{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T21:20:21Z","timestamp":1772659221495,"version":"3.50.1"},"reference-count":78,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T00:00:00Z","timestamp":1763337600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation","award":["OMA-2120757"],"award-info":[{"award-number":["OMA-2120757"]}]},{"DOI":"10.13039\/100000015","name":"Department of Energy","doi-asserted-by":"crossref","award":["DE-SC0020312"],"award-info":[{"award-number":["DE-SC0020312"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000015","name":"Department of Energy","doi-asserted-by":"crossref","award":["DE-SC0025341"],"award-info":[{"award-number":["DE-SC0025341"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000015","name":"Department of Energy","doi-asserted-by":"crossref","award":["DE- SC0019040"],"award-info":[{"award-number":["DE- SC0019040"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000015","name":"Department of Energy","doi-asserted-by":"crossref","award":["DE-SC0024220"],"award-info":[{"award-number":["DE-SC0024220"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]},{"name":"National Science Foundation","award":["DMR-2345644"],"award-info":[{"award-number":["DMR-2345644"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>\n                    Recent experimental progress in controlling open quantum systems enables the pursuit of mixed-state nonequilibrium quantum phases. We investigate whether open quantum systems hosting mixed-state symmetry-protected topological states as steady states retain this property under symmetric perturbations. Focusing on the\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">decohered cluster state<\/mml:mtext>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    \u2013 a mixed-state symmetry-protected topological state protected by a combined strong and weak symmetry \u2013 we construct a parent Lindbladian that hosts it as a steady state. This Lindbladian can be mapped onto exactly solvable reaction-diffusion dynamics, even in the presence of certain perturbations, allowing us to solve the parent Lindbladian in detail and reveal previously-unknown steady states. Using both analytical and numerical methods, we find that typical symmetric perturbations cause strong-to-weak spontaneous symmetry breaking at arbitrarily small perturbations, destabilize the steady-state mixed-state symmetry-protected topological order. However, when perturbations introduce only weak symmetry defects, the steady-state mixed-state symmetry-protected topological order remains stable. Additionally, we construct a quantum channel which replicates the essential physics of the Lindbladian and can be efficiently simulated using only Clifford gates, Pauli measurements, and feedback.\n                  <\/jats:p>","DOI":"10.22331\/q-2025-11-17-1912","type":"journal-article","created":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T13:30:42Z","timestamp":1763386242000},"page":"1912","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Instability of steady-state mixed-state symmetry-protected topological order to strong-to-weak spontaneous symmetry breaking"],"prefix":"10.22331","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5873-8129","authenticated-orcid":false,"given":"Jeet","family":"Shah","sequence":"first","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2590-8221","authenticated-orcid":false,"given":"Christopher","family":"Fechisin","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2848-1216","authenticated-orcid":false,"given":"Yu-Xin","family":"Wang","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3383-1946","authenticated-orcid":false,"given":"Joseph