory of quantum neural network loss landscapes&apos;&apos; (2025). arXiv:2408.11901."},{"key":"13","doi-asserted-by":"publisher","unstructured":"Chenfeng Cao, Filippo Maria Gambetta, Ashley Montanaro, and Raul A. Santos. ``Unveiling quantum phase transitions from traps in variational quantum algorithms&apos;&apos;. npj Quantum Information 11, 93 (2025).","DOI":"10.1038\/s41534-025-01038-5"},{"key":"14","unstructured":"Antonio Anna Mele, Armando Angrisani, Soumik Ghosh, Sumeet Khatri, Jens Eisert, Daniel Stilck Fran\u00e7a, and Yihui Quek. ``Noise-induced shallow circuits and absence of barren plateaus&apos;&apos; (2024). arXiv:2403.13927."},{"key":"15","doi-asserted-by":"publisher","unstructured":"Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles. ``Noise-induced barren plateaus in variational quantum algorithms&apos;&apos;. Nature Communications 12, 6961 (2021).","DOI":"10.1038\/s41467-021-27045-6"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Yihui Quek, Daniel Stilck Fran\u00e7a, Sumeet Khatri, Johannes Jakob Meyer, and Jens Eisert. ``Exponentially tighter bounds on limitations of quantum error mitigation&apos;&apos;. Nature Physics 20, 1648\u20131658 (2024).","DOI":"10.1038\/s41567-024-02536-7"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Daniel Stilck Fran\u00e7a and Raul Garc\u00eda-Patr\u00f3n. ``Limitations of optimization algorithms on noisy quantum devices&apos;&apos;. Nature Physics 17, 1221\u20131227 (2021).","DOI":"10.1038\/s41567-021-01356-3"},{"key":"18","doi-asserted-by":"publisher","unstructured":"M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles. ``Challenges and opportunities in quantum machine learning&apos;&apos;. Nature Computational Science 2, 567\u2013576 (2022).","DOI":"10.1038\/s43588-022-00311-3"},{"key":"19","doi-asserted-by":"publisher","unstructured":"D. Gross, S. T. Flammia, and J. Eisert. ``Most quantum states are too entangled to be useful as computational resources&apos;&apos;. Phys. Rev. Lett. 102, 190501 (2009).","DOI":"10.1103\/PhysRevLett.102.190501"},{"key":"20","doi-asserted-by":"publisher","unstructured":"H. J. Briegel, D. E. Browne, W. D\u00fcr, R. Raussendorf, and M. Van den Nest. ``Measurement-based quantum computation&apos;&apos;. Nature Physics 5, 19\u201326 (2009).","DOI":"10.1038\/nphys1157"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles. ``Effect of barren plateaus on gradient-free optimization&apos;&apos;. Quantum 5, 558 (2021).","DOI":"10.22331\/q-2021-10-05-558"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Andrew Arrasmith, Zo\u00eb Holmes, M Cerezo, and Patrick J Coles. ``Equivalence of quantum barren plateaus to cost concentration and narrow gorges&apos;&apos;. Quantum Science and Technology 7, 045015 (2022).","DOI":"10.1088\/2058-9565\/ac7d06"},{"key":"23","doi-asserted-by":"publisher","unstructured":"M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick 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":"24","doi-asserted-by":"publisher","unstructured":"Hao-Kai Zhang, Shuo Liu, and Shi-Xin Zhang. ``Absence of barren plateaus in finite local-depth circuits with long-range entanglement&apos;&apos;. Phys. Rev. Lett. 132, 150603 (2024).","DOI":"10.1103\/PhysRevLett.132.150603"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Roeland Wiersema, Cunlu Zhou, Yvette de Sereville, Juan Felipe Carrasquilla, Yong Baek Kim, and Henry Yuen. ``Exploring entanglement and optimization within the Hamiltonian variational ansatz&apos;&apos;. PRX Quantum 1, 020319 (2020).","DOI":"10.1103\/PRXQuantum.1.020319"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Arthur Pesah, M. Cerezo, Samson Wang, Tyler Volkoff, Andrew T. Sornborger, and Patrick J. Coles. ``Absence of barren plateaus in quantum convolutional neural networks&apos;&apos;. Phys. Rev. X 11, 