Fast topological pumping for the generation of large-scale Greenberger−Horne−Zeilinger states in a superconducting circuit

Abstract

Topological pumping of edge states in the finite lattice with nontrivial topological phases provides a powerful means for robust excitation transfer, requiring extremely slow evolution to follow an adiabatic transfer. Here, we propose fast topological pumping via edge channels to generate large-scale Greenberger−Horne−Zeilinger (GHZ) states in a topological superconducting circuit with a sped-up evolution process. The scheme indicates a conceptual way of designing fast topological pumping related to the instantaneous energy spectrum characteristics rather than relying on the shortcuts to adiabaticity. Based on fast topological pumping, large-scale GHZ states show greater robustness against on-site potential defects, the fluctuation of couplings and losses of the system in comparison with the conventional adiabatic topological pumping. With experimentally feasible qutrit-resonator coupling strengths and moderate decay rates of qutrits and resonators, fast topological pumping drastically improves the scalability of GHZ states with a high fidelity. Our work opens up prospects for the realization of large-scale GHZ states based on fast topological pumping in the superconducting quantum circuit system, which provides potential applications of topological matters in quantum information processing.

Graphical abstract

Keywords

topological pumping / superconducting ciruit / large-scale / Greenberger−Horne−Zeilinger states

References

[1] A.Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Uspekhi . 44(103), 131 ( 2001)

[2] C.L. Kane E.J. Mele, Z₂ topological order and the quantum spin Hall effect , Phys. Rev. Lett . 95(14), 146802 ( 2005)

[3] D.J. Thouless. Quantization of particle transport. Phys. Rev. B , 1983, 27( 10): 6083

[4] Q.Niu, D.J. Thouless. Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. J. Phys. Math. Gen. , 1984, 17( 12): 2453

[5] Y.E. Kraus, Y.Lahini, Z.Ringel, M.Verbin, O.Zilberberg. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. , 2012, 109( 10): 106402

[6] M.Verbin, O.Zilberberg, Y.Lahini, Y.E. Kraus, Y.Silberberg. Topological pumping over a photonic Fibonacci quasicrystal. Phys. Rev. B , 2015, 91( 6): 064201

[7] N.Lang H.P. Büchler, Topological networks for quantum commuication between distant qubits, npj Quantum Inf. 3, 47 ( 2017)

[8] F.Mei, G.Chen, L.Tian, S.L. Zhu, S.Jia. Robust quantum state transfer via topological edge states in superconducting qubit chains. Phys. Rev. A , 2018, 98( 1): 012331

[9] J.L. Tambasco, G.Corrielli, R.J. Chapman, A.Crespi, O.Zilberberg, R.Osellame, A.Peruzzo. Quantum interference of topological states of light. Sci. Adv. , 2018, 4( 9): eaat3187

[10] P.Boross, J.K. Asbóth, G.Széchenyi, L.Oroszlány, A.Pályi. Poor man’s topological quantum gate based on the Su−Schrieffer−Heeger model. Phys. Rev. B , 2019, 100( 4): 045414

[11] S.Longhi. Topological pumping of edge states via adiabatic passage. Phys. Rev. B , 2019, 99( 15): 155150

[12] N.E. Palaiodimopoulos, I.Brouzos, F.K. Diakonos, G.Theocharis. Fast and robust quantum state transfer via a topological chain. Phys. Rev. A , 2021, 103( 5): 052409

[13] F.M. D’Angelis, F.A. Pinheiro, D.Guéry-Odelin, S.Longhi, F.Impens. Fast and robust quantum state transfer in a topological Su−Schrieffer−Heeger chain with next-to-nearest-neighbor interactions. Phys. Rev. Res. , 2020, 2( 3): 033475

[14] F.Mei, G.Chen, L.Tian, S.L. Zhu, S.Jia. Topology-dependent quantum dynamics and entanglement-dependent topological pumping in superconducting qubit chains. Phys. Rev. A , 2018, 98( 3): 032323

[15] J.X. Han, J.L. Wu, Y.Wang, Y.Xia, Y.Y. Jiang, J.Song. Large-scale Greenberger−Horne−Zeilinger states through a topologically protected zero-energy mode in a superconducting qutrit-resonator chain. Phys. Rev. A , 2021, 103( 3): 032402

