Similar articles

Abstract: Quantum simulation, as a practical application of noisy quantum computing, has aided the study of exotic quantum matters and the implementation of algorithms that outperform classical approaches. Superconducting qubits, one of the most promising candidates for realizing universal quantum computing, possess state-of-the-art features like easy integration of qubits, long coherence time, and high-fidelity single- and two-qubit gates. These characteristics have enabled applications of digital quantum simulation in the fields of physics, chemistry, and computer science. In this review, we first present the basic concepts of superconducting qubits, quantum gates, and digital quantum simulations. We also explore recent progress in digital quantum simulations using superconducting qubits, especially in relation to quantum chemistry, quantum matters, combinatorial optimization, and quantum machine learning. Finally, we address the current challenges of digital quantum simulation with superconducting qubits, and provide a perspective on the future of the field.

超导数字量子模拟的近期应用

作者：姚云焱^1,2，王震^1,2
机构：¹浙江大学，物理学院，中国杭州，310058；²浙江大学杭州国际科创中心，中国杭州，311200
概要：量子模拟是当前含噪声量子计算的主要实用应用，且有效地促进了对奇异量子材料的研究和超越经典方法的算法实现。超导量子比特是实现通用量子计算最有希望的物理平台之一。它们具备了易于集成、长相干时间和高保真度的单比特及两比特门等优异特性。这些特性使得数字量子模拟在物理、化学以及计算机科学领域都得到了应用。在这篇综述中，我们首先介绍了超导量子比特、量子门和数字量子模拟的基本概念。基于超导量子比特，我们围绕量子化学、量子材料、组合优化和量子机器学习等方面，讨论了相关数字量子模拟的最新实验研究进展。最后，我们阐述了基于超导量子比特的数字量子模拟在当前所面临的挑战，并展望了该领域的未来研究方向。

关键词：超导量子电路；数字量子模拟；量子化学；量子物相

[1]AbramsDM, DidierN, JohnsonBR, et al., 2020. Implementation of XY entangling gates with a single calibrated pulse. Nature Electronics, 3(12):744-750.

[2]AcharyaR, Aghababaie-BeniL, AleinerI, et al., 2024. Quantum error correction below the surface code threshold. arXiv:2408.13687.

[3]AnandA, SchleichP, Alperin-LeaS, et al., 2022. A quantum computing view on unitary coupled cluster theory. Chemical Society Reviews, 51(5):1659-1684.

[4]AndersenTI, AstrakhantsevN, KaramlouAH, et al., 2024. Thermalization and criticality on an analog-digital quantum simulator. arXiv:2405.17385.

[5]ArovasDP, BergE, KivelsonSA, et al., 2022. The Hubbard model. Annual Review of Condensed Matter Physics, 13:239-274.

[6]AruteF, AryaK, BabbushR, et al., 2019. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505-510.

[7]Aspuru-GuzikA, DutoiAD, LovePJ, et al., 2005. Simulated quantum computation of molecular energies. Science, 309(5741):1704-1707.

[8]Aspuru-GuzikA, WaltherP, 2012. Photonic quantum simulators. Nature Physics, 8(4):285-291.

[9]AtaidesJPB, TuckettDK, BartlettSD, et al., 2021. The XZZX surface code. Nature Communications, 12(1):2172.

[10]AzsesD, HaenelR, NavehY, et al., 2020. Identification of symmetry-protected topological states on noisy quantum computers. Physical Review Letters, 125(12):120502.

[11]BaoZH, XuSB, SongZX, et al., 2024. Schrödinger cats growing up to 60 qubits and dancing in a cat scar enforced discrete time crystal. arXiv:2401.08284.

[12]BarendsR, KellyJ, MegrantA, et al., 2013. Coherent Josephson qubit suitable for scalable quantum integrated circuits. Physical Review Letters, 111(8):080502.

[13]BarendsR, LamataL, KellyJ, et al., 2015. Digital quantum simulation of fermionic models with a superconducting circuit. Nature Communications, 6(1):7654.