T.","family":"Iosue","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6077-4898","authenticated-orcid":false,"given":"James D.","family":"Watson","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3900-0836","authenticated-orcid":false,"given":"Yan-Qi","family":"Wang","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3321-3198","authenticated-orcid":false,"given":"Brayden","family":"Ware","sequence":"additional","affiliation":[{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Google Quantum AI, California, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0509-3421","authenticated-orcid":false,"given":"Alexey V.","family":"Gorshkov","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7898-0211","authenticated-orcid":false,"given":"Cheng-Ju","family":"Lin","sequence":"additional","affiliation":[{"name":"Joint Quantum Institute, NIST\/University of Maryland, College Park, MD, 20742, USA"},{"name":"Joint Center for Quantum Information and Computer Science, NIST\/University of Maryland, College Park, MD, 20742, USA"}]}],"member":"9598","published-online":{"date-parts":[[2025,11,17]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Ruichao Ma, Brendan Saxberg, Clai Owens, Nelson Leung, Yao Lu, Jonathan Simon, and David I. Schuster. ``A dissipatively stabilized Mott insulator of photons&apos;&apos;. Nature 566, 51\u201357 (2019).","DOI":"10.1038\/s41586-019-0897-9"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Jeffrey M. Gertler, Brian Baker, Juliang Li, Shruti Shirol, Jens Koch, and Chen Wang. ``Protecting a bosonic qubit with autonomous quantum error correction&apos;&apos;. Nature 590, 243\u2013248 (2021).","DOI":"10.1038\/s41586-021-03257-0"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Patrick M. Harrington, Erich Mueller, and Kater Murch. ``Engineered Dissipation for Quantum Information Science&apos;&apos;. Nat. Rev. Phys. 4, 660\u2013671 (2022).","DOI":"10.1038\/s42254-022-00494-8"},{"key":"3","doi-asserted-by":"publisher","unstructured":"T. Brown, E. Doucet, D. Rist\u00e9, G. Ribeill, K. Cicak, J. Aumentado, R. Simmonds, L. Govia, A. Kamal, and L. Ranzani. ``Trade off-free entanglement stabilization in a superconducting qutrit-qubit system&apos;&apos;. Nat. Commun. 13, 3994 (2022).","DOI":"10.1038\/s41467-022-31638-0"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Daniel C. Cole, Stephen D. Erickson, Giorgio Zarantonello, Karl P. Horn, Pan-Yu Hou, Jenny J. Wu, Daniel H. Slichter, Florentin Reiter, Christiane P. Koch, and Dietrich Leibfried. ``Resource-efficient dissipative entanglement of two trapped-ion qubits&apos;&apos;. Phys. Rev. Lett. 128, 080502 (2022).","DOI":"10.1103\/PhysRevLett.128.080502"},{"key":"5","doi-asserted-by":"publisher","unstructured":"M. Malinowski, C. Zhang, V. Negnevitsky, I. Rojkov, F. Reiter, T.-L. Nguyen, M. Stadler, D. Kienzler, K. K. Mehta, and J. P. Home. ``Generation of a maximally entangled state using collective optical pumping&apos;&apos;. Phys. Rev. Lett. 128, 080503 (2022).","DOI":"10.1103\/PhysRevLett.128.080503"},{"key":"6","doi-asserted-by":"publisher","unstructured":"M. W. van Mourik, E. Zapusek, P. Hrmo, L. Gerster, R. Blatt, T. Monz, P. Schindler, and F. Reiter. ``Experimental realization of nonunitary multiqubit operations&apos;&apos;. Phys. Rev. Lett. 132, 040602 (2024).","DOI":"10.1103\/PhysRevLett.132.040602"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Andrea Coser and David P\u00e9rez-Garc\u00eda. ``Classification of phases for mixed states via fast dissipative evolution&apos;&apos;. Quantum 