041011 (2021).","DOI":"10.1103\/PhysRevX.11.041011"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Chen Zhao and Xiao-Shan Gao. ``Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus&apos;&apos;. Quantum 5, 466 (2021).","DOI":"10.22331\/q-2021-06-04-466"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Chae-Yeun Park and Nathan Killoran. ``Hamiltonian variational ansatz without barren plateaus&apos;&apos;. Quantum 8, 1239 (2024).","DOI":"10.22331\/q-2024-02-01-1239"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Xin Wang, Bo Qi, Yabo Wang, and Daoyi Dong. ``Entanglement-variational hardware-efficient ansatz for eigensolvers&apos;&apos;. Phys. Rev. Appl. 21, 034059 (2024).","DOI":"10.1103\/PhysRevApplied.21.034059"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Roeland Wiersema, Cunlu Zhou, Juan Felipe Carrasquilla, and Yong Baek Kim. ``Measurement-induced entanglement phase transitions in variational quantum circuits&apos;&apos;. SciPost Phys. 14, 147 (2023).","DOI":"10.21468\/SciPostPhys.14.6.147"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Antonio A. Mele, Glen B. Mbeng, Giuseppe E. Santoro, Mario Collura, and Pietro Torta. ``Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz&apos;&apos;. Phys. Rev. A 106, L060401 (2022).","DOI":"10.1103\/PhysRevA.106.L060401"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn. ``Avoiding barren plateaus using classical shadows&apos;&apos;. PRX Quantum 3, 020365 (2022).","DOI":"10.1103\/PRXQuantum.3.020365"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Ricard Puig, Marc Drudis, Supanut Thanasilp, and Zo\u00eb Holmes. ``Variational quantum simulation: A case study for understanding warm starts&apos;&apos;. PRX Quantum 6, 010317 (2025).","DOI":"10.1103\/PRXQuantum.6.010317"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin. ``Entanglement devised barren plateau mitigation&apos;&apos;. Phys. Rev. Res. 3, 033090 (2021).","DOI":"10.1103\/PhysRevResearch.3.033090"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Yabo Wang, Bo Qi, Chris Ferrie, and Daoyi Dong. ``Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization&apos;&apos;. Phys. Rev. Appl. 22, 054005 (2024).","DOI":"10.1103\/PhysRevApplied.22.054005"},{"key":"36","doi-asserted-by":"publisher","unstructured":"M. Cerezo, Martin Larocca, Diego Garc\u00eda-Mart\u00edn, N. L. Diaz, Paolo Braccia, Enrico Fontana, Manuel S. Rudolph, Pablo Bermejo, Aroosa Ijaz, Supanut Thanasilp, Eric R. Anschuetz, and Zo\u00eb Holmes. ``Does provable absence of barren plateaus imply classical simulability?&apos;&apos;. Nature Communications 16, 7907 (2025).","DOI":"10.1038\/s41467-025-63099-6"},{"key":"37","unstructured":"Hela Mhiri, Ricard Puig, Sacha Lerch, Manuel S. Rudolph, Thiparat Chotibut, Supanut Thanasilp, and Zo\u00eb Holmes. ``A unifying account of warm start guarantees for patches of quantum landscapes&apos;&apos; (2025). arXiv:2502.07889."},{"key":"38","unstructured":"Sacha Lerch, Ricard Puig, Manuel S. Rudolph, Armando Angrisani, Tyson Jones, M. Cerezo, Supanut Thanasilp, and Zo\u00eb Holmes. ``Efficient quantum-enhanced classical simulation for patches of quantum landscapes&apos;&apos; (2024). arXiv:2411.19896."},{"key":"39","doi-asserted-by":"publisher","unstructured":"P. W. Anderson. ``Absence of diffusion in certain random lattices&apos;&apos;. Phys. Rev. 109, 1492\u20131505 (1958).","DOI":"10.1103\/PhysRev.109.1492"},{"key":"40","doi-asserted-by":"publisher","unstructured":"D.M. Basko, I.L. Aleiner, and B.L. Altshuler. ``Metal\u2013insulator transition in a weakly interacting many-electron system with localized single-particle states&apos;&apos;. Annals of Physics 321, 1126\u20131205 (2006).","DOI":"10.1016\/j.aop.2005.11.014"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Rahul Nandkishore and David