[1] [1] GROVER L K. Quantum mechanics helps in searching for a needle in a haystack[J]. Physical Review Letters, 1997. 79(2): 2‒14. DOI: 10.1103/PhysRevLett.79.325.

[2] [2] SHOR P W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer[J]. Siam Review, 1999, 41(2): 303‒332. DOI: 10.1137/S0097539795293172.

[3] [3] PRESKILL J. Quantum computing in the NISQ era and beyond[J]. Quantum, 2018, 2: 79‒98. DOI: 10.22331/q-2018-08-06-79.

[4] [4] BERRY M V. Quantal phase factors accompanying adiabatic changes[J]. Proceedings of the Royal Society, 1984, 392(1802): 45‒57. DOI: 10.1098/rspa.1984.0023.

[5] [5] WILCZEK F, ZEE A. Appearance of gauge structure in simple dynamical systems[J]. Physical Review Letters, 1984, 52(24): 2111‒2114. DOI: 10.1103/PhysRevLett.52.2111.

[6] [6] AHARONOV Y, ANANDAN J. Phase change during a cyclic quantum evolution[J]. Physical Review Letters, 1987, 58(16): 1593‒1596. DOI: 10.1103/PhysRevLett.58.1593.

[7] [7] ANANDAN J. Non-adiabatic non-abelian geometric phase[J]. Physics Letters A, 1988, 133(4-5): 171‒175. DOI: 10.1016/0375-9601(88)91010-9.

[8] [8] JONES J A, VEDRAL V, EKERT A, et al. Geometric quantum computation using nuclear magnetic resonance[J]. Nature, 2000, 403: 869‒871. DOI: 10.1038/35002528.

[9] [9] ZANARDI P, RASETTI M. Holonomic quantum computation[J]. Physics Letters A, 1999, 264(2-3): 94‒99. DOI: 10.1016/S0375-9601(99)00803-8.

[10] [10] DUAN L M, CIRAC J I, ZOLLER P. Geometric manipulation of trapped ions for quantum computation[J]. Science, 2001, 292(5522): 1695‒1697. DOI: 10.1126/science.1058835.

[11] [11] WANG X B, MATSUMOTO K. Nonadiabatic conditional geometric phase shift with NMR[J]. Physical Review Letters, 2002, 87(9): 097901. DOI: 10.1103/PhysRevLett.87.097901.

[12] [12] ZHU S L, WANG Z D. Implementation of universal quantum gates based on nonadiabatic geometric phases[J]. Physical Review Letters, 2002, 89(9): 097902. DOI: 10.1103/PhysRevLett.89.097902.

[13] [13] SJQVIST E, TONG D M, HESSMO B, et al. Non-adiabatic holonomic quantum computation[J]. New Journal of Physics, 2012, 14(17): 103035. DOI: 10.1088/1367-2630/14/10/103035.

[14] [14] XU G F, ZHANG J, TONG D M, et al. Nonadiabatic holonomic quantum computation in decoherence-free subspaces[J]. Physical Review Letters, 2012, 109(17): 170501. DOI: 10.1103/PhysRevLett.109.170501.

[15] [15] ZHANG Z, ZHAO P Z, WANG T, et al. Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit[J]. New Journal of Physics, 2019, 21(7): 073024. DOI: 10.1088/1367-2630/AB2E26.

[16] [16] CHEN T, SHEN P, XUE Z Y. Robust and fast holonomic quantum gates with encoding on superconducting circuits[J]. Physical Review Applied, 2020, 14(3): 034038. DOI: 10.1103/PhysRevApplied.14.034038.

[17] [17] XU K, NING W, HUANG X J, et al. Demonstration of a Non-Abelian geometric controlled-NOT gate in a superconducting circuit[J]. Optica, 2021, 8(7): 972‒976. DOI: 10.1364/OPTICA.416264.

[18] [18] SONG C, ZHENG S B, ZHANG P, et al. Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit[J]. Nature Communications, 2017, 8(1): 1‒7. DOI: 10.1038/s41467-017-01156-5.

[19] [19] WANG T, ZHANG Z, XIANG L, et al. The Experimental realization of high-fidelity 'Shortcut-to-Adiabaticity' quantum gates in a superconducting Xmon qubit[J]. New Journal of Physics, 2018, 20(6): 065003. DOI: 10.1088/1367-2630/aac9e7.