3, 174 (2019).","DOI":"10.22331\/q-2019-08-12-174"},{"key":"8","doi-asserted-by":"publisher","unstructured":"D. A. Lidar, I. L. Chuang, and K. B. Whaley. ``Decoherence-free subspaces for quantum computation&apos;&apos;. Phys. Rev. Lett. 81, 2594\u20132597 (1998).","DOI":"10.1103\/PhysRevLett.81.2594"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Daniel A. Lidar and K. Birgitta Whaley. ``Decoherence-free subspaces and subsystems&apos;&apos;. Pages 83\u2013120. Springer Berlin Heidelberg. Berlin, Heidelberg (2003).","DOI":"10.1007\/3-540-44874-8_5"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Lian-Ao Wu Mark S. Byrd and Daniel A. Lidar. ``Overview of quantum error prevention and leakage elimination&apos;&apos;. J. Mod. Opt. 51, 2449\u20132460 (2004).","DOI":"10.1080\/09500340408231803"},{"key":"11","unstructured":"Victor V. Albert. ``Lindbladians with multiple steady states: theory and applications&apos;&apos; (2018). arXiv:1802.00010."},{"key":"12","doi-asserted-by":"publisher","unstructured":"Simon Lieu, Ron Belyansky, Jeremy T. Young, Rex Lundgren, Victor V. Albert, and Alexey V. Gorshkov. ``Symmetry breaking and error correction in open quantum systems&apos;&apos;. Phys. Rev. Lett. 125, 240405 (2020).","DOI":"10.1103\/PhysRevLett.125.240405"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Tibor Rakovszky, Sarang Gopalakrishnan, and Curt von Keyserlingk. ``Defining stable phases of open quantum systems&apos;&apos;. Phys. Rev. X 14, 041031 (2024).","DOI":"10.1103\/PhysRevX.14.041031"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Ruochen Ma and Chong Wang. ``Average symmetry-protected topological phases&apos;&apos;. Phys. Rev. X 13, 031016 (2023).","DOI":"10.1103\/PhysRevX.13.031016"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, and Chong Wang. ``Topological phases with average symmetries: The decohered, the disordered, and the intrinsic&apos;&apos;. Phys. Rev. X 15, 021062 (2025).","DOI":"10.1103\/PhysRevX.15.021062"},{"key":"16","unstructured":"Yinning Niu and Yang Qi. ``Strange Correlator for 1D Fermionic Symmetry-Protected Topological Phases&apos;&apos; (2023). arXiv:2312.01310."},{"key":"17","doi-asserted-by":"publisher","unstructured":"Sanket Chirame, Fiona J. Burnell, Sarang Gopalakrishnan, and Abhinav Prem. ``Stable symmetry-protected topological phases in systems with heralded noise&apos;&apos;. Phys. Rev. Lett. 134, 010403 (2025).","DOI":"10.1103\/PhysRevLett.134.010403"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Yuchen Guo, Jian-Hao Zhang, Hao-Ran Zhang, Shuo Yang, and Zhen Bi. ``Locally purified density operators for symmetry-protected topological phases in mixed states&apos;&apos;. Phys. Rev. X 15, 021060 (2025).","DOI":"10.1103\/PhysRevX.15.021060"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Dawid Paszko, Dominic C. Rose, Marzena H. Szyma\u0144ska, and Arijeet Pal. ``Edge modes and symmetry-protected topological states in open quantum systems&apos;&apos;. PRX Quantum 5, 030304 (2024).","DOI":"10.1103\/PRXQuantum.5.030304"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Luan M. Ver\u00edssimo, Marcelo L. Lyra, and Roman Orus. ``Dissipative Symmetry-Protected Topological Order&apos;&apos;. Phys. Rev. B 107, L241104 (2023).","DOI":"10.1103\/PhysRevB.107.L241104"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Jian-Hao Zhang, Ke Ding, Shuo Yang, and Zhen Bi. ``Fractonic higher-order topological phases in open quantum systems&apos;&apos;. Phys. Rev. B 108, 155123 (2023).","DOI":"10.1103\/PhysRevB.108.155123"},{"key":"22","unstructured":"Hanyu Xue, Jong Yeon Lee, and Yimu Bao. ``Tensor network formulation of symmetry protected topological phases in mixed states&apos;&apos; (2024). arXiv:2403.17069."