[16] S.Das Sarma, M.Freedman, C.Nayak. Topological quantum computation. Phys. Today , 2006, 59( 7): 32

[17] M.H. Devoret, R.J. Schoelkopf. Superconducting circuits for quantum information: An outlook. Science , 2013, 339( 6124): 1169

[18] J.Q. You F.Nori, Atomic physics and quantum optics using superconducting circuits, Nature 474(189), 8 ( 2005)

[19] J.M. Martinis, Qubit metrdogy for building a fault−tolerant quantum computer, npj Quantum Inf. 1, 15005 ( 2015)

[20] C.Song, K.Xu, H.Li, Y.R. Zhang, X.Zhang, W.Liu, Q.Guo, Z.Wang, W.Ren, J.Hao, H.Feng, H.Fan, D.Zheng, D.W. Wang, H.Wang, S.Y. Zhu. Generation of multicomponent atomic Schrödinger cat states of up to 20 qubits. Science , 2019, 365( 6453): 574

[21] J.Kelly, R.Barends, A.G. Fowler, A.Megrant, E.Jeffrey, T.C. White, D.Sank, J.Y. Mutus, B.Campbell, Y.Chen, Z.Chen, B.Chiaro, A.Dunsworth, I.C. Hoi, C.Neill, P.J. J. O’Malley, C.Quintana, P.Roushan, A.Vainsencher, J.Wenner, A.N. Cleland, J.M. Martinis. State preservation by repetitive error detection in a superconducting quantum circuit. Nature , 2015, 519( 7541): 66

[22] J.S. Otterbach R.Manenti N.Alidoust A.Bestwick M.Block B.Bloom S.Caldwell N.Didier E.S. Fried S.Hong P.Karalekas C.B. Osborn A.Papageorge E.C. Peterson G.Prawiroatmodjo N.Rubin C.A. Ryan D.Scarabelli M.Scheer E.A. Sete P.Sivarajah R.S. Smith N.T. A. Staley W.J. Zeng A.Hudson B.R. Johnson M.Reagor M.P. da Silva C.Rigetti, Unsupervised machine learning on a hybrid quantum computer, arXiv: 1712.05771 ( 2017)

[23] A.Kandala, A.Mezzacapo, K.Temme, M.Takita, M.Brink, J.M. Chow, J.M. Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature , 2017, 549( 7671): 242

[24] C.Neill P.Roushan K.Kechedzhi S.Boixo S.V. Isakov V.Smelyanskiy A.Megrant B.Chiaro A.Dunsworth K.Arya R.Barends B.Burkett Y.Chen Z.Chen A.Fowler B.Foxen M.Giustina R.Graff E.Jeffrey T.Huang J.Kelly P.Klimov E.Lucero J.Mutus M.Neeley C.Quintana D.Sank A.Vainsencher J.Wenner T.C. White H.Neven J.M. Martinis, A blueprint for demonstrating quantum supremacy with superconducting qubits, Science 360(6385), 195 ( 2018)

[25] K.X. Wei, I.Lauer, S.Srinivasan, N.Sundaresan, D.T. McClure, D.Toyli, D.C. McKay, J.M. Gambetta, S.Sheldon. Verifying multipartite entangled Greenberger−Horne−Zeilinger states via multiple quantum coherences. Phys. Rev. A , 2020, 101( 3): 032343

[26] Y.Salathé, M.Mondal, M.Oppliger, J.Heinsoo, P.Kurpiers, A.Potočnik, A.Mezzacapo, U.Las Heras, L.Lamata, E.Solano, S.Filipp, A.Wallraff. Digital quantum simulation of spin models with circuit quantum electrodynamics. Phys. Rev. X , 2015, 5( 2): 021027

[27] S.Hacohen-Gourgy, V.V. Ramasesh, C.De Grandi, I.Siddiqi, S.M. Girvin. Cooling and autonomous feedback in a Bose−Hubbard chain with attractive interactions. Phys. Rev. Lett. , 2015, 115( 24): 240501