A. Huse. ``Many-body localization and thermalization in quantum statistical mechanics&apos;&apos;. Annual Review of Condensed Matter Physics 6, 15\u201338 (2015).","DOI":"10.1146\/annurev-conmatphys-031214-014726"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn. ``Colloquium: Many-body localization, thermalization, and entanglement&apos;&apos;. Rev. Mod. Phys. 91, 021001 (2019).","DOI":"10.1103\/RevModPhys.91.021001"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Piotr Sierant, Maciej Lewenstein, Antonello Scardicchio, Lev Vidmar, and Jakub Zakrzewski. ``Many-body localization in the age of classical computing*&apos;&apos;. Reports on Progress in Physics 88, 026502 (2025).","DOI":"10.1088\/1361-6633\/ad9756"},{"key":"44","doi-asserted-by":"publisher","unstructured":"J. Eisert, M. Cramer, and M. B. Plenio. ``Colloquium: Area laws for the entanglement entropy&apos;&apos;. Rev. Mod. Phys. 82, 277\u2013306 (2010).","DOI":"10.1103\/RevModPhys.82.277"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Chenfeng Cao, Jian Xue, Nic Shannon, and Robert Joynt. ``Speedup of the quantum adiabatic algorithm using delocalization catalysis&apos;&apos;. Phys. Rev. Res. 3, 013092 (2021).","DOI":"10.1103\/PhysRevResearch.3.013092"},{"key":"46","doi-asserted-by":"publisher","unstructured":"Shuo Liu, Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao. ``Probing many-body localization by excited-state variational quantum eigensolver&apos;&apos;. Phys. Rev. B 107, 024204 (2023).","DOI":"10.1103\/PhysRevB.107.024204"},{"key":"47","doi-asserted-by":"publisher","unstructured":"Jirawat Tangpanitanon, Supanut Thanasilp, Ninnat Dangniam, Marc-Antoine Lemonde, and Dimitris G. Angelakis. ``Expressibility and trainability of parametrized analog quantum systems for machine learning applications&apos;&apos;. Phys. Rev. Res. 2, 043364 (2020).","DOI":"10.1103\/PhysRevResearch.2.043364"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Kaixiang Su and Michael J. Lawler. ``Signatures of interacting floquet phases in shallow quantum circuits&apos;&apos;. Phys. Rev. B 106, 214310 (2022).","DOI":"10.1103\/PhysRevB.106.214310"},{"key":"49","doi-asserted-by":"publisher","unstructured":"D. Zhu, S. Johri, N. H. Nguyen, C. Huerta Alderete, K. A. Landsman, N. M. Linke, C. Monroe, and A. Y. Matsuura. ``Probing many-body localization on a noisy quantum computer&apos;&apos;. Phys. Rev. A 103, 032606 (2021).","DOI":"10.1103\/PhysRevA.103.032606"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Oles Shtanko, Derek S. Wang, Haimeng Zhang, Nikhil Harle, Alireza Seif, Ramis Movassagh, and Zlatko Minev. ``Uncovering local integrability in quantum many-body dynamics&apos;&apos;. Nature Communications 16, 2552 (2025).","DOI":"10.1038\/s41467-025-57623-x"},{"key":"51","unstructured":"Chae-Yeun Park, Minhyeok Kang, and Joonsuk Huh. ``Hardware-efficient ansatz without barren plateaus in any depth&apos;&apos; (2024). arXiv:2403.04844."},{"key":"52","doi-asserted-by":"crossref","unstructured":"Kasidit Srimahajariyapong, Supanut Thanasilp, and Thiparat Chotibut. ``Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms&apos;&apos; (2025). arXiv:2506.13865.","DOI":"10.1038\/s42005-026-02528-4"},{"key":"53","doi-asserted-by":"publisher","unstructured":"Pedro Ponte, Z. Papi\u0107, Fran\u00e7ois Huveneers, and Dmitry A. Abanin. ``Many-body localization in periodically driven systems&apos;&apos;. Phys. Rev. Lett. 114, 140401 (2015).","DOI":"10.1103\/PhysRevLett.114.140401"},{"key":"54","doi-asserted-by":"publisher","unstructured":"Achilleas Lazarides, Arnab Das, and Roderich Moessner. ``Fate of many-body localization under periodic driving&apos;&apos;. Phys. Rev. Lett. 115, 030402 (2015).","DOI":"10.1103\/PhysRevLett.115.030402"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Dmitry