[14]BartlettRJ, KucharskiSA, NogaJ, 1989. Alternative coupled-cluster ansätze II. The unitary coupled-cluster method. Chemical Physics Letters, 155(1):133-140.

[15]BauerCW, DavoudiZ, BalantekinAB, et al., 2023. Quantum simulation for high-energy physics. PRX Quantum, 4(2):027001.

[16]BengtssonA, VikstålP, WarrenC, et al., 2020. Improved success probability with greater circuit depth for the quantum approximate optimization algorithm. Physical Review Applied, 14(3):034010.

[17]BhartiK, Cervera-LiertaA, KyawTH, et al., 2022. Noisy intermediate-scale quantum algorithms. Reviews of Modern Physics, 94(1):015004.

[18]BiamonteJ, WittekP, PancottiN, et al., 2017. Quantum machine learning. Nature, 549(7671):195-202.

[19]BlaisA, HuangRS, WallraffA, et al., 2004. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Physical Review A, 69(6):062320.

[20]BlattR, RoosCF, 2012. Quantum simulations with trapped ions. Nature Physics, 8(4):277-284.

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

[22]BluvsteinD, EveredSJ, GeimAA, et al., 2024. Logical quantum processor based on reconfigurable atom arrays. Nature, 626(7997):58-65.

[23]BravyiSB, KitaevAY, 1998. Quantum codes on a lattice with boundary. arXiv:quant-ph/9811052.

[24]BrowaeysA, LahayeT, 2020. Many-body physics with individually controlled rydberg atoms. Nature Physics, 16(2):132-142.

[25]CaiWZ, MaYW, WangWT, et al., 2021. Bosonic quantum error correction codes in superconducting quantum circuits. Fundamental Research, 1(1):50-67.

[26]CaiZY, BabbushR, BenjaminSC, et al., 2023. Quantum error mitigation. Reviews of Modern Physics, 95(4):045005.

[27]ChaiYH, HanYJ, WuYC, et al., 2022. Shortcuts to the quantum approximate optimization algorithm. Physical Review A, 105(4):042415.

[28]ChenIC, BurdickB, YaoYX, et al., 2022. Error-mitigated simulation of quantum many-body scars on quantum computers with pulse-level control. Physical Review Research, 4(4):043027.

[29]ChooK, von KeyserlingkCW, RegnaultN, et al., 2018. Measurement of the entanglement spectrum of a symmetry-protected topological state using the IBM quantum computer. Physical Review Letters, 121(8):086808.

[30]CiracJI, ZollerP, 2012. Goals and opportunities in quantum simulation. Nature Physics, 8(4):264-266.

[31]ClarkeJ, WilhelmFK, 2008. Superconducting quantum bits. Nature, 453(7198):1031-1042.

[32]ClarkeJ, ClelandAN, DevoretMH, et al., 1988. Quantum mechanics of a macroscopic variable: the phase difference of a josephson junction. Science, 239(4843):992-997.

[33]CollessJI, RamaseshVV, DahlenD, et al., 2018. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Physical Review X, 8(1):011021.

[34]CongI, ChoiS, LukinMD, 2019. Quantum convolutional neural networks. Nature Physics, 15(12):1273-1278.

[35]DaleyAJ, BlochI, KokailC, et al., 2022. Practical quantum advantage in quantum simulation. Nature, 607(7920):667-676.

[36]Dallaire-DemersPL, KilloranN, 2018. Quantum generative adversarial networks. Physical Review A, 98(1):012324.

[37]DengYH, GongSQ, GuYC, et al., 2023. Solving graph problems using Gaussian boson sampling. Physical Review Letters, 130(19):190601.

[38]DennisE, KitaevA, LandahlA, et al., 2002. Topological quantum memory. Journal of Mathematical Physics, 43(9):4452-4505.

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

[40]DiVincenzoDP, 2000. The physical implementation of quantum computation. Fortschritte der Physik, 48(9-11):771-783. https://doi.‍org/10.1002/1521-3978(200009)48:‍9/11<771::AID-PROP771>3.0.CO;2-E

[41]DunjkoV, BriegelHJ, 2018. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Reports on Progress in Physics, 81(7):074001.

