Khemani, V., Lazarides, A., Moessner, R. & Sondhi, S. L. Phase structure of driven quantum systems. Phys. Rev. Lett. 116, 250401 (2016).
Else, D. V., Bauer, B. & Nayak, C. Floquet time crystals. Phys. Rev. Lett. 117, 090402 (2016).
Yao, N., Potter, A., Potirniche, I.-D. & Vishwanath, A. Discrete time crystals: rigidity, criticality, and realizations. Phys. Rev. Lett. 118, 030401 (2017).
Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).
D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).
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).
Eisert, J., Friesdorf, M. & Gogolin, C. Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124–130 (2015).
Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk–edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).
Aidelsburger, M. et al. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111, 185301 (2013).
Bastidas, V. M., Emary, C., Regler, B. & Brandes, T. Nonequilibrium quantum phase transitions in the Dicke model. Phys. Rev. Lett. 108, 043003 (2012).
Yang, K. et al. Floquet dynamical quantum phase transitions. Phys. Rev. B 100, 085308 (2019).
Schweizer, C. et al. Floquet approach to \({{\mathbb{Z}}}_{2}\) lattice gauge theories with ultracold atoms in optical lattices. Nat. Phys. 15, 1168–1173 (2019).
Geier, S. et al. Floquet Hamiltonian engineering of an isolated many-body spin system. Science 374, 1149–1152 (2021).
Liu, Y. et al. Interplay between disorder and topology in Thouless pumping on a superconducting quantum processor. Nat. Commun. 16, 108 (2025).
Viebahn, K. et al. Suppressing dissipation in a Floquet–Hubbard system. Phys. Rev. X 11, 011057 (2021).
Bai, S.-Y. & An, J.-H. Floquet engineering to overcome no-go theorem of noisy quantum metrology. Phys. Rev. Lett. 131, 050801 (2023).
Weitenberg, C. & Simonet, J. Tailoring quantum gases by Floquet engineering. Nat. Phys. 17, 1342–1348 (2021).
Maskara, N. et al. Discrete time-crystalline order enabled by quantum many-body scars: entanglement steering via periodic driving. Phys. Rev. Lett. 127, 090602 (2021).
Nandy, S., Sen, A. & Sen, D. Aperiodically driven integrable systems and their emergent steady states. Phys. Rev. X 7, 031034 (2017).
Crowley, P., Martin, I. & Chandran, A. Half-integer quantized topological response in quasiperiodically driven quantum systems. Phys. Rev. Lett. 125, 100601 (2020).
Zhao, H., Mintert, F., Moessner, R. & Knolle, J. Random multipolar driving: tunably slow heating through spectral engineering. Phys. Rev. Lett. 126, 040601 (2021).
Schmid, H., Peng, Y., Refael, G. & von Oppen, F. Self-similar phase diagram of the Fibonacci-driven quantum Ising model. Phys. Rev. Lett. 134, 240404 (2025).
Pilatowsky-Cameo, S., Choi, S. & Ho, W. W. Critically slow Hilbert-space ergodicity in quantum morphic drives. Phys. Rev. Lett. 135, 140402 (2025).
Dumitrescu, P. T., Vasseur, R. & Potter, A. C. Logarithmically slow relaxation in quasiperiodically driven random spin chains. Phys. Rev. Lett. 120, 070602 (2018).
Zhao, H., Mintert, F. & Knolle, J. Floquet time spirals and stable discrete-time quasicrystals in quasiperiodically driven quantum many-body systems. Phys. Rev. B 100, 134302 (2019).
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).
He, G. et al. Experimental realization of discrete time quasicrystals. Phys. Rev. X 15, 011055 (2025).
Moon, L. J. I. et al. Experimental observation of a time rondeau crystal. Nat. Phys. 21, 1813–1819 (2025).
Page, D. N. Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993).
Bordia, P., Lüschen, H., Schneider, U., Knap, M. & Bloch, I. Periodically driving a many-body localized quantum system. Nat. Phys. 13, 460–464 (2017).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Berges, J., Borsányi, S. & Wetterich, C. Prethermalization. Phys. Rev. Lett. 93, 142002 (2004).