A. Abanin, Wojciech De Roeck, and Fran\u00e7ois Huveneers. ``Theory of many-body localization in periodically driven systems&apos;&apos;. Annals of Physics 372, 1\u201311 (2016).","DOI":"10.1016\/j.aop.2016.03.010"},{"key":"56","doi-asserted-by":"publisher","unstructured":"D. R. Hartree. ``The wave mechanics of an atom with a non-coulomb central field. part i. theory and methods&apos;&apos;. Mathematical Proceedings of the Cambridge Philosophical Society 24, 89\u2013110 (1928).","DOI":"10.1017\/S0305004100011919"},{"key":"57","doi-asserted-by":"publisher","unstructured":"F. Verstraete, V. Murg, and J.I. Cirac. ``Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems&apos;&apos;. Advances in Physics 57, 143\u2013224 (2008).","DOI":"10.1080\/14789940801912366"},{"key":"58","doi-asserted-by":"publisher","unstructured":"Ulrich Schollw\u00f6ck. ``The density-matrix renormalization group in the age of matrix product states&apos;&apos;. Annals of Physics 326, 96\u2013192 (2011).","DOI":"10.1016\/j.aop.2010.09.012"},{"key":"59","doi-asserted-by":"publisher","unstructured":"Liangsheng Zhang, Vedika Khemani, and David A. Huse. ``A floquet model for the many-body localization transition&apos;&apos;. Phys. Rev. B 94, 224202 (2016).","DOI":"10.1103\/PhysRevB.94.224202"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Takashi Mori, Tomotaka Kuwahara, and Keiji Saito. ``Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems&apos;&apos;. Phys. Rev. Lett. 116, 120401 (2016).","DOI":"10.1103\/PhysRevLett.116.120401"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Dmitry A. Abanin, Wojciech De Roeck, Wen Wei Ho, and Fran\u00e7ois Huveneers. ``Effective hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems&apos;&apos;. Phys. Rev. B 95, 014112 (2017).","DOI":"10.1103\/PhysRevB.95.014112"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Takashi Mori, Tatsuhiko N Ikeda, Eriko Kaminishi, and Masahito Ueda. ``Thermalization and prethermalization in isolated quantum systems: a theoretical overview&apos;&apos;. Journal of Physics B: Atomic, Molecular and Optical Physics 51, 112001 (2018).","DOI":"10.1088\/1361-6455\/aabcdf"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, and Dacheng Tao. ``Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits&apos;&apos;. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems. Volume 35, pages 18612\u201318627. Curran Associates, Inc. (2022).","DOI":"10.48550\/arXiv.2203.09376"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Zo\u00eb Holmes, Kunal Sharma, M. Cerezo, and Patrick 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":"65","doi-asserted-by":"publisher","unstructured":"Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, and Carlton M. Caves. ``Symmetric informationally complete quantum measurements&apos;&apos;. Journal of Mathematical Physics 45, 2171\u20132180 (2004).","DOI":"10.1063\/1.1737053"},{"key":"66","doi-asserted-by":"publisher","unstructured":"D. Gross, K. Audenaert, and J. Eisert. ``Evenly distributed unitaries: On the structure of unitary designs&apos;&apos;. Journal of Mathematical Physics 48 (2007).","DOI":"10.1063\/1.2716992"},{"key":"67","doi-asserted-by":"publisher","unstructured":"Christoph Dankert, Richard Cleve, Joseph Emerson, and Etera Livine. ``Exact and approximate unitary 2-designs and their application to fidelity estimation&apos;&apos;. Phys. Rev. A 80, 012304 (2009).","DOI":"10.1103\/PhysRevA.80.012304"},{"key":"68","doi-asserted-by":"publisher","unstructured":"John Watrous. ``The theory of quantum information&apos;&apos;. Cambridge University Press. (2018).","DOI":"10.1017\/9781316848142"},{"key":"69","doi-asserted-by":"publisher","unstructured":"Antonio Anna Mele. ``Introduction to Haar Measure Tools in Quantum Information: A Beginner&apos;s Tutorial&apos;&apos;. Quantum 