[28] P.J. J. O’Malley, R.Babbush, I.D. Kivlichan, J.Romero, J.R. McClean, R.Barends, J.Kelly, P.Roushan, A.Tranter, N.Ding, B.Campbell, Y.Chen, Z.Chen, B.Chiaro, A.Dunsworth, A.G. Fowler, E.Jeffrey, E.Lucero, A.Megrant, J.Y. Mutus, M.Neeley, C.Neill, C.Quintana, D.Sank, A.Vainsencher, J.Wenner, T.C. White, P.V. Coveney, P.J. Love, H.Neven, A.Aspuru-Guzik, J.M. Martinis. Scalable quantum simulation of molecular energies. Phys. Rev. X , 2016, 6( 3): 031007

[29] Y.P. Zhong, D.Xu, P.Wang, C.Song, Q.J. Guo, W.X. Liu, K.Xu, B.X. Xia, C.Y. Lu, S.Han, J.W. Pan, H.Wang. Emulating anyonic fractional statistical behavior in a superconducting quantum circuit. Phys. Rev. Lett. , 2016, 117( 11): 110501

[30] M.Fitzpatrick, N.M. Sundaresan, A.C. Y. Li, J.Koch, A.A. Houck. Observation of a dissipative phase transition in a one-dimensional circuit QED lattice. Phys. Rev. X , 2017, 7( 1): 011016

[31] J.Q. You, X.F. Shi, X.Hu, F.Nori. Quantum emulation of a spin system with topologically protected ground states using superconducting quantum circuits. Phys. Rev. B , 2010, 81( 1): 014505

[32] J.Koch, A.A. Houck, K.L. Hur, S.M. Girvin. Time−reversal-symmetry breaking in circuit-QED-based photon lattices. Phys. Rev. A , 2010, 82( 4): 043811

[33] M.Hafezi, P.Adhikari, J.M. Taylor. Engineering three-body interaction and Pfaffian states in circuit QED systems. Phys. Rev. B , 2014, 90( 6): 060503

[34] E.Kapit, M.Hafezi, S.H. Simon. Induced self-stabilization in fractional quantum Hall states of light. Phys. Rev. X , 2014, 4( 3): 031039

[35] D.I. Tsomokos, S.Ashhab, F.Nori. Using superconducting qubit circuits to engineer exotic lattice systems. Phys. Rev. A , 2010, 82( 5): 052311

[36] F.Mei, Z.Y. Xue, D.W. Zhang, L.Tian, C.Lee, S.L. Zhu. Witnessing topological Weyl semimetal phase in a minimal circuit-QED lattice. Quantum Sci. Technol. , 2016, 1( 1): 015006

[37] J.Tangpanitanon, V.M. Bastidas, S.Al-Assam, P.Roushan, D.Jaksch, D.G. Angelakis. Topological pumping of photons in nonlinear resonator arrays. Phys. Rev. Lett. , 2016, 117( 21): 213603

[38] T.Goren, K.Plekhanov, F.Appas, K.Le Hur. Topological Zak phase in strongly coupled LC circuits. Phys. Rev. B , 2018, 97( 4): 041106

[39] V.V. Ramasesh, E.Flurin, M.Rudner, I.Siddiqi, N.Y. Yao. Direct probe of topological invariants using Bloch oscillating quantum walks. Phys. Rev. Lett. , 2017, 118( 13): 130501

[40] E.Flurin, V.V. Ramasesh, S.Hacohen-Gourgy, L.S. Martin, N.Y. Yao, I.Siddiqi. Observing topological invariants using quantum walks in superconducting circuits. Phys. Rev. X , 2017, 7( 3): 031023

[41] M.D. Schroer, M.H. Kolodrubetz, W.F. Kindel, M.Sandberg, J.Gao, M.R. Vissers, D.P. Pappas, A.Polkovnikov, K.W. Lehnert. Measuring a topological transition in an artificial spin-1/2 system. Phys. Rev. Lett. , 2014, 113( 5): 050402

[42] P.Roushan C.Neill Y.Chen M.Kolodrubetz C.Quintana N.Leung M.Fang R.Barends B.Campbell Z.Chen B.Chiaro A.Dunsworth E.Jeffrey J.Kelly A.Megrant J.Mutus P.J. J. O’Malley D.Sank A.Vainsencher J.Wenner T.White A.Polkovnikov A.N. Cleland J.M. Martinis, Observation of topological transitions in interacting quantum circuits, Nature 515(7526), 241 ( 2014)

