Chiu, C.-K., Teo, J. C. Y., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).
Wen, X.-G. Colloquium: zoo of quantum-topological phases of matter. Rev. Mod. Phys. 89, 041004 (2017).
Chen, X., Gu, Z.-C., Liu, Z.-X. & Wen, X.-G. Symmetry-protected topological orders in interacting bosonic systems. Science 338, 1604–1606 (2012).
Senthil, T. Symmetry-protected topological phases of quantum matter. Annu. Rev. Condens. Matter Phys. 6, 299–324 (2015).
Landau, L. D. & Lifshitz, E. M. Statistical Physics 3rd edn, Vol. 5 (Elsevier, 2013).
Pollmann, F., Berg, E., Turner, A. M. & Oshikawa, M. Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B 85, 075125 (2012).
Fidkowski, L. & Kitaev, A. Topological phases of fermions in one dimension. Phys. Rev. B 83, 075103 (2011).
Ma, R. & Wang, C. Average symmetry-protected topological phases. Phys. Rev. X 13, 031016 (2023).
Brown, B. J., Loss, D., Pachos, J. K., Self, C. N. & Wootton, J. R. Quantum memories at finite temperature. Rev. Mod. Phys. 88, 045005 (2016).
Hastings, M. B. Topological order at nonzero temperature. Phys. Rev. Lett. 107, 210501 (2011).
Roberts, S., Yoshida, B., Kubica, A. & Bartlett, S. D. Symmetry-protected topological order at nonzero temperature. Phys. Rev. A 96, 022306 (2017).
Else, D. V., Fendley, P., Kemp, J. & Nayak, C. Prethermal strong zero modes and topological qubits. Phys. Rev. X 7, 041062 (2017).
Parker, D. E., Vasseur, R. & Scaffidi, T. Topologically protected long edge coherence times in symmetry-broken phases. Phys. Rev. Lett. 122, 240605 (2019).
Kemp, J., Yao, N. Y. & Laumann, C. R. Symmetry-enhanced boundary qubits at infinite temperature. Phys. Rev. Lett. 125, 200506 (2020).
Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153–185 (2014).
Tacchino, F., Chiesa, A., Carretta, S. & Gerace, D. Quantum computers as universal quantum simulators: state-of-the-art and perspectives. Adv. Quantum Technol. 3, 1900052 (2020).
Daley, A. J. et al. Practical quantum advantage in quantum simulation. Nature 607, 667–676 (2022).
Fauseweh, B. Quantum many-body simulations on digital quantum computers: state-of-the-art and future challenges. Nat. Commun. 15, 2123 (2024).
Bravyi, S., Hastings, M. B. & Michalakis, S. Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010).
Nandkishore, R. & Huse, D. A. Many-body localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15–38 (2015).
Kjäll, J. A., Bardarson, J. H. & Pollmann, F. Many-body localization in a disordered quantum Ising chain. Phys. Rev. Lett. 113, 107204 (2014).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: many-body localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Huse, D. A., Nandkishore, R., Oganesyan, V., Pal, A. & Sondhi, S. L. Localization-protected quantum order. Phys. Rev. B 88, 014206 (2013).
Chandran, A., Khemani, V., Laumann, C. R. & Sondhi, S. L. Many-body localization and symmetry-protected topological order. Phys. Rev. B 89, 144201 (2014).
Bahri, Y., Vosk, R., Altman, E. & Vishwanath, A. Localization and topology protected quantum coherence at the edge of hot matter. Nat. Commun. 6, 7341 (2015).
Schreiber, M. et al. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science 349, 842–845 (2015).
Choi, J.-Y. et al. Exploring the many-body localization transition in two dimensions. Science 352, 1547–1552 (2016).
Smith, J. et al. Many-body localization in a quantum simulator with programmable random disorder. Nat. Phys. 12, 907–911 (2016).
Morningstar, A., Colmenarez, L., Khemani, V., Luitz, D. J. & Huse, D. A. Avalanches and many-body resonances in many-body localized systems. Phys. Rev. B 105, 174205 (2022).
Ha, H., Morningstar, A. & Huse, D. A. Many-body resonances in the avalanche instability of many-body localization. Phys. Rev. Lett. 130, 250405 (2023).
Long, D. M., Crowley, P. J. D., Khemani, V. & Chandran, A. Phenomenology of the prethermal many-body localized regime. Phys. Rev. Lett. 131, 106301 (2023).