[42]EisertJ, FriesdorfM, GogolinC, 2015. Quantum many-body systems out of equilibrium. Nature Physics, 11(2):124-130.

[43]ElseDV, BauerB, NayakC, 2016. Floquet time crystals. Physical Review Letters, 117(9):090402.

[44]ElseDV, BauerB, NayakC, 2017. Prethermal phases of matter protected by time-translation symmetry. Physical Review X, 7(1):011026.

[45]ElseDV, MonroeC, NayakC, et al., 2020. Discrete time crystals. Annual Review of Condensed Matter Physics, 11:467-499.

[46]FarhiE, GoldstoneJ, GutmannS, 2014. A quantum approximate optimization algorithm. arXiv:1411.4028.

[47]FeynmanRP, 1982. Simulating physics with computers. International Journal of Theoretical Physics, 21(6):467-488.

[48]FoulkesWMC, MitasL, NeedsRJ, et al., 2001. Quantum Monte Carlo simulations of solids. Reviews of Modern Physics, 73(1):33-83.

[49]FowlerAG, MariantoniM, MartinisJM, et al., 2012. Surface codes: towards practical large-scale quantum computation. Physical Review A, 86(3):032324.

[50]FoxenB, NeillC, DunsworthA, et al., 2020. Demonstrating a continuous set of two-qubit gates for near-term quantum algorithms. Physical Review Letters, 125(12):120504.

[51]FreedmanMH, 2001. Quantum computation and the localization of modular functors. Foundations of Computational Mathematics, 1(2):183-204.

[52]FreyP, RachelS, 2022. Realization of a discrete time crystal on 57 qubits of a quantum computer. Science Advances, 8(9):eabm7652.

[53]GanzhornM, EggerDJ, BarkoutsosP, et al., 2019. Gate-efficient simulation of molecular eigenstates on a quantum computer. Physical Review Applied, 11(4):044092.

[54]GeorgescuIM, AshhabS, NoriF, 2014. Quantum simulation. Reviews of Modern Physics, 86(1):153-185.

[55]GongM, HuangHL, WangSY, et al., 2023. Quantum neuronal sensing of quantum many-body states on a 61-qubit programmable superconducting processor. Science Bulletin, 68(9):906-912.

[56]GoodfellowIJ, Pouget-AbadieJ, MirzaM, et al., 2014. Generative adversarial networks. arXiv:1406.2661.

[57]Google AI Quantum and Collaborators, AruteF, AryaK, et al., 2020. Hartree-Fock on a superconducting qubit quantum computer. Science, 369(6507):1084-1089.

[58]GoogleQuantum AI, 2023. Suppressing quantum errors by scaling a surface code logical qubit. Nature, 614(7949):676-681.

[59]Google Quantum AI and Collaborators, 2023. Non-Abelian braiding of graph vertices in a superconducting processor. Nature, 618(7964):264-269.

[60]GreinerM, MandelO, EsslingerT, et al., 2002. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature, 415(6867):39-44.

[61]GuoSA, WuYK, YeJ, et al., 2024. A site-resolved two-dimensional quantum simulator with hundreds of trapped ions. Nature, 630(8017):613-618.

[62]GuoSJ, SunJZ, QianHR, et al., 2024. Experimental quantum computational chemistry with optimized unitary coupled cluster ansatz. Nature Physics, 20(8):1240-1246.

[63]HaldarA, SenD, MoessnerR, et al., 2021. Dynamical freezing and scar points in strongly driven Floquet matter: resonance vs emergent conservation laws. Physical Review X, 11(2):021008.

[64]HarriganMP, SungKJ, NeeleyM, et al., 2021. Quantum approximate optimization of non-planar graph problems on a planar superconducting processor. Nature Physics, 17(3):332-336.

[65]HavlíčekV, CórcolesAD, TemmeK, et al., 2019. Supervised learning with quantum-enhanced feature spaces. Nature, 567(7747):209-212.