[43] Z.Zhang, T.Wang, L.Xiang, J.Yao, J.Wu, Y.Yin. Measuring the Berry phase in a superconducting phase qubit by a shortcut to adiabaticity. Phys. Rev. A , 2017, 95( 4): 042345

[44] X.Tan, D.W. Zhang, Q.Liu, G.Xue, H.F. Yu, Y.Q. Zhu, H.Yan, S.L. Zhu, Y.Yu. Topological Maxwell metal bands in a superconducting qutrit. Phys. Rev. Lett. , 2018, 120( 13): 130503

[45] C.Song, D.Xu, P.Zhang, J.Wang, Q.Guo, W.Liu, K.Xu, H.Deng, K.Huang, D.Zheng, S.B. Zheng, H.Wang, X.Zhu, C.Y. Lu, J.W. Pan. Demonstration of topological robustness of anyonic braiding statistics with a superconducting quantum circuit. Phys. Rev. Lett. , 2018, 121( 3): 030502

[46] W.Cai, J.Han, F.Mei, Y.Xu, Y.Ma, X.Li, H.Wang, Y.P. Song, Z.Y. Xue, Z.Q. Yin, S.Jia, L.Sun. Observation of topological magnon insulator states in a superconducting circuit. Phys. Rev. Lett. , 2019, 123( 8): 080501

[47] O.Viyuela A.Rivas S.Gasparinetti A.Wallraff A.Wallraff S.Filipp M.A. Martin-Delgado, Observation of topological Uhlmann phases with superconduction qubits, npj Quantum Inform. 4, 10 ( 2018)

[48] R.Horodecki, P.Horodecki, M.Horodecki, K.Horodecki. Quantum entanglement. Rev. Mod. Phys. , 2009, 81( 2): 865

[49] J.W. Pan, Z.B. Chen, C.Y. Lu, H.Weinfurter, A.Zeilinger, M.Żukowski. Multiphoton entanglement and interferometry. Rev. Mod. Phys. , 2012, 84( 2): 777

[50] L.Pezzè, A.Smerzi, M.K. Oberthaler, R.Schmied, P.Treutlein. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys. , 2018, 90( 3): 035005

[51] M.C. Rechtsman, Y.Lumer, Y.Plotnik, A.Perez-Leija, A.Szameit, M.Segev. Topological protection of photonic path entanglement. Optica , 2016, 3( 9): 925

[52] A.Blanco-Redondo, B.Bell, D.Oren, B.J. Eggleton, M.Segev. Topological protection of biphoton states. Science , 2018, 362( 6414): 568

[53] M.Wang, C.Doyle, B.Bell, M.J. Collins, E.Magi, B.J. Eggleton, M.Segev, A.Blanco-Redondo. Topologically protected entangled photonic states. Nanophotonics , 2019, 8( 8): 1327

[54] C.K. Hong, Z.Y. Ou, L.Mandel. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. , 1987, 59( 18): 2044

[55] I.Brouzos, I.Kiorpelidis, F.K. Diakonos, G.Theocharis. Fast, robust, and amplified transfer of topological edge modes on a time-varying mechanical chain. Phys. Rev. B , 2020, 102( 17): 174312

[56] Y.X. Shen, L.S. Zeng, Z.G. Geng, D.G. Zhao, Y.G. Peng, X.F. Zhu. Acoustic adiabatic propagation based on topological pumping in a coupled multicavity chain lattice. Phys. Rev. Appl. , 2020, 14( 1): 014043

[57] J.L. Wu, Y.Wang, J.X. Han, Y.K. Feng, S.L. Su, Y.Xia, Y.Jiang, J.Song. One-step implementation of Rydberg-antiblockade SWAP and controlled-SWAP gates with modified robustness. Photon. Res. , 2021, 9( 5): 814

[58] X.R. Huang, Z.X. Ding, C.S. Hu, L.T. Shen, W.Li, H.Wu, S.B. Zheng. Robust Rydberg gate via Landau−Zener control of Förster resonance. Phys. Rev. A , 2018, 98( 5): 052324

