Similar articles

Abstract: Quantum many-body systems lie at the heart of modern fundamental physics. The study of these systems has revealed a plethora of fascinating phenomena, such as quantum thermalization, many-body localization, and quantum many-body scars. This review provides a comprehensive overview of the recent advances in understanding quantum many-body scars and non-ergodic dynamics in quantum systems on superconducting-circuit platforms, ranging from theoretical mechanisms and effective models to experimental observations.

量子海森堡XY模型和超导电路模拟中的多体疤痕

作者：郭泽贤，马金楼，高宇，应磊
机构：浙江大学，物理学院，浙江省微纳量子芯片与量子控制重点实验室，中国杭州，310027
概要：量子多体系统是现代基础物理学的核心。对这些系统的研究揭示了许多有趣的现象，如量子热化、多体局域和量子多体疤痕。本综述聚焦于超导电路平台，系统地回顾了该平台上量子多体疤痕和非遍历动力学的前沿进展，并深入探讨了其理论基础、有效模型和关键的实验观测。

关键词：量子多体疤痕；量子混沌；量子模拟；超导电路

[1]AdlerD, WeiD, WillM, et al., 2024. Observation of Hilbert space fragmentation and fractonic excitations in 2D. Nature, 636(8041):80-85.

[2]AtasYY, BogomolnyE, GiraudO, et al., 2013. Distribution of the ratio of consecutive level spacings in random matrix ensembles. Physical Review Letters, 110(8):084101.

[3]BabukhinDV, ZhukovAA, PogosovWV, 2020. Hybrid digital-analog simulation of many-body dynamics with superconducting qubits. Physical Review A, 101(5):052337.

[4]Bar LevY, ReichmanDR, 2014. Dynamics of many-body localization. Physical Review B, 89(22):220201.

[5]BaskoDM, AleinerIL, AltshulerBL, 2006. Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states. Annals of Physics, 321(5):1126-1205.

[6]BernienH, SchwartzS, KeeslingA, et al., 2017. Probing many-body dynamics on a 51-atom quantum simulator. Nature, 551(7682):579-584.

[7]BerryMV, TaborM, 1977. Level clustering in the regular spectrum. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 356(1686):‍375-394.

[8]BertiniB, EsslerFHL, GrohaS, et al., 2015. Prethermalization and thermalization in models with weak integrability breaking. Physical Review Letters, 115(18):180601.

[9]BeugelingW, MoessnerR, HaqueM, 2014. Finite-size scaling of eigenstate thermalization. Physical Review E, 89(4):042112.

[10]BhattacharjeeB, SurS, NandyP, 2022. Probing quantum scars and weak ergodicity breaking through quantum complexity. Physical Review B, 106(20):205150.

[11]BluvsteinD, OmranA, LevineH, et al., 2021. Controlling quantum many-body dynamics in driven Rydberg atom arrays. Science, 371(6536):1355-1359.

[12]BohigasO, GiannoniMJ, SchmitC, 1984. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Physical Review Letters, 52(1):1-4.

[13]BučaB, 2023. Unified theory of local quantum many-body dynamics: eigenoperator thermalization theorems. Physical Review X, 13(3):031013.

[14]BuddeT, Krstic MarinkovicM, Pinto BarrosJC, 2024. Quantum many-body scars for arbitrary integer spin in 2+1D Abelian gauge theories. Physical Review D, 110(9):094506.

[15]BullK, MartinI, PapićZ, 2019. Systematic construction of scarred many-body dynamics in 1D lattice models. Physical Review Letters, 123(3):030601.

[16]KANBull, 2022. The Anatomy of Quantum Many-Body Scars: Origins and Implementations. PhD Thesis, University of Leeds, Leeds, UK.

[17]BurkePC, DooleyS, 2025. Taking the temperature of quantum many-body scars. arXiv: 2503.21884.

[18]CalajóG, CataldiG, RigobelloM, et al., 2025. Quantum many-body scarring in a non-Abelian lattice gauge theory. Physical Review Research, 7(1):013322.

[19]CaoH, AngelakisDG, LeykamD, 2024. Unsupervised learning of quantum many-body scars using intrinsic dimension. Machine Learning: Science and Technology, 5(2):025049.

[20]CauxJS, MosselJ, 2011. Remarks on the notion of quantum integrability. Journal of Statistical Mechanics: Theory and Experiment, 2011(2):P02023.

[21]CazalillaMA, RigolM, 2010. Focus on dynamics and thermalization in isolated quantum many-body systems. New Journal of Physics, 12(5):055006.