[66]HendersonM, ShakyaS, PradhanS, et al., 2020. Quanvolutional neural networks: powering image recognition with quantum circuits. Quantum Machine Intelligence, 2(1):2.

[67]HerrmannJ, LlimaSM, RemmA, et al., 2022. Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases. Nature Communications, 13(1):4144.

[68]HeyaK, NakanishiKM, MitaraiK, et al., 2023. Subspace variational quantum simulator. Physical Review Research, 5(2):023078.

[69]HuL, WuSH, CaiWZ, et al., 2019. Quantum generative adversarial learning in a superconducting quantum circuit. Science Advances, 5(1):eaav2761.

[70]HuangB, 2023. Analytical theory of cat scars with discrete time-crystalline dynamics in Floquet systems. Physical Review B, 108(10):104309.

[71]HuangHL, DuYX, GongM, et al., 2021. Experimental quantum generative adversarial networks for image generation. Physical Review Applied, 16(2):024051.

[72]HuangKX, WangZA, SongC, et al., 2021. Quantum generative adversarial networks with multiple superconducting qubits. npj Quantum Information, 7(1):165.

[73]HuangKX, CaiXX, LiH, et al., 2022. Variational quantum computation of molecular linear response properties on a superconducting quantum processor. The Journal of Physical Chemistry Letters, 13(39):9114-9121.

[74]HugginsWJ, O’GormanBA, RubinNC, et al., 2022. Unbiasing fermionic quantum Monte Carlo with a quantum computer. Nature, 603(7901):416-420.

[75]IppolitiM, KechedzhiK, MoessnerR, et al., 2021. Many-body physics in the NISQ era: quantum programming a discrete time crystal. PRX Quantum, 2(3):030346.

[76]JerbiS, FidererLJ, NautrupHP, et al., 2023. Quantum machine learning beyond kernel methods. Nature Communications, 14(1):517.

[77]JinYX, XuHZ, WangZA, et al., 2024. Quafu-RL: the cloud quantum computers based quantum reinforcement learning. Chinese Physics B, 33(5):050301.

[78]KandalaA, MezzacapoA, TemmeK, et al., 2017. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549(7671):242-246.

[79]KandalaA, TemmeK, CórcolesAD, et al., 2019. Error mitigation extends the computational reach of a noisy quantum processor. Nature, 567(7749):491-495.

[80]KaramlouAH, SimonWA, KatabarwaA, et al., 2021. Analyzing the performance of variational quantum factoring on a superconducting quantum processor. npj Quantum Information, 7(1):156.

[81]KellyJ, BarendsR, FowlerAG, et al., 2015. State preservation by repetitive error detection in a superconducting quantum circuit. Nature, 519(7541):66-69.

[82]KerenidisI, LandmanJ, PrakashA, 2020. Quantum algorithms for deep convolutional neural networks. Proceedings of the 8th International Conference on Learning Representations.

[83]KilloranN, LeeLJ, DelongA, et al., 2017. Generating and designing DNA with deep generative models. arXiv:1712.06148.

[84]KimY, EddinsA, AnandS, et al., 2023. Evidence for the utility of quantum computing before fault tolerance. Nature, 618(7965):500-505.

[85]KirmaniA, BullK, HouCY, et al., 2022. Probing geometric excitations of fractional quantum hall states on quantum computers. Physical Review Letters, 129(5):056801.

[86]KitaevA, PreskillJ, 2006. Topological entanglement entropy. Physical Review Letters, 96(11):110404.

[87]KitaevAY, 1997. Quantum computations: algorithms and error correction. Russian Mathematical Surveys, 52(6):1191-1249.

[88]KitaevAY, 2001. Unpaired Majorana fermions in quantum wires. Physics-Uspekhi, 44(10S):131-136.

[89]KitaevAY, 2003. Fault-tolerant quantum computation by anyons. Annals of Physics, 303(1):2-30.

[90]KochJ, YuTM, GambettaJ, et al., 2007. Charge-insensitive qubit design derived from the Cooper pair box. Physical Review A, 76(4):042319.

