Noncyclic nonadiabatic holonomic quantum gates via shortcuts to adiabaticity

Abstract

High-fidelity quantum gates are essential for large-scale quantum computation. However, any quantum manipulation will inevitably affected by noises, systematic errors and decoherence effects, which lead to infidelity of a target quantum task. Therefore, implementing high-fidelity, robust and fast quantum gates is highly desired. Here, we propose a fast and robust scheme to construct high-fidelity holonomic quantum gates for universal quantum computation based on resonant interaction of three-level quantum systems via shortcuts to adiabaticity. In our proposal, the target Hamiltonian to induce noncyclic non-Abelian geometric phases can be inversely engineered with less evolution time and demanding experimentally, leading to high-fidelity quantum gates in a simple setup. Besides, our scheme is readily realizable in physical system currently pursued for implementation of quantum computation. Therefore, our proposal represents a promising way towards fault-tolerant geometric quantum computation.

Graphical abstract

Keywords

noncyclic / holonomic quantum gates / shortcuts to adiabaticity / Lewis–Riesenfeld invariant

References

[1] R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21(6–7), 467 (1982)

[2] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. Ser. A 392, 45 (1984)

[3] F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52(24), 2111 (1984)

[4] Y. Aharonov and J. Anandan, Phase change during a cyclic quantum evolution, Phys. Rev. Lett. 58(16), 1593 (1987)

[5] P. Solinas, P. Zanardi, and N. Zanghì, Robustness of non-Abelian holonomic quantum gates against parametric noise, Phys. Rev. A 70(4), 042316 (2004)

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

[7] C. Lupo, P. Aniello, M. Napolitano, and G. Florio, Robustness against parametric noise of nonideal holonomic gates, Phys. Rev. A 76(1), 012309 (2007)

[8] S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, Experimental demonstration of the stability of Berry’s phase for a spin- 1/2 particle, Phys. Rev. Lett. 102(3), 030404 (2009)

[9] J. Xu, S. Li, T. Chen, and Z. Y. Xue, Nonadiabatic geometric quantum computation with optimal control on superconducting circuits, Front. Phys. 15(4), 41503 (2020)

[10] Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. Song, Z. Y. Xue, and L. Sun, Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit, Phys. Rev. Lett. 124(23), 230503 (2020)

[11] E. Sjöqvist, D. M. Tong, L. Mauritz Andersson, B. Hessmo, M. Johansson, and K. Singh, Non-adiabatic holonomic quantum computation, New J. Phys. 14(10), 103035 (2012)

[12] G. F. Xu, J. Zhang, D. M. Tong, E. Sjöqvist, and L. C. Kwek, Nonadiabatic holonomic quantum computation in decoherence-free subspaces, Phys. Rev. Lett. 109(17), 170501 (2012)

[13] Y. C. Zheng and T. A. Brun, Fault-tolerant holonomic quantum computation in surface codes, Phys. Rev. A 91(2), 022302 (2015)

[14] G. F. Xu, C. L. Liu, P. Z. Zhao, and D. M. Tong, Nonadiabatic holonomic gates realized by a single-shot implementation, Phys. Rev. A 92(5), 052302 (2015)

[15] E. Herterich and E. Sjöqvist, Single-loop multiple-pulse nonadiabatic holonomic quantum gates, Phys. Rev. A 94(5), 052310 (2016)

[16] Z. P. Hong, B. J. Liu, J. Q. Cai, X. D. Zhang, Y. Hu, Z. D. Wang, and Z. Y. Xue, Implementing universal nonadiabatic holonomic quantum gates with transmons, Phys. Rev. A 97(2), 022332 (2018)

[17] J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, Holonomic surface codes for fault-tolerant quantum computation, Phys. Rev. A 97(2), 022335 (2018)

[18] T. Chen, J. Zhang, and Z. Y. Xue, Nonadiabatic holonomic quantum computation on coupled transmons with ancillaries, Phys. Rev. A 98(5), 052314 (2018)

[19] N. Ramberg and E. Sjöqvist, Environment-assisted holonomic quantum maps, Phys. Rev. Lett. 122(14), 140501 (2019)

[20] C. Wu, Y. Wang, X. L. Feng, and J. L. Chen, Holonomic quantum computation in surface codes, Phys. Rev. Appl. 13(1), 014055 (2020)