[59] Q.Guo, S.B. Zheng, J.Wang, C.Song, P.Zhang, K.Li, W.Liu, H.Deng, K.Huang, D.Zheng, X.Zhu, H.Wang, C.Y. Lu, J.W. Pan. Dephasing-insensitive quantum information storage and processing with superconducting qubits. Phys. Rev. Lett. , 2018, 121( 13): 130501

[60] V.Balachandran, J.Gong. Adiabatic quantum transport in a spin chain with a moving potential. Phys. Rev. A , 2008, 77( 1): 012303

[61] J.Allcock, N.Linden. Quantum communication beyond the localization length in disordered spin chains. Phys. Rev. Lett. , 2009, 102( 11): 110501

[62] J.L. Wu, Y.Wang, J.X. Han, S.L. Su, Y.Xia, Y.Jiang, J.Song. Unselective ground-state blockade of Rydberg atoms for implementing quantum gates. Front. Phys. , 2022, 17( 2): 22501

[63] R.R. Agundez, C.D. Hill, L.C. L. Hollenberg, S.Rogge, M.Blaauboer. Superadiabatic quantum state transfer in spin chains. Phys. Rev. A , 2017, 95( 1): 012317

[64] J.L. Wu, Y.Wang, J.X. Han, Y.Jiang, J.Song, Y.Xia, S.L. Su, W.Li. Systematic-error-tolerant multiqubit holonomic entangling gates. Phys. Rev. Appl. , 2021, 16( 6): 064031

[65] Y.Zhou, D.Y. Lü, W.Y. Zeng. Chiral single-photon switch-assisted quantum logic gate with a nitrogen-vacancy center in a hybrid system. Photon. Res. , 2021, 9( 3): 405

[66] D.M. Greenberger M.A. Horne A.Zeilinger, Bell’s theorem, Quantum Theory, and Conceptions of the Universe, Kluwer Dordrecht, 1989

[67] M.Hillery, V.Bužek, A.Berthiaume. Quantum secret sharing. Phys. Rev. A , 1999, 59( 3): 1829

[68] S.Bose, V.Vedral, P.L. Knight. Multiparticle generalization of entanglement swapping. Phys. Rev. A , 1998, 57( 2): 822

[69] E.Knill. Quantum computing with realistically noisy devices. Nature , 2005, 434( 7029): 39

[70] V.Giovannetti, S.Lloyd, L.Maccone. Quantum-enhanced measurements: Beating the standard quantum limit. Science , 2004, 306( 5700): 1330

[71] D.Leibfried, M.Barrett, T.Schaetz, J.Britton, J.Chiaverini, W.Itano, J.Jost, C.Langer, D.Wineland. Toward Heisenberg-limited spectroscopy with multiparticle entangled states. Science , 2004, 304( 5676): 1476

[72] C.P. Yang, Q.P. Su, S.B. Zheng, F.Nori. Entangling superconducting qubits in a multi-cavity system. New J. Phys. , 2016, 18( 1): 013025

[73] S.Matsuo S.Ashhab T.Fujii F.Nori K.Nagai N.Hatakenaka, Generation of Bell states and Greenberger−Horne−Zeilinger states in superconducting phase qubits, in: Quantum Communication, Measurement and Computing, No. 8, Ed.: O. Hirota et al. , Tokyo: NICT, 2006

[74] L.F. Wei, Y.Liu, F.Nori. Generation and control of Greenberger−Horne−Zeilinger entanglement in superconducting circuits. Phys. Rev. Lett. , 2006, 96( 24): 246803

[75] S.L. Zhu, Z.D. Wang, P.Zanardi. Geometric quantum computation and multiqubit entanglement with superconducting qubits inside a cavity. Phys. Rev. Lett. , 2005, 94( 10): 100502

[76] C.P. Yang, Q.P. Su, S.Han. Generation of Greenberger−Horne−Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction. Phys. Rev. A , 2012, 86( 2): 022329

[77] C.P. Yang, Q.P. Su, S.B. Zheng, S.Han. Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit. Phys. Rev. A , 2013, 87( 2): 022320

[78] S.Aldana, Y.D. Wang, C.Bruder. Greenberger−Horne−Zeilinger generation protocol forN superconducting transmon qubits capacitively coupled to a quantum bus. Phys. Rev. B , 2011, 84( 13): 134519