[22]ChandranA, IadecolaT, KhemaniV, et al., 2023. Quantum many-body scars: a quasiparticle perspective. Annual Review of Condensed Matter Physics, 14:443-469.

[23]ChenLY, LiHX, LuY, et al., 2023. Transmon qubit readout fidelity at the threshold for quantum error correction without a quantum-limited amplifier. npj Quantum Information, 9(1):26.

[24]ChoiS, TurnerCJ, PichlerH, et al., 2019. Emergent SU(2) dynamics and perfect quantum many-body scars. Physical Review Letters, 122(22):220603.

[25]D’AlessioL, KafriY, PolkovnikovA, et al., 2016. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Advances in Physics, 65(3):239-362.

[26]DavoudiZ, HafeziM, MonroeC, et al., 2020. Towards analog quantum simulations of lattice gauge theories with trapped ions. Physical Review Research, 2(2):023015.

[27]de TomasiG, HetterichD, SalaP, et al., 2019. Dynamics of strongly interacting systems: from Fock-space fragmentation to many-body localization. Physical Review B, 100(21):214313.

[28]DesaulesJY, PietracaprinaF, PapićZ, et al., 2022. Extensive multipartite entanglement from SU(2) quantum many-body scars. Physical Review Letters, 129(2):020601.

[29]DeutschJM, 1991. Quantum statistical mechanics in a closed system. Physical Review A, 43(4):2046-2049.

[30]DeutschJM, 2018. Eigenstate thermalization hypothesis. Reports on Progress in Physics, 81(8):082001.

[31]di MeglioA, JansenK, TavernelliI, et al., 2024. Quantum computing for high-energy physics: state of the art and challenges. PRX Quantum, 5(3):037001.

[32]DingDS, BaiZY, LiuZK, et al., 2024. Ergodicity breaking from Rydberg clusters in a driven-dissipative many-body system. Science Advances, 10(9):eadl5893.

[33]DingL, HaysM, SungY, et al., 2023. High-fidelity, frequency-flexible two-qubit fluxonium gates with a transmon coupler. Physical Review X, 13(3):031035.

[34]DongH, DesaulesJY, GaoY, et al., 2023. Disorder-tunable entanglement at infinite temperature. Science Advances, 9(51):eadj3822.

[35]DongH, ZhangPF, DağCB, et al., 2025. Measuring the spectral form factor in many-body chaotic and localized phases of quantum processors. Physical Review Letters, 134(1):010402.

[36]DooleyS, 2021. Robust quantum sensing in strongly interacting systems with many-body scars. PRX Quantum, 2(2):020330.

[37]DooleyS, PappalardiS, GooldJ, 2023. Entanglement enhanced metrology with quantum many-body scars. Physical Review B, 107(3):035123.

[38]DysonFJ, 1962. Statistical theory of the energy levels of complex systems. I. Journal of Mathematical Physics, 3(1):140-156.

[39]EbadiS, WangTT, LevineH, et al., 2021. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature, 595(7866):227-232.

[40]ErmakovI, LychkovskiyO, FineBV, 2024. Periodic classical trajectories and quantum scars in many-spin systems. arXiv: 2409.00258.

[41]EvrardB, PizziA, MistakidisSI, et al., 2024. Quantum many-body scars from unstable periodic orbits. Physical Review B, 110(14):144302.

[42]FengJJ, ZhangBZ, YangZC, et al., 2025. Uncovering quantum many-body scars with quantum machine learning. npj Quantum Information, 11(1):42.

[43]FogartyT, García-MarchMÁ, SantosLF, et al., 2021. Probing the edge between integrability and quantum chaos in interacting few-atom systems. Quantum, 5:486.

[44]FriedmanAJ, ChanA, de LucaA, et al., 2019. Spectral statistics and many-body quantum chaos with conserved charge. Physical Review Letters, 123(21):210603.

[45]GarrisonJR, GroverT, 2018. Does a single eigenstate encode the full Hamiltonian? Physical Review X, 8(2):021026.

[46]GrabowskiMP, MathieuP, 1995. Structure of the conservation laws in quantum integrable spin chains with short range interactions. Annals of Physics, 243(2):299-371.

[47]GuoQJ, ChengC, SunZH, et al., 2021a. Observation of energy-resolved many-body localization. Nature Physics, 17(2):234-239.

[48]GuoQJ, ChengC, LiHK, et al., 2021b. Stark many-body localization on a superconducting quantum processor. Physical Review Letters, 127(24):240502.