Mallayya, K., Rigol, M. & De Roeck, W. Prethermalization and thermalization in isolated quantum systems. Phys. Rev. X 9, 021027 (2019).
Rubio-Abadal, A. et al. Floquet prethermalization in a Bose–Hubbard system. Phys. Rev. X 10, 021044 (2020).
Mori, T., Zhao, H., Mintert, F., Knolle, J. & Moessner, R. Rigorous bounds on the heating rate in Thue–Morse quasiperiodically and randomly driven quantum many-body systems. Phys. Rev. Lett. 127, 050602 (2021).
Peng, P., Yin, C., Huang, X., Ramanathan, C. & Cappellaro, P. Floquet prethermalization in dipolar spin chains. Nat. Phys. 17, 444–447 (2021).
Beatrez, W. et al. Critical prethermal discrete time crystal created by two-frequency driving. Nat. Phys. 19, 407–413 (2023).
He, G. et al. Quasi-Floquet prethermalization in a disordered dipolar spin ensemble in diamond. Phys. Rev. Lett. 131, 130401 (2023).
Morningstar, A. et al. Simulation of quantum many-body dynamics with tensor processing units: Floquet prethermalization. PRX Quantum 3, 020331 (2022).
Yang, Y. et al. Simulating prethermalization using near-term quantum computers. PRX Quantum 4, 030320 (2023).
Deng, C.-L. et al. High-order topological pumping on a superconducting quantum processor. Phys. Rev. Lett. 133, 140402 (2024).
Yan, F. et al. Tunable coupling scheme for implementing high-fidelity two-qubit gates. Phys. Rev. Appl. 10, 054062 (2018).
Krantz, P. et al. A quantum engineer’s guide to superconducting qubits. Appl. Phys. Rev. 6, 021318 (2019).
Liang, G.-H. et al. Tunable-coupling architectures with capacitively connecting pads for large-scale superconducting multiqubit processors. Phys. Rev. Appl. 20, 044028 (2023).
Ho, W. W., Mori, T., Abanin, D. A. & Dalla Torre, E. G. Quantum and classicalFloquet prethermalization. Ann. Phys. 454, 169297 (2023).
Machado, F., Kahanamoku-Meyer, G. D., Else, D. V., Nayak, C. & Yao, N. Y. Exponentially slow heating in short and long-range interacting floquet systems. Phys. Rev. Res. 1, 033202 (2019).
Osterloh, A., Amico, L., Falci, G. & Fazio, R. Scaling of entanglement close to a quantum phase transition. Nature 416, 608–610 (2002).
Eisert, J., Cramer, M. & Plenio, M. B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys. 82, 277 (2010).
Karamlou, A. H. et al. Probing entanglement in a 2D hard-core Bose–Hubbard lattice. Nature 629, 561–566 (2024).
Xu, K. et al. Emulating many-body localization with a superconducting quantum processor. Phys. Rev. Lett. 120, 050507 (2018).
Cirac, J. I., Pérez-García, D., Schuch, N. & Verstraete, F. Matrix product states and projected entangled pair states: concepts, symmetries, theorems. Rev. Mod. Phys. 93, 045003 (2021).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Zhong, H.-S. et al. Quantum computational advantage using photons. Science 370, 1460–1463 (2020).
Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).
Daley, A. J. et al. Practical quantum advantage in quantum simulation. Nature 607, 667–676 (2022).
Gao, D. et al. Establishing a new benchmark in quantum computational advantage with 105-qubit Zuchongzhi 3.0 processor. Phys. Rev. Lett. 134, 090601 (2025).
King, A. D. et al. Beyond-classical computation in quantum simulation. Science 388, 199–204 (2025).
Zhao, H., Rudner, M. S., Moessner, R. & Knolle, J. Anomalous random multipolar driven insulators. Phys. Rev. B 105, 245119 (2022).