[79] W.Feng, P.Wang, X.Ding, L.Xu, X.Q. Li. Generating and stabilizing the Greenberger−Horne−Zeilinger state in circuit QED: Joint measurement, Zeno effect, and feedback. Phys. Rev. A , 2011, 83( 4): 042313

[80] J.L. Wu, C.Song, J.Xu, L.Yu, X.Ji, S.Zhang. Adiabatic passage for one-step generation of n-qubit Greenberger–Horne–Zeilinger states of superconducting qubits via quantum Zeno dynamics. Quantum Inform. Process. , 2016, 15( 9): 3663

[81] X.T. Mo, Z.Y. Xue. Single-step multipartite entangled states generation from coupled circuit cavities. Front. Phys. , 2019, 14( 3): 31602

[82] Y.H. Kang, Z.C. Shi, B.H. Huang, J.Song, Y.Xia. Deterministic conversions between Greenberger−Horne−Zeilinger states and W states of spin qubits via Lie-transform-based inverse Hamiltonian engineering. Phys. Rev. A , 2019, 100( 1): 012332

[83] T.Liu, Q.P. Su, Y.Zhang, Y.L. Fang, C.P. Yang. Generation of quantum entangled states of multiple groups of qubits distributed in multiple cavities. Phys. Rev. A , 2020, 101( 1): 012337

[84] J.M. Chow, J.M. Gambetta, E.Magesan, D.W. Abraham, A.W. Cross, B.R. Johnson, N.A. Masluk, C.A. Ryan, J.A. Smolin, S.J. Srinivasan, M.Steffen. Implementing a strand of a scalable fault-tolerant quantum computing fabric. Nat. Commun. , 2014, 5( 1): 4015

[85] L.DiCarlo M.D. Reed L.Sun B.R. Johnson J.M. Chow J.M. Gambetta L.Frunzio M.H. Girvin, S. M. and Devoret, and R. J. Schoelkopf, Preparation and measurement of three-qubit entanglement in a superconduction circuit, Nature 467(7315), 574 ( 2010)

[86] R.Barends, J.Kelly, A.Megrant, A.Veitia, D.Sank, E.Jeffrey, T.C. White, J.Mutus, A.G. Fowler, B.Campbell, Y.Chen, Z.Chen, B.Chiaro, A.Dunsworth, C.Neill, P.O’Malley, P.Roushan, A.Vainsencher, J.Wenner, A.N. Korotkov, A.N. Cleland, J.M. Martinis. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature , 2014, 508( 7497): 500

[87] C.Song K.Xu W. Liu C.Yang S.B. Zheng H.Deng Q.Xie K.Huang Q.Guo L.Zhang P.Zhang D.Xu D. Zheng X.Zhu H.Wang Y.A. Chen C.Y. Lu S.Han J.W. Pan, 10-qubit entanglement and parallel logic operations with a superconducting circuit, Phys. Rev. Lett . 119(18), 180511 ( 2017)

[88] A.Cervera-Lierta, M.Krenn, A.Aspuru-Guzik, A.Galda. Experimental high-dimensional Greenberger−Horne−Zeilinger entanglement with superconducting transmon qutrits. Phys. Rev. Appl. , 2022, 17( 2): 024062

[89] X.Chen, I.Lizuain, A.Ruschhaupt, D.Guéry-Odelin, J.G. Muga. Shortcut to adiabatic passage in two- and three-level atoms. Phys. Rev. Lett. , 2010, 105( 12): 123003

[90] D.Guéry-Odelin, A.Ruschhaupt, A.Kiely, E.Torrontegui, S.Martínez-Garaot, J.G. Muga. Shortcuts to adiabaticity: Concepts, methods, and applications. Rev. Mod. Phys. , 2019, 91( 4): 045001

[91] A.Altland, M.R. Zirnbauer. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys. Rev. B , 1997, 55( 2): 1142

[92] S.Ryu, Y.Hatsugai. Topological origin of zero-energy edge states in particle−hole symmetric systems. Phys. Rev. Lett. , 2002, 89( 7): 077002

[93] J.K. Asbóth L.Oroszlány A.Pályi, A short course on topological insulators, Lect. Notes Phys . 919, 85 ( 2016)

[94] A.Coutant, V.Achilleos, O.Richoux, G.Theocharis, V.Pagneux. Robustness of topological corner modes against disorder with application to acoustic networks. Phys. Rev. B , 2020, 102( 21): 214204