[49]GuoZX, LiuBB, GaoY, et al., 2023. Origin of Hilbert-space quantum scars in unconstrained models. Physical Review B, 108(7):075124.

[50]HaakeF, 1991. Quantum signatures of chaos. In: Kramer B (Ed.), Quantum Coherence in Mesoscopic Systems. Springer, New York, USA, p.583-595.

[51]HallamA, DesaulesJY, PapićZ, 2023. Embedding semiclassical periodic orbits into chaotic many-body Hamiltonians. Physical Review Letters, 131(11):110401.

[52]HarrisJ, YanB, SinitsynNA, 2022. Benchmarking information scrambling. Physical Review Letters, 129(5):050602.

[53]HoWW, ChoiS, PichlerH, et al., 2019. Periodic orbits, entanglement, and quantum many-body scars in constrained models: matrix product state approach. Physical Review Letters, 122(4):040603.

[54]HuangJY, YeLL, LaiYC, 2025. Floquet quantum many-body scars in the tilted Fermi-Hubbard chain. arXiv: 2504.02152.

[55]HudomalA, VasićI, RegnaultN, et al., 2020. Quantum scars of bosons with correlated hopping. Communications Physics, 3(1):99.

[56]HuseDA, NandkishoreR, OganesyanV, 2014. Phenomenology of fully many-body-localized systems. Physical Review B, 90(17):174202.

[57]ImaiS, TsujiN, 2025. Quantum many-body scars with unconventional superconducting pairing symmetries via multibody interactions. Physical Review Research, 7(1):013064.

[58]ImbrieJZ, 2016. On many-body localization for quantum spin chains. Journal of Statistical Physics, 163(5):998-1048.

[59]IvanovAN, MotrunichOI, 2025. Many exact area-law scar eigenstates in the nonintegrable PXP and related models. arXiv: 2503.16327.

[60]JoshiLK, ElbenA, VikramA, et al., 2022. Probing many-body quantum chaos with quantum simulators. Physical Review X, 12(1):011018.

[61]KanekoR, KunimiM, DanshitaI, 2024. Quantum many-body scars in the Bose-Hubbard model with a three-body constraint. Physical Review A, 109(1):L011301.

[62]KaplanHB, GuoLZ, TanWL, et al., 2020. Many-body dephasing in a trapped-ion quantum simulator. Physical Review Letters, 125(12):120605.

[63]KaufmanAM, TaiME, LukinA, et al., 2016. Quantum thermalization through entanglement in an isolated many-body system. Science, 353(6301):794-800.

[64]KerschbaumerA, LjubotinaM, SerbynM, et al., 2025. Quantum many-body scars beyond the PXP model in Rydberg simulators. Physical Review Letters, 134(16):160401.

[65]KhemaniV, LaumannCR, ChandranA, 2019. Signatures of integrability in the dynamics of Rydberg-blockaded chains. Physical Review B, 99(16):161101.

[66]KolbP, PakrouskiK, 2023. Stability of the many-body scars in fermionic spin-1/2 models. PRX Quantum, 4(4):040348.

[67]KounalakisM, DickelC, BrunoA, et al., 2018. Tuneable hopping and nonlinear cross-Kerr interactions in a high-coherence superconducting circuit. npj Quantum Information, 4(1):38.

[68]KrantzP, KjaergaardM, YanF, et al., 2019. A quantum engineer’s guide to superconducting qubits. Applied Physics Reviews, 6(2):021318.

[69]LamataL, Parra-RodriguezA, SanzM, et al., 2018. Digital-analog quantum simulations with superconducting circuits. Advances in Physics: X, 3(1):1457981.

[70]LanyonBP, MaierC, HolzäpfelM, et al., 2017. Efficient tomography of a quantum many-body system. Nature Physics, 13(12):1158-1162.

[71]LarsenPG, NielsenAEB, 2024. Phase transitions in quantum many-body scars. Physical Review Research, 6(4):L042007.

[72]LeBlondT, MallayyaK, VidmarL, et al., 2019. Entanglement and matrix elements of observables in interacting integrable systems. Physical Review E, 100(6):062134.

[73]LeroseA, ParoliniT, FazioR, et al., 2025. Theory of robust quantum many-body scars in long-range interacting systems. Physical Review X, 15(1):011020.

[74]LiKM, DongH, SongC, et al., 2019. Approaching the chaotic regime with a fully connected superconducting quantum processor. Physical Review A, 100(6):062302.