},{"key":"23","doi-asserted-by":"publisher","unstructured":"Zhehao Zhang, Utkarsh Agrawal, and Sagar Vijay. ``Quantum communication and mixed-state order in decohered symmetry-protected topological states&apos;&apos;. Phys. Rev. B 111, 115141 (2025).","DOI":"10.1103\/PhysRevB.111.115141"},{"key":"24","unstructured":"Jian-Hao Zhang, Yang Qi, and Zhen Bi. ``Strange correlation function for average symmetry-protected topological phases&apos;&apos; (2024). arXiv:2210.17485."},{"key":"25","doi-asserted-by":"publisher","unstructured":"Jong Yeon Lee, Yi-Zhuang You, and Cenke Xu. ``Symmetry protected topological phases under decoherence&apos;&apos;. Quantum 9, 1607 (2025).","DOI":"10.22331\/q-2025-01-23-1607"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Ruochen Ma and Alex Turzillo. ``Symmetry-protected topological phases of mixed states in the doubled space&apos;&apos;. PRX Quantum 6, 010348 (2025).","DOI":"10.1103\/PRXQuantum.6.010348"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Yoshihito Kuno, Takahiro Orito, and Ikuo Ichinose. ``Strong-to-weak symmetry breaking states in stochastic dephasing stabilizer circuits&apos;&apos;. Phys. Rev. B 110, 094106 (2024).","DOI":"10.1103\/PhysRevB.110.094106"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Sanket Chirame, Abhinav Prem, Sarang Gopalakrishnan, and Fiona J. Burnell. ``Stabilizing non-abelian topological order against heralded noise via local lindbladian dynamics&apos;&apos;. PRX Quantum 6, 030363 (2025).","DOI":"10.1103\/zf7y-hxtq"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Po-Shen Hsin, Zhu-Xi Luo, and Hao-Yu Sun. ``Anomalies of average symmetries: entanglement and open quantum systems&apos;&apos;. J. High Energy Phys. 2024, 134 (2024).","DOI":"10.1007\/JHEP10(2024)134"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Leonardo A. Lessa, Meng Cheng, and Chong Wang. ``Mixed-state quantum anomaly and multipartite entanglement&apos;&apos;. Phys. Rev. X 15, 011069 (2025).","DOI":"10.1103\/PhysRevX.15.011069"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Zijian Wang and Linhao Li. ``Anomaly in open quantum systems and its implications on mixed-state quantum phases&apos;&apos;. PRX Quantum 6, 010347 (2025).","DOI":"10.1103\/PRXQuantum.6.010347"},{"key":"32","unstructured":"Yimu Bao, Ruihua Fan, Ashvin Vishwanath, and Ehud Altman. ``Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions&apos;&apos; (2023) arXiv:2301.05687."},{"key":"33","doi-asserted-by":"publisher","unstructured":"Jong Yeon Lee, Chao-Ming Jian, and Cenke Xu. ``Quantum criticality under decoherence or weak measurement&apos;&apos;. PRX Quantum 4, 030317 (2023).","DOI":"10.1103\/PRXQuantum.4.030317"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Tyler D. Ellison and Meng Cheng. ``Toward a classification of mixed-state topological orders in two dimensions&apos;&apos;. PRX Quantum 6, 010315 (2025).","DOI":"10.1103\/PRXQuantum.6.010315"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Ramanjit Sohal and Abhinav Prem. ``Noisy approach to intrinsically mixed-state topological order&apos;&apos;. PRX Quantum 6, 010313 (2025).","DOI":"10.1103\/PRXQuantum.6.010313"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Jos\u00e9 Garre-Rubio, Laurens Lootens, and Andr\u00e1s Moln\u00e1r. ``Classifying phases protected by matrix product operator symmetries using matrix product states&apos;&apos;. Quantum 7, 927 (2023).","DOI":"10.22331\/q-2023-02-21-927"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Berislav