8, 1340 (2024).","DOI":"10.22331\/q-2024-05-08-1340"},{"key":"70","doi-asserted-by":"publisher","unstructured":"Beno\u0131\u0302t Collins and Piotr \u015aniady. ``Integration with respect to the haar measure on unitary, orthogonal and symplectic group&apos;&apos;. Communications in Mathematical Physics 264, 773\u2013795 (2006).","DOI":"10.1007\/s00220-006-1554-3"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Luca D&apos;Alessio and Marcos Rigol. ``Long-time behavior of isolated periodically driven interacting lattice systems&apos;&apos;. Phys. Rev. X 4, 041048 (2014).","DOI":"10.1103\/PhysRevX.4.041048"},{"key":"72","doi-asserted-by":"publisher","unstructured":"A. De Luca and A. Scardicchio. ``Ergodicity breaking in a model showing many-body localization&apos;&apos;. Europhysics Letters 101, 37003 (2013).","DOI":"10.1209\/0295-5075\/101\/37003"},{"key":"73","doi-asserted-by":"publisher","unstructured":"David J. Luitz, Xavier Plat, Fabien Alet, and Nicolas Laflorencie. ``Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries&apos;&apos;. Phys. Rev. B 91, 155145 (2015).","DOI":"10.1103\/PhysRevB.91.155145"},{"key":"74","doi-asserted-by":"publisher","unstructured":"Sukin Sim, Peter D. Johnson, and Al\u00e1n 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":"75","unstructured":"Aram W. Harrow. ``The church of the symmetric subspace&apos;&apos; (2013). arXiv:1308.6595."},{"key":"76","doi-asserted-by":"publisher","unstructured":"Arijeet Pal and David A. Huse. ``Many-body localization phase transition&apos;&apos;. Phys. Rev. B 82, 174411 (2010).","DOI":"10.1103\/PhysRevB.82.174411"},{"key":"77","doi-asserted-by":"publisher","unstructured":"Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. ``Stabilizer r\u00e9nyi entropy&apos;&apos;. Phys. Rev. Lett. 128, 050402 (2022).","DOI":"10.1103\/PhysRevLett.128.050402"},{"key":"78","doi-asserted-by":"publisher","unstructured":"Salvatore F. E. Oliviero, Lorenzo Leone, and Alioscia Hamma. ``Magic-state resource theory for the ground state of the transverse-field ising model&apos;&apos;. Phys. Rev. A 106, 042426 (2022).","DOI":"10.1103\/PhysRevA.106.042426"},{"key":"79","doi-asserted-by":"publisher","unstructured":"Davide Rattacaso, Lorenzo Leone, Salvatore F. E. Oliviero, and Alioscia Hamma. ``Stabilizer entropy dynamics after a quantum quench&apos;&apos;. Phys. Rev. A 108, 042407 (2023).","DOI":"10.1103\/PhysRevA.108.042407"},{"key":"80","doi-asserted-by":"publisher","unstructured":"Nikolas P. Breuckmann and Jens Niklas Eberhardt. ``Quantum low-density parity-check codes&apos;&apos;. PRX Quantum 2, 040101 (2021).","DOI":"10.1103\/PRXQuantum.2.040101"},{"key":"81","doi-asserted-by":"publisher","unstructured":"Don N. Page. ``Average entropy of a subsystem&apos;&apos;. Phys. Rev. Lett. 71, 1291\u20131294 (1993).","DOI":"10.1103\/PhysRevLett.71.1291"},{"key":"82","doi-asserted-by":"publisher","unstructured":"S. K. Foong and S. Kanno. ``Proof of page&apos;s conjecture on the average entropy of a subsystem&apos;&apos;. Phys. Rev. Lett. 72, 1148\u20131151 (1994).","DOI":"10.1103\/PhysRevLett.72.1148"},{"key":"83","doi-asserted-by":"publisher","unstructured":"M B Hastings. ``An area law for one-dimensional quantum systems&apos;&apos;. Journal of Statistical Mechanics: Theory and Experiment 2007, P08024 (2007).","DOI":"10.1088\/1742-5468\/2007\/08\/P08024"},{"key":"84","unstructured":"Serge Aubry and Gilles Andr\u00e9. ``Analyticity breaking and anderson localization in incommensurate lattices&apos;&apos;. Ann. Israel Phys. Soc 3, 18 (1980)."