[75]LiangXH, YueZP, ChaoYX, et al., 2025. Observation of anomalous information scrambling in a Rydberg atom array. Physical Review Letters, 135(5):050201.

[76]LinCJ, MotrunichOI, 2019. Exact quantum many-body scar states in the Rydberg-blockaded atom chain. Physical Review Letters, 122(17):173401.

[77]LinCJ, CalveraV, HsiehTH, 2020a. Quantum many-body scar states in two-dimensional Rydberg atom arrays. Physical Review B, 101(22):220304.

[78]LinCJ, ChandranA, MotrunichOI, 2020b. Slow thermalization of exact quantum many-body scar states under perturbations. Physical Review Research, 2(3):033044.

[79]LiskaD, GritsevV, VleeshouwersW, et al., 2023. Holographic quantum scars. SciPost Physics, 15(3):106.

[80]LjubotinaM, RoosB, AbaninDA, et al., 2022. Optimal steering of matrix product states and quantum many-body scars. PRX Quantum, 3(3):030343.

[81]MaJL, GuoZX, GaoY, et al., 2025. Liouvillean spectral transition in noisy quantum many-body scars. arXiv: 2504.12291.

[82]MaldacenaJ, StanfordD, 2016. Remarks on the Sachdev-Ye-Kitaev model. Physical Review D, 94(10):106002.

[83]MaldacenaJ, ShenkerSH, StanfordD, 2016. A bound on chaos. Journal of High Energy Physics, 2016(8):106.

[84]MarkDK, MotrunichOI, 2020. η-pairing states as true scars in an extended Hubbard model. Physical Review B, 102(7):075132.

[85]McClartyPA, HaqueM, SenA, et al., 2020. Disorder-free localization and many-body quantum scars from magnetic frustration. Physical Review B, 102(22):224303.

[86]MehtaML, 2004. Random Matrices. 3rd Edition. Academic Press, London, UK.

[87]MichailidisAA, TurnerCJ, PapićZ, et al., 2020. Slow quantum thermalization and many-body revivals from mixed phase space. Physical Review X, 10(1):011055.

[88]MizutaK, TakasanK, KawakamiN, 2020. Exact Floquet quantum many-body scars under Rydberg blockade. Physical Review Research, 2(3):033284.

[89]MondainiR, FratusKR, SrednickiM, et al., 2016. Eigenstate thermalization in the two-dimensional transverse field Ising model. Physical Review E, 93(3):032104.

[90]Mondragon-ShemI, VavilovMG, MartinI, 2021. Fate of quantum many-body scars in the presence of disorder. PRX Quantum, 2(3):030349.

[91]MonroeC, CampbellWC, DuanLM, et al., 2021. Programmable quantum simulations of spin systems with trapped ions. Reviews of Modern Physics, 93(2):025001.

[92]MoudgalyaS, RachelS, BernevigBA, et al., 2018a. Exact excited states of nonintegrable models. Physical Review B, 98(23):235155.

[93]MoudgalyaS, RegnaultN, BernevigBA, 2018b. Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis. Physical Review B, 98(23):235156.

[94]MoudgalyaS, RegnaultN, BernevigBA, 2020. η‍-pairing in Hubbard models: from spectrum generating algebras to quantum many-body scars. Physical Review B, 102(8):085140.

[95]MoudgalyaS, BernevigBA, RegnaultN, 2022. Quantum many-body scars and Hilbert space fragmentation: a review of exact results. Reports on Progress in Physics, 85(8):086501.

[96]MukherjeeB, NandyS, SenA, et al., 2020a. Collapse and revival of quantum many-body scars via Floquet engineering. Physical Review B, 101(24):245107.

[97]MukherjeeB, SenA, SenD, et al., 2020b. Restoring coherence via aperiodic drives in a many-body quantum system. Physical Review B, 102(1):014301.

[98]NandkishoreR, HuseDA, 2015. Many-body localization and thermalization in quantum statistical mechanics. Annual Review of Condensed Matter Physics, 6:15-38.

[99]NandyS, MukherjeeB, BhattacharyyaA, et al., 2024. Quantum state complexity meets many-body scars. Journal of Physics: Condensed Matter, 36(15):155601.

[100]NeillC, RoushanP, FangM, et al., 2016. Ergodic dynamics and thermalization in an isolated quantum system. Nature Physics, 12(11):1037-1041.

[101]O’DeaN, BurnellF, ChandranA, et al., 2020. From tunnels to towers: quantum scars from lie algebras and q-deformed lie algebras. Physical Review Research, 2(4):043305.