[95] D.M. Greenberger, M.A. Horne, A.Shimony, A.Zeilinger. Bell’s theorem without inequalities. Am. J. Phys. , 1990, 58( 12): 1131

[96] R.Mukherjee, H.Xie, F.Mintert. Bayesian optimal control of Greenberger−Horne−Zeilinger states in Rydberg lattices. Phys. Rev. Lett. , 2020, 125( 20): 203603

[97] F.Reiter, D.Reeb, A.S. Sørensen. Scalable dissipative preparation of many-body entanglement. Phys. Rev. Lett. , 2016, 117( 4): 040501

[98] S.B. Zheng. One-step synthesis of multiatom Greenberger−Horne−Zeilinger states. Phys. Rev. Lett. , 2001, 87( 23): 230404

[99] P.Mundada, G.Zhang, T.Hazard, A.Houck. Suppression of qubit crosstalk in a tunable coupling superconducting circuit. Phys. Rev. Appl. , 2019, 12( 5): 054023

[100] W.P. Su, J.R. Schrieffer, A.J. Heeger. Solitons in Polyacetylene. Phys. Rev. Lett. , 1979, 42( 25): 1698

[101] J.Pachos, H.Walther. Quantum computation with trapped ions in an optical cavity. Phys. Rev. Lett. , 2002, 89( 18): 187903

[102] A.P. M. Place, L.V. H. Rodgers, P.Mundada, B.M. Smitham, M.Fitzpatrick, Z.Leng, A.Premkumar, J.Bryon, A.Vrajitoarea, S.Sussman, G.Cheng, T.Madhavan, H.K. Babla, X.H. Le, Y.Gang, B.Jäck, A.Gyenis, N.Yao, R.J. Cava, N.P. de Leon, A.A. Houck. New material platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nat. Commun. , 2021, 12( 1): 1779

[103] H.Zhang, S.Chakram, T.Roy, N.Earnest, Y.Lu, Z.Huang, D.K. Weiss, J.Koch, D.I. Schuster. Universal fast-flux control of a coherent, low-frequency qubit. Phys. Rev. X , 2021, 11( 1): 011010

[104] I.M. Pop, M.Ansmann, G.Catelani, R.J. Schoelkopf, L.I. Glazman, M.H. Devoret. Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles. Nature , 2014, 508( 7496): 7496

[105] X.Gu, A.F. Kockum, A.Miranowicz, Y.X. Liu, F.Nori. Microwave photonics with superconducting quantum circuits. Phys. Rep. , 2017, 718– 719,1

[106] M.Reagor, H.Paik, G.Catelani, L.Sun, C.Axline, E.Holland, I.M. Pop, N.A. Masluk, T.Brecht, L.Frunzio, M.H. Devoret, L.Glazman, R.J. Schoelkopf. Reaching 10 ms single photon lifetimes for superconducting aluminum cavities. Appl. Phys. Lett. , 2013, 102( 19): 192604

[107] M.Reagor, W.Pfaff, C.Axline, R.W. Heeres, N.Ofek, K.Sliwa, E.Holland, C.Wang, J.Blumoff, K.Chou, M.J. Hatridge, L.Frunzio, M.H. Devoret, L.Jiang, R.J. Schoelkopf. Quantum memory with millisecond coherence in circuit QED. Phys. Rev. B , 2016, 94( 1): 014506

[108] C.Axline, M.Reagor, R.Heeres, P.Reinhold, C.Wang, K.Shain, W.Pfaff, Y.Chu, L.Frunzio, R.J. Schoelkopf. An architecture for integrating planar and 3D cQED devices. Appl. Phys. Lett. , 2016, 109( 4): 042601

[109] J.E. Mooij, T.P. Orlando, L.Levitov, L.Tian, C.H. van der Wal, S.Lloyd. Josephson persistent-current qubit. Science , 1999, 285( 5430): 1036

[110] C.H. van der Wal, A.C. J. ter Haar, F.K. Wilhelm, R.N. Schouten, C.J. P. M. Harmans, T.P. Orlando, S.Lloyd, J.E. Mooij. Quantum superposition of macroscopic persistent-current states. Science , 2000, 290( 5492): 773