},{"key":"85","doi-asserted-by":"publisher","unstructured":"Shankar Iyer, Vadim Oganesyan, Gil Refael, and David A. Huse. ``Many-body localization in a quasiperiodic system&apos;&apos;. Phys. Rev. B 87, 134202 (2013).","DOI":"10.1103\/PhysRevB.87.134202"},{"key":"86","doi-asserted-by":"publisher","unstructured":"P. Jordan and E. Wigner. ``\u00dcber das paulische \u00e4quivalenzverbot&apos;&apos;. Zeitschrift f\u00fcr Physik 47, 631\u2013651 (1928).","DOI":"10.1007\/BF01331938"},{"key":"87","doi-asserted-by":"publisher","unstructured":"Rom\u00e1n Or\u00fas. ``Tensor networks for complex quantum systems&apos;&apos;. Nature Reviews Physics 1, 538\u2013550 (2019).","DOI":"10.1038\/s42254-019-0086-7"},{"key":"88","doi-asserted-by":"publisher","unstructured":"Guifr\u00e9 Vidal. ``Efficient simulation of one-dimensional quantum many-body systems&apos;&apos;. Phys. Rev. Lett. 93, 040502 (2004).","DOI":"10.1103\/PhysRevLett.93.040502"},{"key":"89","doi-asserted-by":"publisher","unstructured":"Christa Zoufal, Aur\u00e9lien Lucchi, and Stefan Woerner. ``Quantum generative adversarial networks for learning and loading random distributions&apos;&apos;. npj Quantum Information 5, 103 (2019).","DOI":"10.1038\/s41534-019-0223-2"},{"key":"90","doi-asserted-by":"publisher","unstructured":"Xiao Mi et al. ``Information scrambling in quantum circuits&apos;&apos;. Science 374, 1479\u20131483 (2021).","DOI":"10.1126\/science.abg5029"},{"key":"91","doi-asserted-by":"publisher","unstructured":"Yunlong Yu, Chenfeng Cao, Carter Dewey, Xiang-Bin Wang, Nic Shannon, and Robert Joynt. ``Quantum approximate optimization algorithm with adaptive bias fields&apos;&apos;. Phys. Rev. Res. 4, 023249 (2022).","DOI":"10.1103\/PhysRevResearch.4.023249"},{"key":"92","doi-asserted-by":"publisher","unstructured":"Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon, and Robert Joynt. ``Solution of sat problems with the adaptive-bias quantum approximate optimization algorithm&apos;&apos;. Phys. Rev. Res. 5, 023147 (2023).","DOI":"10.1103\/PhysRevResearch.5.023147"},{"key":"93","doi-asserted-by":"publisher","unstructured":"Yunlong Yu, Xiang-Bin Wang, Nic Shannon, and Robert Joynt. ``Warm-start adaptive-bias quantum approximate optimization algorithm&apos;&apos;. Phys. Rev. A 112, 012422 (2025).","DOI":"10.1103\/nt3w-j4mj"},{"key":"94","doi-asserted-by":"publisher","unstructured":"Zheng-Hang Sun, Yong-Yi Wang, Jian Cui, and Heng Fan. ``Improving the performance of quantum approximate optimization for preparing non-trivial quantum states without translational symmetry&apos;&apos;. New Journal of Physics 25, 013015 (2023).","DOI":"10.1088\/1367-2630\/acb22c"},{"key":"95","doi-asserted-by":"publisher","unstructured":"Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. ``Theory of variational quantum simulation&apos;&apos;. Quantum 3, 191 (2019).","DOI":"10.22331\/q-2019-10-07-191"},{"key":"96","doi-asserted-by":"publisher","unstructured":"Marcello Benedetti, Mattia Fiorentini, and Michael Lubasch. ``Hardware-efficient variational quantum algorithms for time evolution&apos;&apos;. Phys. Rev. Res. 3, 033083 (2021).","DOI":"10.1103\/PhysRevResearch.3.033083"},{"key":"97","doi-asserted-by":"publisher","unstructured":"Shi-Ju Ran. ``Encoding of matrix product states into quantum circuits of one- and two-qubit gates&apos;&apos;. Phys. Rev. A 101, 032310 (2020).","DOI":"10.1103\/PhysRevA.101.032310"},{"key":"98","doi-asserted-by":"publisher","unstructured":"Manuel S Rudolph, Jing Chen, Jacob Miller, Atithi Acharya, and Alejandro Perdomo-Ortiz. ``Decomposition of matrix product states into shallow quantum circuits&apos;&apos;. Quantum Science and Technology 9, 015012 (2023).","DOI":"10.1088\/2058-9565\/ad04e6"},{"key":"99","doi-asserted-by":"publisher","unstructured":"Daniel Malz, Georgios Styliaris, Zhi-Yuan Wei, and J. Ignacio Cirac. ``Preparation of matrix product states with log-depth quantum circuits&apos;&apos;. Phys. Rev. Lett. 132, 040404 (2024).","DOI":"10.1103\/PhysRevLett.132.040404"},{"key":"100","doi-asserted-by":"publisher","unstructured":"Elisa B\u00e4umer, Vinay Tripathi, Derek S. Wang, Patrick Rall, Edward H. Chen, Swarnadeep Majumder, Alireza Seif, and Zlatko K. Minev. ``Efficient long-range entanglement using dynamic circuits&apos;&apos;. PRX Quantum 5, 030339 (2024).","DOI":"10.1103\/PRXQuantum.5.030339"},{"key":"101","unstructured":"Rahul Sahay and Ruben Verresen. ``Finite-depth preparation of tensor network states from measurement&apos;&apos; (2024). arXiv:2404.17087."