Bu\u010da and Toma\u017e Prosen. ``A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains&apos;&apos;. New J. Phys. 14, 073007 (2012).","DOI":"10.1088\/1367-2630\/14\/7\/073007"},{"key":"38","doi-asserted-by":"publisher","unstructured":"Victor V. Albert and Liang Jiang. ``Symmetries and conserved quantities in Lindblad master equations&apos;&apos;. Phys. Rev. A 89, 022118 (2014).","DOI":"10.1103\/PhysRevA.89.022118"},{"key":"39","doi-asserted-by":"publisher","unstructured":"Caroline de Groot, Alex Turzillo, and Norbert Schuch. ``Symmetry protected topological order in open quantum systems&apos;&apos;. Quantum 6, 856 (2022).","DOI":"10.22331\/q-2022-11-10-856"},{"key":"40","doi-asserted-by":"publisher","unstructured":"Leonardo A. Lessa, Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, and Chong Wang. ``Strong-to-weak spontaneous symmetry breaking in mixed quantum states&apos;&apos;. PRX Quantum 6, 010344 (2025).","DOI":"10.1103\/PRXQuantum.6.010344"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Pablo Sala, Sarang Gopalakrishnan, Masaki Oshikawa, and Yizhi You. ``Spontaneous strong symmetry breaking in open systems: Purification perspective&apos;&apos;. Phys. Rev. B 110, 155150 (2024).","DOI":"10.1103\/PhysRevB.110.155150"},{"key":"42","doi-asserted-by":"publisher","unstructured":"M. B. Hastings and Xiao-Gang Wen. ``Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance&apos;&apos;. Phys. Rev. B 72, 045141 (2005).","DOI":"10.1103\/PhysRevB.72.045141"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, and Xiao-Gang Wen. ``Symmetry protected topological orders and the group cohomology of their symmetry group&apos;&apos;. Phys. Rev. B 87, 155114 (2013).","DOI":"10.1103\/PhysRevB.87.155114"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Hans J. Briegel and Robert Raussendorf. ``Persistent entanglement in arrays of interacting particles&apos;&apos;. Phys. Rev. Lett. 86, 910\u2013913 (2001).","DOI":"10.1103\/PhysRevLett.86.910"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Wonmin Son, Luigi Amico, and Vlatko Vedral. ``Topological order in 1d cluster state protected by symmetry&apos;&apos;. Quantum Inf. Process. 11, 1961\u20131968 (2012).","DOI":"10.1007\/s11128-011-0346-7"},{"key":"46","unstructured":"AA Lushnikov. ``Binary reaction $1+ 1 \\to 0$ in one dimension&apos;&apos;. Zh. Eksp. Teor. Fiz 91, 1376\u20131386 (1986). url: http:\/\/www.jetp.ras.ru\/cgi-bin\/dn\/e_064_04_0811.pdf."},{"key":"47","doi-asserted-by":"publisher","unstructured":"Jiannis K. Pachos and Martin B. Plenio. ``Three-spin interactions in optical lattices and criticality in Cluster Hamiltonians&apos;&apos;. Phys. Rev. Lett. 93, 056402 (2004).","DOI":"10.1103\/PhysRevLett.93.056402"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Andrew C. Doherty and Stephen D. Bartlett. ``Identifying phases of quantum many-body systems that are universal for quantum computation&apos;&apos;. Phys. Rev. Lett. 103, 020506 (2009).","DOI":"10.1103\/PhysRevLett.103.020506"},{"key":"49","unstructured":"Daniel Gottesman. ``The Heisenberg representation of quantum computers&apos;&apos; (1998). arXiv:quant-ph\/9807006."},{"key":"50","doi-asserted-by":"publisher","unstructured":"Xie Chen, Yuan-Ming Lu, and Ashvin Vishwanath. ``Symmetry-protected topological phases from decorated domain walls&apos;&apos;. Nat. Commun. 5, 3507 (2014).","DOI":"10.1038\/ncomms4507"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Toby S Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia. ``Stability of local quantum dissipative systems&apos;&apos;. Commun. Math. Phys. 337, 1275\u20131315 (2015).","DOI":"10.1007\/s00220-015-2355-3"},{"key":"52","unstructured":"Emilio Onorati, Cambyse Rouz\u00e9, Daniel Stilck Fran\u00e7a, and James D Watson. ``Provably efficient learning of phases of matter via dissipative evolutions&apos;&apos; (2023). arXiv:2311.07506."