[91]KohJM, TaiT, LeeCH, 2022a. Simulation of interaction-induced chiral topological dynamics on a digital quantum computer. Physical Review Letters, 129(14):140502.

[92]KohJM, TaiT, PheeYH, et al., 2022b. Stabilizing multiple topological fermions on a quantum computer. npj Quantum Information, 8(1):16.

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

[94]KrinnerS, LacroixN, RemmA, et al., 2022. Realizing repeated quantum error correction in a distance-three surface code. Nature, 605(7911):669-674.

[95]KrizhevskyA, SutskeverI, HintonGE, 2017. ImageNet classification with deep convolutional neural networks. Communications of the ACM, 60(6):84-90.

[96]KyprianidisA, MachadoF, MorongW, et al., 2021. Observation of a prethermal discrete time crystal. Science, 372(6547):1192-1196.

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

[98]LanyonBP, HempelC, NiggD, et al., 2011. Universal digital quantum simulation with trapped ions. Science, 334(6052):57-61.

[99]LecunY, BottouL, BengioY, et al., 1998. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324.

[100]LeCunY, BengioY, HintonG, 2015. Deep learning. Nature, 521(7553):436-444.

[101]LedigC, TheisL, HuszárF, et al., 2017. Photo-realistic single image super-resolution using a generative adversarial network. IEEE Conference on Computer Vision and Pattern Recognition, p.105-114.

[102]LevinM, WenXG, 2006. Detecting topological order in a ground state wave function. Physical Review Letters, 96(11):110405.

[103]LiZK, LiuXM, XuNY, et al., 2015. Experimental realization of a quantum support vector machine. Physical Review Letters, 114(14):140504.

[104]LianB, SunXQ, VaeziA, et al., 2018. Topological quantum computation based on chiral Majorana fermions. Proceedings of the National Academy of Sciences of the United States of America, 115(43):10938-10942.

[105]LiuJH, LimKH, WoodKL, et al., 2021. Hybrid quantum-classical convolutional neural networks. Science China Physics, Mechanics & Astronomy, 64(9):290311.

[106]LloydS, 1996. Universal quantum simulators. Science, 273(5278):1073-1078.

[107]LloydS, WeedbrookC, 2018. Quantum generative adversarial learning. Physical Review Letters, 121(4):040502.

[108]LossD, DiVincenzoDP, 1998. Quantum computation with quantum dots. Physical Review A, 57(1):120-126.

[109]MaWL, PuriS, SchoelkopfRJ, et al., 2021. Quantum control of bosonic modes with superconducting circuits. Science Bulletin, 66(17):1789-1805.

[110]MachadoF, ElseDV, Kahanamoku-MeyerGD, et al., 2020. Long-range prethermal phases of nonequilibrium matter. Physical Review X, 10(1):011043.

[111]MarquesJF, VarbanovBM, MoreiraMS, et al., 2022. Logical-qubit operations in an error-detecting surface code. Nature Physics, 18(1):80-86.

[112]MartinisJM, NamS, AumentadoJ, et al., 2002. Rabi oscillations in a large josephson-junction qubit. Physical Review Letters, 89(11):117901.

[113]MaskaraN, MichailidisAA, HoWW, et al., 2021. Discrete time-crystalline order enabled by quantum many-body scars: entanglement steering via periodic driving. Physical Review Letters, 127(9):090602.

[114]MathieuM, CouprieC, LeCunY, 2016. Deep multi-scale video prediction beyond mean square error. The 4th International Conference on Learning Representations.

[115]McArdleS, EndoS, Aspuru-GuzikA, et al., 2020. Quantum computational chemistry. Reviews of Modern Physics, 92(1):015003.

[116]McCleanJR, BoixoS, SmelyanskiyVN, et al., 2018. Barren plateaus in quantum neural network training landscapes. Nature Communications, 9(1):4812.

[117]MiX, SonnerM, NiuMY, et al., 2022a. Noise-resilient edge modes on a chain of superconducting qubits. Science, 378(6621):785-790.

[118]MiX, IppolitiM, QuintanaC, et al., 2022b. Time-crystalline eigenstate order on a quantum processor. Nature, 601(7894):531-536.