[20] [20] XU Y, CAI W, MA Y, et al. Single-loop realization of arbitrary nonadiabatic holonomic single-qubit quantum gates in a superconducting circuit[J]. Physical Review Letters, 2018, 121(11): 110501. DOI: 10.1103/PhysRevLett.121.110501.

[21] [21] AI M Z, LI S, HE R, et al. Experimental realization of nonadiabatic holonomic single-qubit quantum gates with two dark paths in a trapped ion[J]. Fundamental Research, 2022, 2(5): 661‒666. DOI: 10.1016/j.fmre.2021.11.031.

[22] [22] NAOKI I, TAKAAKI N, TOUTA T, et al. Universal holonomic single quantum gates over a geometric spin with phase-modulated polarized light[J]. Optics Letters, 2018, 43(10): 2380‒2383. DOI: 10.1364/OL.43.002380.

[23] [23] ZHANG J, KYAW T H, FILIPP S, et al. Geometric and holonomic quantum computation [J]. Physics Reports, 2023, 1027: 1‒53. DOI: 10.1016/j.physrep.2023.07.004.

[24] [24] JOHANSSON A, SJQVIST E, ANDERSSON M, et al. Robustness of nonadiabatic holonomic gates[J]. Physical Review A, 2012, 86(6): 062322. DOI: 10.1103/PhysRevA.86.062322.

[25] [25] ZHENG S B, YANG C P, NORI F. Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts[J]. Physical Review A, 2012, 93(3): 032313. DOI: 10.1103/PhysRevA.93.032313.

[26] [26] COLMENAR R K L, GNGRD U, KESTNER J P. Conditions for equivalent noise sensitivity of geometric and dynamical quantum gates[J]. PRX Quantum, 2022, 3(3): 030310. DOI: 10.1103/PRXQuantum.3.030310.

[27] [27] LIANG Y, WU X Y, XUE Z Y. Nonadiabatic geometric quantum gates that are robust against systematic errors[J]. Physical Review Applied, 2024, 22(2): 024061. DOI: 10.1103/PhysRevApplied.22.024061.

[28] [28] VITANOV N V, RANGELOV A A, SHORE B W, et al. Stimulated raman adiabatic passage in physics, chemistry, and beyond[J]. Reviews of Modern Physics, 2017, 89(1): 015006. DOI: 10.1103/RevModPhys.89.015006.

[30] [30] CEN L X, WANG Z D, WANG S J. Scalable quantum computation in decoherence-free subspaces with trapped ions[J]. Physical Review A, 2006, 74(3): 032321. DOI: 10.1103/PhysRevA.74.032321.

[31] [31] FENG X L, WANG Z S, WU C F, et al. Scheme for unconventional geometric quantum computation in cavity QED[J]. Physical Review A, 2007, 75(3): 052312. DOI: 10.1103/PhysRevA.75.052312.

[32] [32] WU C F, WANG Z S, FENG X L, et al. Unconventional geometric quantum computation in a two-mode cavity[J]. Physical Review A, 2007, 76(2): 024302. DOI: 10.1103/PhysRevA.76.024302.

[33] [33] KIM K, ROOS C F, AOLITA L, et al. Geometric phase gate on an optical transition for ion trap quantum computation[J]. Physical Review A, 2008, 77(5): 050303. DOI: 10.1103/PhysRevA.77.050303.

[34] [34] FENG X L, WU C F, SUN H, et al. Geometric entangling gates in decoherence-free subspaces with minimal requirements[J]. Physical Review Letters, 2009, 103(20): 200501. DOI: 10.1103/PhysRevLett.103.200501.

[35] [35] CHEN Y Y, FENG X L, OH C H. Geometric entangling gates for coupled cavity system in decoherence-free subspaces[J]. Optics Communications, 2012, 285(24): 5554‒5557. DOI: 10.1016/j.optcom.2012.08.008.

[36] [36] ZHAO P Z, XU G F, TONG D M. Nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases[J]. Physical Review A, 2016, 94(6): 062327. DOI: 10.1103/PhysRevA.94.062327.

[37] [37] RUSCHHAUPT A, CHEN X, ALONSO D, et al. Optimally robust shortcuts to population inversion in two-level quantum systems[J]. New Journal of Physics, 2012, 14(9): 093040. DOI: 10.1088/1367-2630/14/9/093040.

[38] [38] DAEMS D, RUSCHHAUPT A, SUGNY D, et al. Robust quantum control by a single-shot shaped pulse[J]. Physical Review Letters, 2013, 111(5): 050404. DOI: 10.1103/PhysRevLett.111.050404.