},{"key":"53","doi-asserted-by":"publisher","unstructured":"Elliott H Lieb and Derek W Robinson. ``The finite group velocity of quantum spin systems&apos;&apos;. Commun. Math. Phys. 28, 251\u2013257 (1972).","DOI":"10.1007\/BF01645779"},{"key":"54","doi-asserted-by":"publisher","unstructured":"David Poulin. ``Lieb-Robinson bound and locality for general Markovian quantum dynamics&apos;&apos;. Phys. Rev. Lett. 104, 190401 (2010).","DOI":"10.1103\/PhysRevLett.104.190401"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Kasper Duivenvoorden and Thomas Quella. ``From symmetry-protected topological order to Landau order&apos;&apos;. Phys. Rev. B 88, 125115 (2013).","DOI":"10.1103\/PhysRevB.88.125115"},{"key":"56","doi-asserted-by":"publisher","unstructured":"Marcel den Nijs and Koos Rommelse. ``Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains&apos;&apos;. Phys. Rev. B 40, 4709\u20134734 (1989).","DOI":"10.1103\/PhysRevB.40.4709"},{"key":"57","doi-asserted-by":"publisher","unstructured":"Tom Kennedy and Hal Tasaki. ``Hidden $\\mathbb{Z}_{2}\\times\\mathbb{Z}_{2}$ symmetry breaking in Haldane-gap antiferromagnets&apos;&apos;. Phys. Rev. B 45, 304\u2013307 (1992).","DOI":"10.1103\/PhysRevB.45.304"},{"key":"58","doi-asserted-by":"publisher","unstructured":"Frank Pollmann and Ari M. Turner. ``Detection of symmetry-protected topological phases in one dimension&apos;&apos;. Phys. Rev. B 86, 125441 (2012).","DOI":"10.1103\/PhysRevB.86.125441"},{"key":"59","doi-asserted-by":"publisher","unstructured":"Dominic V. Else, Stephen D. Bartlett, and Andrew C. Doherty. ``Hidden symmetry-breaking picture of symmetry-protected topological order&apos;&apos;. Phys. Rev. B 88, 085114 (2013).","DOI":"10.1103\/PhysRevB.88.085114"},{"key":"60","doi-asserted-by":"publisher","unstructured":"Yasaman Bahri and Ashvin Vishwanath. ``Detecting Majorana fermions in quasi-one-dimensional topological phases using nonlocal order parameters&apos;&apos;. Phys. Rev. B 89, 155135 (2014).","DOI":"10.1103\/PhysRevB.89.155135"},{"key":"61","doi-asserted-by":"publisher","unstructured":"Goran Lindblad. ``On the generators of quantum dynamical semigroups&apos;&apos;. Commun. Math. Phys. 48, 119\u2013130 (1976).","DOI":"10.1007\/BF01608499"},{"key":"62","doi-asserted-by":"publisher","unstructured":"Vittorio Gorini, Andrzej Kossakowski, and Ennackal Chandy George Sudarshan. ``Completely positive dynamical semigroups of N-level systems&apos;&apos;. J. Math. Phys. 17, 821\u2013825 (1976).","DOI":"10.1063\/1.522979"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Cheng-Ju Lin and Liujun Zou. ``Reaction-diffusion dynamics in a Fibonacci chain: Interplay between classical and quantum behavior&apos;&apos;. Phys. Rev. B 103, 174305 (2021).","DOI":"10.1103\/PhysRevB.103.174305"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Matthew Fishman, Steven R. White, and E. Miles Stoudenmire. ``The ITensor Software Library for Tensor Network Calculations&apos;&apos;. SciPost Phys. CodebasesPage 4 (2022).","DOI":"10.21468\/SciPostPhysCodeb.4"},{"key":"65","doi-asserted-by":"publisher","unstructured":"M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert. ``Dissipative quantum Church-Turing theorem&apos;&apos;. Phys. Rev. Lett. 107, 120501 (2011).","DOI":"10.1103\/PhysRevLett.107.120501"},{"key":"66","doi-asserted-by":"publisher","unstructured":"Dong An, Jin-Peng