[111] B.Peropadre, P.Forn-Díaz, E.Solano, J.J. García-Ripoll. Switchable ultrastrong coupling in circuit QED. Phys. Rev. Lett. , 2010, 105( 2): 023601

[112] M.S. Allman F.Altomare J.D. Whittaker K.Cicak D.Li A. Sirois J.Strong J.D. Teufel R.W. Simmonds, RF-SQUID-mediated coherent tunable coupling between a superconducting phase qubit and a lumped-element resonator, Phys. Rev. Lett . 104(17), 177004 ( 2010)

[113] M.S. Allman, J.D. Whittaker, M.Castellanos-Beltran, K.Cicak, F.da Silva, M.P. DeFeo, F.Lecocq, A.Sirois, J.D. Teufel, J.Aumentado, R.W. Simmonds. Tunable resonant and nonresonant interactions between a phase qubit and LC resonator. Phys. Rev. Lett. , 2014, 112( 12): 123601

[114] J.Bourassa, J.M. Gambetta, A.A. Abdumalikov, O.Astafiev, Y.Nakamura, A.Blais. Ultrastrong coupling regime of cavity QED with phase-biased flux qubits. Phys. Rev. A , 2009, 80( 3): 032109

[115] T.P. Orlando, J.E. Mooij, L.Tian, C.H. van der Wal, L.S. Levitov, S.Lloyd, J.J. Mazo. Superconducting persistent-current qubit. Phys. Rev. B , 1999, 60( 22): 15398

[116] Y.X. Liu, J.Q. You, L.F. Wei, C.P. Sun, F.Nori. Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit. Phys. Rev. Lett. , 2005, 95( 8): 087001

[117] M.Devoret B.Huard R.Schoelkopf L.F. Cugliandolo, Quantum Machines: Measurement and Control of Engineered Quantum Systems, Oxford University Press, USA, 2014

[118] V.E. Manucharyan, J.Koch, L.I. Glazman, M.H. Devoret. Fluxonium: Single Cooper-pair circuit free of charge offsets. Science , 2009, 326( 5949): 113

[119] L.DiCarlo, J.M. Chow, J.M. Gambetta, L.S. Bishop, B.R. Johnson, D.I. Schuster, J.Majer, A.Blais, L.Frunzio, S.M. Girvin, R.J. Schoelkopf. Demonstration of two-qubit algorithms with a superconducting quantum processor. Nature , 2009, 460( 7252): 240

[120] T.Wang, Z.Zhang, L.Xiang, Z.Jia, P.Duan, Z.Zong, Z.Sun, Z.Dong, J.Wu, Y.Yin, G.Guo. Experimental realization of a fast controlled-Z gate via a shortcut to adiabaticity. Phys. Rev. Appl. , 2019, 11( 3): 034030

[121] A.A. Clerk, K.W. Lehnert, P.Bertet, J.R. Petta, Y.Nakamura. Hybrid quantum systems with circuit quantum electrodynamics. Nat. Phys. , 2020, 16( 3): 257

[122] T.Niemczyk, F.Deppe, H.Huebl, E.P. Menzel, F.Hocke, M.J. Schwarz, J.J. Garcia-Ripoll, D.Zueco, T.Hümmer, E.Solano, A.Marx, R.Gross. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. , 2010, 6( 10): 772

[123] A.Blais, S.M. Girvin, W.D. Oliver. Quantum information processing and quantum optics with circuit quantum electrodynamics. Nat. Phys. , 2020, 16( 3): 247

[124] G.J. Mooney, G.A. L. White, C.D. Hill, L.C. L. Hollenberg. Whole-device entanglement in a 65-qubit superconducting quantum computer. Adv. Quantum Technol. , 2021, 4( 10): 2100061

[125] A.Blais, R.S. Huang, A.Wallraff, S.M. Girvin, R.J. Schoelkopf. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A , 2004, 69( 6): 062320

[126] J.Clarke, F.K. Wilhelm. Superconducting quantum bits. Nature , 2008, 453( 7198): 1031

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11675046), the Program for Innovation Research of Science in Harbin Institute of Technology (Grant No. A201412), and the Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (Grant No. LBH-Q15060).

RIGHTS & PERMISSIONS

2022 Higher Education Press