},{"key":"102","doi-asserted-by":"publisher","unstructured":"Chenfeng Cao and Jens Eisert. ``Measurement-driven quantum advantages in shallow circuits&apos;&apos; (2025). arXiv:2505.04705.","DOI":"10.1103\/4b99-xmqn"},{"key":"103","doi-asserted-by":"publisher","unstructured":"Chenfeng Cao, Yunlong Yu, Zipeng Wu, Nic Shannon, Bei Zeng, and Robert Joynt. ``Mitigating algorithmic errors in quantum optimization through energy extrapolation&apos;&apos;. Quantum Science and Technology 8, 015004 (2022).","DOI":"10.1088\/2058-9565\/ac969c"},{"key":"104","doi-asserted-by":"publisher","unstructured":"Chenfeng Cao, Hiroshi Yano, and Yuya O. Nakagawa. ``Accelerated variational quantum eigensolver with joint bell measurement&apos;&apos;. Phys. Rev. Res. 6, 013205 (2024).","DOI":"10.1103\/PhysRevResearch.6.013205"},{"key":"105","doi-asserted-by":"publisher","unstructured":"Lana Mineh and Ashley Montanaro. ``Accelerating the variational quantum eigensolver using parallelism&apos;&apos;. Quantum Science and Technology 8, 035012 (2023).","DOI":"10.1088\/2058-9565\/acd0d2"},{"key":"106","unstructured":"Robert R. Tucci. ``An introduction to cartan&apos;s kak decomposition for qc programmers&apos;&apos; (2005). arXiv:quant-ph\/0507171."},{"key":"107","doi-asserted-by":"publisher","unstructured":"A Yu Kitaev. ``Quantum computations: algorithms and error correction&apos;&apos;. Russian Mathematical Surveys 52, 1191 (1997).","DOI":"10.1070\/RM1997v052n06ABEH002155"},{"key":"108","doi-asserted-by":"publisher","unstructured":"S. Datta, S. Howard, and Douglas Cochran. ``Geometry of the welch bounds&apos;&apos;. Linear Algebra and Its Applications 437, 2455\u20132470 (2012).","DOI":"10.1016\/j.laa.2012.05.036"},{"key":"109","doi-asserted-by":"publisher","unstructured":"Richard N. Youngworth, Benjamin B. Gallagher, and Brian L. Stamper. ``An overview of power spectral density (PSD) calculations&apos;&apos;. In H. Philip Stahl, editor, Optical Manufacturing and Testing VI. Volume 5869, page 58690U. International Society for Optics and PhotonicsSPIE (2005).","DOI":"10.1117\/12.618478"},{"key":"110","doi-asserted-by":"publisher","unstructured":"Steven R. White. ``Density matrix formulation for quantum renormalization groups&apos;&apos;. Phys. Rev. Lett. 69, 2863\u20132866 (1992).","DOI":"10.1103\/PhysRevLett.69.2863"},{"key":"111","unstructured":"Diederik P. Kingma and Jimmy Ba. ``Adam: A method for stochastic optimization&apos;&apos; (2017). arXiv:1412.6980."},{"key":"112","doi-asserted-by":"publisher","unstructured":"Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay 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":"113","doi-asserted-by":"publisher","unstructured":"Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. ``Efficient quantum measurement of Pauli operators in the presence of finite sampling error&apos;&apos;. Quantum 5, 385 (2021).","DOI":"10.22331\/q-2021-01-20-385"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-12-12-1942\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T09:15:45Z","timestamp":1772097345000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-12-12-1942\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,12]]},"references-count":114,"URL":"https:\/\/doi.org\/10.22331\/q-2025-12-12-1942","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,12]]},"article-number":"1942"}}