Liu, and Lin Lin. ``Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost&apos;&apos;. Phys. Rev. Lett. 131, 150603 (2023).","DOI":"10.1103\/PhysRevLett.131.150603"},{"key":"67","doi-asserted-by":"publisher","unstructured":"Archak Purkayastha, Giacomo Guarnieri, Steve Campbell, Javier Prior, and John Goold. ``Periodically refreshed baths to simulate open quantum many-body dynamics&apos;&apos;. Phys. Rev. B 104, 045417 (2021).","DOI":"10.1103\/PhysRevB.104.045417"},{"key":"68","doi-asserted-by":"publisher","unstructured":"Craig Gidney. ``Stim: a fast stabilizer circuit simulator&apos;&apos;. Quantum 5, 497 (2021).","DOI":"10.22331\/q-2021-07-06-497"},{"key":"69","doi-asserted-by":"publisher","unstructured":"Carlos S\u00e1nchez Mu\u00f1oz, Berislav Bu\u010da, Joseph Tindall, Alejandro Gonz\u00e1lez-Tudela, Dieter Jaksch, and Diego Porras. ``Symmetries and conservation laws in quantum trajectories: Dissipative freezing&apos;&apos;. Phys. Rev. A 100, 042113 (2019).","DOI":"10.1103\/PhysRevA.100.042113"},{"key":"70","doi-asserted-by":"publisher","unstructured":"Joseph Tindall, Dieter Jaksch, and Carlos S\u00e1nchez Mu\u00f1oz. ``On the generality of symmetry breaking and dissipative freezing in quantum trajectories&apos;&apos;. SciPost Phys. Core 6, 004 (2023).","DOI":"10.21468\/SciPostPhysCore.6.1.004"},{"key":"71","doi-asserted-by":"publisher","unstructured":"Carlos S\u00e1nchez Mu\u00f1oz, Berislav Bu\u010da, Joseph Tindall, Alejandro Gonz\u00e1lez-Tudela, Dieter Jaksch, and Diego Porras. ``Symmetries and conservation laws in quantum trajectories: Dissipative freezing&apos;&apos;. Phys. Rev. A 100, 042113 (2019).","DOI":"10.1103\/PhysRevA.100.042113"},{"key":"72","doi-asserted-by":"publisher","unstructured":"Catalin-Mihai Halati, Ameneh Sheikhan, and Corinna Kollath. ``Breaking strong symmetries in dissipative quantum systems: Bosonic atoms coupled to a cavity&apos;&apos;. Phys. Rev. Res. 4, L012015 (2022).","DOI":"10.1103\/PhysRevResearch.4.L012015"},{"key":"73","unstructured":"Chi-Fang Chen, Michael J Kastoryano, and Andr\u00e1s Gily\u00e9n. ``An efficient and exact noncommutative quantum Gibbs sampler&apos;&apos; (2023). arXiv:2311.09207."},{"key":"74","doi-asserted-by":"crossref","unstructured":"Cambyse Rouz\u00e9, Daniel Stilck Fran\u00e7a, and \u00c1lvaro M Alhambra. ``Efficient thermalization and universal quantum computing with quantum Gibbs samplers&apos;&apos; (2024). arXiv:2403.12691.","DOI":"10.1145\/3717823.3718268"},{"key":"75","doi-asserted-by":"publisher","unstructured":"Thiago Bergamaschi, Chi-Fang Chen, and Yunchao Liu. ``Quantum computational advantage with constant-temperature Gibbs sampling&apos;&apos; (2024). arXiv:2404.14639.","DOI":"10.1109\/FOCS61266.2024.00071"},{"key":"76","unstructured":"Joel Rajakumar and James D Watson. ``Gibbs sampling gives quantum advantage at constant temperatures with O(1)-local Hamiltonians&apos;&apos; (2024). arXiv:2408.01516."},{"key":"77","doi-asserted-by":"publisher","unstructured":"Shengqi Sang and Timothy H. Hsieh. ``Stability of mixed-state quantum phases via finite markov length&apos;&apos;. Phys. Rev. Lett. 134, 070403 (2025).","DOI":"10.1103\/PhysRevLett.134.070403"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-11-17-1912\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T13:30:52Z","timestamp":1763386252000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-11-17-1912\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,17]]},"references-count":78,"URL":"https:\/\/doi.org\/10.22331\/q-2025-11-17-1912","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,17]]},"article-number":"1912"}}