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

[120]MottaM, SunC, TanATK, et al., 2020. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics, 16(2):205-210.

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

[122]NakamuraY, PashkinYA, TsaiJS, 1999. Coherent control of macroscopic quantum states in a single-cooper-pair box. Nature, 398(6730):786-788.

[123]NeillC, RoushanP, KechedzhiK, et al., 2018. A blueprint for demonstrating quantum supremacy with superconducting qubits. Science, 360(6385):195-199.

[124]NiZC, LiS, DengXW, et al., 2023. Beating the break-even point with a discrete-variable-encoded logical qubit. Nature, 616(7955):56-60.

[125]O’BrienTE, SenjeanB, SagastizabalR, et al., 2019. Calculating energy derivatives for quantum chemistry on a quantum computer. npj Quantum Information, 5(1):113.

[126]O’BrienTE, AnselmettiG, GkritsisF, et al., 2023. Purification-based quantum error mitigation of pair-correlated electron simulations. Nature Physics, 19(12):1787-1792.

[127]O’MalleyP, BabbushR, KivlichanI, et al., 2016. Scalable quantum simulation of molecular energies. Physical Review X, 6(3):031007.

[128]OtterbachJS, ManentiR, AlidoustN, et al., 2017. Unsupervised machine learning on a hybrid quantum computer. arXiv:1712.05771.

[129]Parra-RodriguezA, LougovskiP, LamataL, et al., 2020. Digital-analog quantum computation. Physical Review A, 101(2):022305.

[130]PechalM, RoyF, WilkinsonSA, et al., 2022. Direct implementation of a perceptron in superconducting circuit quantum hardware. Physical Review Research, 4(3):033190.

[131]PetersE, CaldeiraJ, HoA, et al., 2021. Machine learning of high dimensional data on a noisy quantum processor. npj Quantum Information, 7(1):161.

[132]PlaceAPM, RodgersLVH, MundadaP, et al., 2021. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nature Communications, 12(1):1779.

[133]PreskillJ, 2012. Quantum computing and the entanglement frontier. arXiv:1203.5813.

[134]PreskillJ, 2018. Quantum Computing in the NISQ era and beyond. Quantum, 2:79.

[135]QinMP, SchäferT, AndergassenS, et al., 2022. The hubbard model: a computational perspective. Annual Review of Condensed Matter Physics, 13:275-302.

[136]RahmaniA, SungKJ, PuttermanH, et al., 2020. Creating and manipulating a laughlin-type ν=1/3 fractional quantum hall state on a quantum computer with linear depth circuits. PRX Quantum, 1(2):020309.

[137]RebentrostP, MohseniM, LloydS, 2014. Quantum support vector machine for big data classification. Physical Review Letters, 113(13):130503.

[138]RenWH, LiWK, XuSB, et al., 2022. Experimental quantum adversarial learning with programmable superconducting qubits. Nature Computational Science, 2(11):711-717.

[139]ReuerK, LandgrafJ, FöselT, et al., 2022. Realizing a deep reinforcement learning agent discovering real-time feedback control strategies for a quantum system. arXiv:2210.16715.

[140]RistèD, da SilvaMP, RyanCA, et al., 2017. Demonstration of quantum advantage in machine learning. npj Quantum Information, 3(1):16.

[141]RomeroJ, BabbushR, McCleanJR, et al., 2019. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Science and Technology, 4(1):014008.

[142]RosenbergE, AndersenTI, SamajdarR, et al., 2024. Dynamics of magnetization at infinite temperature in a Heisenberg spin chain. Science, 384(6691):48-53.

[143]RoushanP, NeillC, ChenY, et al., 2014. Observation of topological transitions in interacting quantum circuits. Nature, 515(7526):241-244.

[144]SagastizabalR, Bonet-MonroigX, SinghM, et al., 2019. Experimental error mitigation via symmetry verification in a variational quantum eigensolver. Physical Review A, 100(1):010302.

[145]SalathéY, MondalM, OppligerM, et al., 2015. Digital quantum simulation of spin models with circuit quantum electrodynamics. Physical Review X, 5(2):021027.