Léonard, J. et al. Probing the onset of quantum avalanches in a many-body localized system. Nat. Phys. 19, 481–485 (2023).
Fendley, P. Parafermionic edge zero modes in Zn-invariant spin chains. J. Stat. Mech. 2012, P11020 (2012).
Fendley, P. Strong zero modes and eigenstate phase transitions in the XYZ/interacting Majorana chain. J. Phys. A Math. Theor. 49, 30LT01 (2016).
Kemp, J., Yao, N. Y., Laumann, C. R. & Fendley, P. Long coherence times for edge spins. J. Stat. Mech. 2017, 063105 (2017).
Yates, D. J., Abanov, A. G. & Mitra, A. Dynamics of almost strong edge modes in spin chains away from integrability. Phys. Rev. B 102, 195419 (2020).
Zhang, X. et al. Digital quantum simulation of Floquet symmetry-protected topological phases. Nature 607, 468–473 (2022).
Dumitrescu, P. T. et al. Dynamical topological phase realized in a trapped-ion quantum simulator. Nature 607, 463–467 (2022).
Mi, X. et al. Noise-resilient edge modes on a chain of superconducting qubits. Science 378, 785–790 (2022).
Briegel, H. J. & Raussendorf, R. Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910–913 (2001).
Abanin, D. A., De Roeck, W. & Huveneers, F. Exponentially slow heating in periodically driven many-body systems. Phys. Rev. Lett. 115, 256803 (2015).
Mori, T., Kuwahara, T. & Saito, K. Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems. Phys. Rev. Lett. 116, 120401 (2016).
Abanin, D., De Roeck, W., Ho, W. W. & Huveneers, F. A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems. Commun. Math. Phys. 354, 809–827 (2017).
Mi, X. et al. Time-crystalline eigenstate order on a quantum processor. Nature 601, 531–536 (2022).
Else, D. V., Bauer, B. & Nayak, C. Prethermal phases of matter protected by time-translation symmetry. Phys. Rev. X 7, 011026 (2017).
Yin, C. & Lucas, A. Prethermalization and the local robustness of gapped systems. Phys. Rev. Lett. 131, 050402 (2023).
Roushan, P. et al. Spectroscopic signatures of localization with interacting photons in superconducting qubits. Science 358, 1175–1179 (2017).
Xu, S. et al. Digital simulation of projective non-Abelian anyons with 68 superconducting qubits. Chin. Phys. Lett. 40, 060301 (2023).
Olund, C. T., Yao, N. Y. & Kemp, J. Boundary strong zero modes. Phys. Rev. B 111, L201114 (2025).
Wildeboer, J., Iadecola, T. & Williamson, D. J. Symmetry-protected infinite-temperature quantum memory from subsystem codes. PRX Quantum 3, 020330 (2022).
Jiang, S., Yuan, D., Jiang, W., Deng, D.-L. & Machado, F. Prethermal time-crystalline corner modes. Phys. Rev. Lett. https://doi.org/10.1103/np9w-jsf9 (2025).
Else, D. V., Ho, W. W. & Dumitrescu, P. T. Long-lived interacting phases of matter protected by multiple time-translation symmetries in quasiperiodically driven systems. Phys. Rev. X 10, 021032 (2020).
Friedman, A. J., Ware, B., Vasseur, R. & Potter, A. C. Topological edge modes without symmetry in quasiperiodically driven spin chains. Phys. Rev. B 105, 115117 (2022).
Koch, J. et al. Charge-insensitive qubit design derived from the cooper pair box. Phys. Rev. A 76, 042319 (2007).
Yan, F. et al. Tunable coupling scheme for implementing high-fidelity two-qubit gates. Phys. Rev. Appl. 10, 054062 (2018).
Foxen, B. et al. Demonstrating a continuous set of two-qubit gates for near-term quantum algorithms. Phys. Rev. Lett. 125, 120504 (2020).
Magnus, W. On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954).
Blanes, S., Casas, F., Oteo, J. A. & Ros, J. The magnus expansion and some of its applications. Phys. Rep. 470, 151–238 (2009).
Jin, F. et al. Topological prethermal strong zero modes on superconducting processors. Code Ocean https://codeocean.com/capsule/8397511/tree (2025).