[102]OganesyanV, HuseDA, 2007. Localization of interacting fermions at high temperature. Physical Review B, 75(15):155111.

[103]OkS, ChooK, MudryC, et al., 2019. Topological many-body scar states in dimensions one, two, and three. Physical Review Research, 1(3):033144.

[104]OmiyaK, 2025. Quantum many-body scars as remnants of stable many-body periodic orbits. Physical Review B, 111(24):245158.

[105]OmiyaK, MüllerM, 2023a. Fractionalization paves the way to local projector embeddings of quantum many-body scars. Physical Review B, 108(5):054412.

[106]OmiyaK, MüllerM, 2023b. Quantum many-body scars in bipartite Rydberg arrays originating from hidden projector embedding. Physical Review A, 107(2):023318.

[107]OrellT, MichailidisAA, SerbynM, et al., 2019. Probing the many-body localization phase transition with superconducting circuits. Physical Review B, 100(13):134504.

[108]OsborneJ, McCullochIP, HalimehJC, 2024. Quantum many-body scarring in 2+1D gauge theories with dynamical matter. arXiv: 2403.08858.

[109]PageDN, 1993. Average entropy of a subsystem. Physical Review Letters, 71(9):1291-1294.

[110]PakrouskiK, PallegarPN, PopovFK, et al., 2020. Many-body scars as a group invariant sector of Hilbert space. Physical Review Letters, 125(23):230602.

[111]PetrovaE, LjubotinaM, YalnızG, et al., 2025. Finding periodic orbits in projected quantum many-body dynamics. arXiv: 2504.12472.

[112]Pilatowsky-CameoS, VillaseñorD, Bastarrachea-MagnaniMA, et al., 2021. Ubiquitous quantum scarring does not prevent ergodicity. Nature Communications, 12(1):852.

[113]RenJ, LiangCG, FangC, 2021. Quasisymmetry groups and many-body scar dynamics. Physical Review Letters, 126(12):120604.

[114]RenJ, HallamA, YingL, et al., 2025. ScarFinder: a detector of optimal scar trajectories in quantum many-body dynamics. arXiv: 2504.12383.

[115]RigolM, DunjkoV, YurovskyV, et al., 2007. Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Physical Review Letters, 98(5):050405.

[116]RigolM, DunjkoV, OlshaniiM, 2008. Thermalization and its mechanism for generic isolated quantum systems. Nature, 452(7189):854-858.

[117]SanadaK, MiaoY, KatsuraH, 2023. Quantum many-body scars in spin models with multibody interactions. Physical Review B, 108(15):155102.

[118]SchecterM, IadecolaT, 2019. Weak ergodicity breaking and quantum many-body scars in spin-1 XY magnets. Physical Review Letters, 123(14):147201.

[119]SchergS, KohlertT, SalaP, et al., 2021. Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains. Nature Communications, 12(1):4490.

[120]SchindlerF, RegnaultN, BernevigBA, 2022. Exact quantum scars in the chiral nonlinear Luttinger liquid. Physical Review B, 105(3):035146.

[121]SchmittM, HeylM, 2020. Quantum many-body dynamics in two dimensions with artificial neural networks. Physical Review Letters, 125(10):100503.

[122]SchreiberM, HodgmanSS, BordiaP, et al., 2015. Observation of many-body localization of interacting fermions in a quasirandom optical lattice. Science, 349(6250):842-845.

[123]SemeghiniG, LevineH, KeeslingA, et al., 2021. Probing topological spin liquids on a programmable quantum simulator. Science, 374(6572):1242-1247.

[124]SerbynM, AbaninDA, PapićZ, 2021. Quantum many-body scars and weak breaking of ergodicity. Nature Physics, 17(6):675-685.

[125]ShenRZ, QinF, DesaulesJY, et al., 2024. Enhanced many-body quantum scars from the non-Hermitian Fock skin effect. Physical Review Letters, 133(21):216601.

[126]ShibataN, YoshiokaN, KatsuraH, 2020. Onsager’s scars in disordered spin chains. Physical Review Letters, 124(18):180604.

[127]ShiraishiN, MoriT, 2017. Systematic construction of counterexamples to the eigenstate thermalization hypothesis. Physical Review Letters, 119(3):030601.

[128]SilevitchDM, TangC, AeppliG, et al., 2019. Tuning high-Q nonlinear dynamics in a disordered quantum magnet. Nature Communications, 10(1):4001.