[146]SalimansT, GoodfellowI, ZarembaW, et al., 2016. Improved techniques for training GANs. Proceedings of the 30th International Conference on Neural Information Processing Systems, p.2234-2242.

[147]SarmaSD, FreedmanM, NayakC, 2015. Majorana zero modes and topological quantum computation. npj Quantum Information, 1(1):15001.

[148]SatzingerKJ, LiuYJ, SmithA, et al., 2021. Realizing topologically ordered states on a quantum processor. Science, 374(6572):1237-1241.

[149]SchroerMD, KolodrubetzMH, KindelWF, et al., 2014. Measuring a topological transition in an artificial spin-1/2 system. Physical Review Letters, 113(5):050402.

[150]SchuldM, 2021. Supervised quantum machine learning models are kernel methods. arXiv:2101.11020.

[151]SchuldM, KilloranN, 2019. Quantum machine learning in feature hilbert spaces. Physical Review Letters, 122(4):040504.

[152]SchuldM, SinayskiyI, PetruccioneF, 2015. An introduction to quantum machine learning. Contemporary Physics, 56(2):172-185.

[153]SchusterDI, WallraffA, BlaisA, et al., 2005. ac Stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field. Physical Review Letters, 94(12):123602.

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

[155]SherringtonD, KirkpatrickS, 1975. Solvable model of a spin-glass. Physical Review Letters, 35(26):1792-1796.

[156]SivakVV, EickbuschA, RoyerB, et al., 2023. Real-time quantum error correction beyond break-even. Nature, 616(7955):50-55.

[157]SmithA, JobstB, GreenAG, et al., 2022. Crossing a topological phase transition with a quantum computer. Physical Review Research, 4(2):L022020.

[158]SternA, LindnerNH, 2013. Topological quantum computation—from basic concepts to first experiments. Science, 339(6124):1179-1184.

[159]StreifM, LeibM, 2019. Comparison of QAOA with quantum and simulated annealing. arXiv:1901.01903.

[160]SungY, DingL, BraumüllerJ, et al., 2021. Realization of high-fidelity CZ and ZZ-free iSWAP gates with a tunable coupler. Physical Review X, 11(2):021058.

[161]SuzukiM, 1976. Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Communications in Mathematical Physics, 51(2):183-190.

[162]TanXS, ZhaoYX, LiuQ, et al., 2017. Realizing and manipulating space-time inversion symmetric topological semimetal bands with superconducting quantum circuits. npj Quantum Materials, 2(1):60.

[163]TanXS, ZhangDW, LiuQ, et al., 2018. Topological Maxwell metal bands in a superconducting qutrit. Physical Review Letters, 120(3):130503.

[164]TanXS, ZhaoYX, LiuQ, et al., 2019. Simulation and manipulation of tunable Weyl-semimetal bands using superconducting quantum circuits. Physical Review Letters, 122(1):010501.

[165]TazhigulovRN, SunSN, HaghshenasR, et al., 2022. Simulating models of challenging correlated molecules and materials on the Sycamore quantum processor. PRX Quantum, 3(4):040318.

[166]TrotterHF, 1959. On the product of semi-groups of operators. Proceedings of the American Mathematical Society, 10(4):545-551.

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

[168]ViyuelaO, RivasA, GasparinettiS, et al., 2018. Observation of topological Uhlmann phases with superconducting qubits. npj Quantum Information, 4(1):10.

[169]WangC, GaoYY, ReinholdP, et al., 2016. A Schrödinger cat living in two boxes. Science, 352(6289):1087-1091.

[170]WangCL, LiXG, XuHK, et al., 2022. Towards practical quantum computers: transmon qubit with a lifetime approaching 0.5 milliseconds. npj Quantum Information, 8(1):3.

[171]WangCY, HarringtonJ, PreskillJ, 2003. Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory. Annals of Physics, 303(1):31-58.

[172]WangS, FontanaE, CerezoM, et al., 2021. Noise-induced barren plateaus in variational quantum algorithms. Nature Communications, 12(1):6961.

[173]WangZT, ChenQH, DuYX, et al., 2024. Quantum compiling with reinforcement learning on a superconducting processor. arXiv:2406.12195.

[174]WeberJR, KoehlWF, VarleyJB, et al., 2010. Quantum computing with defects. Proceedings of the National Academy of Sciences of the United States of America, 107(19):8513-8518.

[175]WeiSJ, ChenYH, ZhouZR, et al., 2022. A quantum convolutional neural network on NISQ devices. AAPPS Bulletin, 32(1):2.

[176]WenXG, 2017. Colloquium: zoo of quantum-topological phases of matter. Reviews of Modern Physics, 89(4):041004.

[177]WuYL, BaoWS, CaoS, et al., 2021. Strong quantum computational advantage using a superconducting quantum processor. Physical Review Letters, 127(18):180501.

[178]XiangZC, HuangKX, ZhangYR, et al., 2023. Simulating Chern insulators on a superconducting quantum processor. Nature Communications, 14(1):5433.

[179]XuHK, ZhangJN, HanJX, et al., 2021. Realizing discrete time crystal in an one-dimensional superconducting qubit chain. arXiv:2108.00942.

[180]XuHZ, ZhuangWF, WangZA, et al., 2024. Quafu-Qcover: explore combinatorial optimization problems on cloud-based quantum computers. Chinese Physics B, 33(5):050302.

[181]XuK, NingW, HuangXJ, et al., 2021. Demonstration of a non-Abelian geometric controlled-NOT gate in a superconducting circuit. Optica, 8(7):972-976.

[182]XuSB, SunZZ, WangK, et al., 2023. Digital simulation of projective non-Abelian anyons with 68 superconducting qubits. Chinese Physics Letters, 40(6):060301.

[183]XuSB, SunZZ, WangK, et al., 2024. Non-Abelian braiding of Fibonacci anyons with a superconducting processor. Nature Physics, 20(9):1469-1475.

[184]YanF, KrantzP, SungY, et al., 2018. Tunable coupling scheme for implementing high-fidelity two-qubit gates. Physical Review Applied, 10(5):054062.

[185]YaoYY, XiangL, 2024. Superconducting quantum simulation for many-body physics beyond equilibrium. Entropy, 26(7):592.

[186]YarlooH, KopaeiAE, LangariA, 2020. Homogeneous Floquet time crystal from weak ergodicity breaking. Physical Review B, 102(22):224309.

[187]YingC, GuoQH, LiSW, et al., 2022. Floquet prethermal phase protected by U(1) symmetry on a superconducting quantum processor. Physical Review A, 105(1):012418.

[188]YuYL, CaoCF, DeweyC, et al., 2022. Quantum approximate optimization algorithm with adaptive bias fields. Physical Review Research, 4(2):023249.

[189]YuYL, CaoCF, WangXB, et al., 2023. Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm. Physical Review Research, 5(2):023147.

[190]ZhangJ, HessPW, KyprianidisA, et al., 2017. Observation of a discrete time crystal. Nature, 543(7644):217-220.

[191]ZhangX, JiangWJ, DengJF, et al., 2022. Digital quantum simulation of Floquet symmetry-protected topological phases. Nature, 607(7919):468-473.

[192]ZhaoYW, YeYS, HuangHL, et al., 2022. Realization of an error-correcting surface code with superconducting qubits. Physical Review Letters, 129(3):030501.

[193]ZhongHS, WangH, DengYH, et al., 2020. Quantum computational advantage using photons. Science, 370(6523):1460-1463.

[194]ZhouL, WangST, ChoiS, et al., 2020. Quantum approximate optimization algorithm: performance, mechanism, and implementation on near-term devices. Physical Review X, 10(2):021067.

[195]ZhuLH, TangHL, BarronGS, et al., 2022. Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer. Physical Review Research, 4(3):033029.

[196]ZhuQL, CaoSR, ChenFS, et al., 2022. Quantum computational advantage via 60-qubit 24-cycle random circuit sampling. Science Bulletin, 67(3):240-245.