[129]SmithJ, LeeA, RichermeP, et al., 2016. Many-body localization in a quantum simulator with programmable random disorder. Nature Physics, 12(10):907-911.

[130]SomoroffA, FicheuxQ, MenciaRA, et al., 2023. Millisecond coherence in a superconducting qubit. Physical Review Letters, 130(26):267001.

[131]SrednickiM, 1994. Chaos and quantum thermalization. Physical Review E, 50(2):888-901.

[132]SrednickiM, 1999. The approach to thermal equilibrium in quantized chaotic systems. Journal of Physics A: Mathematical and General, 32(7):1163-1175.

[133]SuGX, SunH, HudomalA, et al., 2023. Observation of many-body scarring in a Bose-Hubbard quantum simulator. Physical Review Research, 5(2):023010.

[134]SuraceFM, DalmonteM, SilvaA, 2023. Quantum local random networks and the statistical robustness of quantum scars. SciPost Physics, 14(6):174.

[135]SutherlandB, 2004. Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems. World Scientific, River Edge, USA.

[136]TakagiR, EndoS, MinagawaS, et al., 2022. Fundamental limits of quantum error mitigation. npj Quantum Information, 8(1):114.

[137]TurnerCJ, MichailidisAA, AbaninDA, et al., 2018a. Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations. Physical Review B, 98(15):155134.

[138]TurnerCJ, MichailidisAA, AbaninDA, et al., 2018b. Weak ergodicity breaking from quantum many-body scars. Nature Physics, 14(7):745-749.

[139]van DammeM, DesaulesJY, PapićZ, et al., 2023. Anatomy of dynamical quantum phase transitions. Physical Review Research, 5(3):033090.

[140]WangHR, YuanD, ZhangSY, et al., 2024. Embedding quantum many-body scars into decoherence-free subspaces. Physical Review Letters, 132(15):150401.

[141]WangJW, ZhouXF, GuoGC, et al., 2024. Quantum many-body scar models in one-dimensional spin chains. Physical Review B, 109(12):125102.

[142]WangYY, ShiYH, SunZH, et al., 2025. Exploring Hilbert-space fragmentation on a superconducting processor. PRX Quantum, 6(1):010325.

[143]WeiKX, RamanathanC, CappellaroP, 2018. Exploring localization in nuclear spin chains. Physical Review Letters, 120(7):070501.

[144]WignerEP, 1955. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics, 62(3):548-564.

[145]WildeboerJ, LanglettCM, YangZC, et al., 2022. Quantum many-body scars from Einstein-Podolsky-Rosen states in bilayer systems. Physical Review B, 106(20):205142.

[146]WrightK, BeckKM, DebnathS, et al., 2019. Benchmarking an 11-qubit quantum computer. Nature Communications, 10(1):5464.

[147]XiangDS, ZhangYW, LiuHX, et al., 2024. Observation of quantum information collapse-and-revival in a strongly-interacting Rydberg atom array. arXiv: 2410.15455.

[148]XuK, ChenJJ, ZengY, et al., 2018. Emulating many-body localization with a superconducting quantum processor. Physical Review Letters, 120(5):050507.

[149]YangKN, ZhangYC, LiKY, et al., 2024. Phantom energy in the nonlinear response of a quantum many-body scar state. Science, 385(6713):1063-1067.

[150]YaoYY, XiangL, GuoZX, et al., 2023. Observation of many-body Fock space dynamics in two dimensions. Nature Physics, 19(10):1459-1465.

[151]YeLL, LaiYC, 2025. Controlling nonergodicity in quantum many-body systems by reinforcement learning. Physical Review Research, 7(1):013256.

[152]YuanD, ZhangSY, WangY, et al., 2022. Quantum information scrambling in quantum many-body scarred systems. Physical Review Research, 4(2):023095.

[153]ZhangPF, DongH, GaoY, et al., 2023. Many-body Hilbert space scarring on a superconducting processor. Nature Physics, 19(1):120-125.

[154]ZhangSY, YuanD, IadecolaT, et al., 2023. Extracting quantum many-body scarred eigenstates with matrix product states. Physical Review Letters, 131(2):020402.

[155]ZhaoHZ, VovroshJ, MintertF, et al., 2020. Quantum many-body scars in optical lattices. Physical Review Letters, 124(16):160604.

[156]ZhaoLH, DatlaPR, TianWK, et al., 2025. Observation of quantum thermalization restricted to Hilbert space fragments and ℤ_2k scars. Physical Review X, 15(1):011035.