[21] A. A. Jr Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, Experimental realization of non-Abelian non-adiabatic geometric gates, Nature 496(7446), 482 (2013)

[22] G. Feng, G. Xu, and G. Long, Experimental realization of nonadiabatic holonomic quantum computation, Phys. Rev. Lett. 110(19), 190501 (2013)

[23] C. Zu, W. B. Wang, L. He, W. G. Zhang, C. Y. Dai, F. Wang, and L. M. Duan, Experimental realization of universal geometric quantum gates with solid-state spins, Nature 514(7520), 72 (2014)

[24] S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin, Nat. Commun. 5(1), 4870 (2014)

[25] Y. Sekiguchi, N. Niikura, R. Kuroiwa, H. Kano, and H. Kosaka, Optical holonomic single quantum gates with a geometric spin under a zero field, Nat. Photonics 11(5), 309 (2017)

[26] B. B. Zhou, P. C. Jerger, V. O. Shkolnikov, F. J. Heremans, G. Burkard, and D. D. Awschalom, Holonomic quantum control by coherent optical excitation in diamond, Phys. Rev. Lett. 119(14), 140503 (2017)

[27] H. Li, Y. Liu, and G. L. Long, Experimental realization of single-shot nonadiabatic holonomic gates in nuclear spins, Sci. China Phys. Mech. Astron. 60(8), 080311 (2017)

[28] Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y. P. Song, Z. Y. Xue, Z. Q. Yin, and L. Sun, Single-loop realization of arbitrary nonadiabatic holonomic singlequbit quantum gates in a superconducting circuit, Phys. Rev. Lett. 121(11), 110501 (2018)

[29] N. Ishida, T. Nakamura, T. Tanaka, S. Mishima, H. Kano, R. Kuroiwa, Y. Sekiguchi, and H. Kosaka, Universal holonomic single quantum gates over a geometric spin with phase-modulated polarized light, Opt. Lett. 43(10), 2380 (2018)

[30] K. Nagata, K. Kuramitani, Y. Sekiguchi, and H. Kosaka, Universal holonomic quantum gates over geometric spin qubits with polarised microwaves, Nat. Commun. 9(1), 3227 (2018)

[31] D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Muller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, Entanglement generation in superconducting qubits using holonomic operations, Phys. Rev. Appl. 11(1), 014017 (2019)

[32] S. B. Zheng, C. P. Yang, and F. Nori, Comparison of the sensitivity to systematic errors between nonadiabatic non- Abelian geometric gates and their dynamical counterparts, Phys. Rev. A 93(3), 032313 (2016)

[33] J. Jing, C. H. Lam, and L. A. Wu, Non-Abelian holonomic transformation in the presence of classical noise, Phys. Rev. A 95(1), 012334 (2017)

[34] Y. C. Zheng and T. A. Brun, Fault-tolerant scheme of holonomic quantum computation on stabilizer codes with robustness to low-weight thermal noise, Phys. Rev. A 89(3), 032317 (2014)

[35] J. Zhang, L. C. Kwek, E. Sjöqvist, D. M. Tong, and P. Zanardi, Quantum computation in noiseless subsystems with fast non-Abelian holonomies, Phys. Rev. A 89(4), 042302 (2014)

[36] Z. T. Liang, Y. X. Du, W. Huang, Z. Y. Xue, and H. Yan, Nonadiabatic holonomic quantum computation in decoherence-free subspaces with trapped ions, Phys. Rev. A 89(6), 062312 (2014)

[37] J. Zhou, W. C. Yu, Y. M. Gao, and Z. Y. Xue, Cavity QED implementation of non-adiabatic holonomies for universal quantum gates in decoherence-free subspaces with nitrogen-vacancy centers, Opt. Express 23(11), 14027 (2015)

[38] Z. Y. Xue, J. Zhou, and Z. D. Wang, Universal holonomic quantum gates in decoherence-free subspace on superconducting circuits, Phys. Rev. A 92(2), 022320 (2015)

[39] Y. Wang, J. Zhang, C. Wu, J. Q. You, and G. Romero, Holonomic quantum computation in the ultrastrongcoupling regime of circuit QED, Phys. Rev. A 94(1), 012328 (2016)

[40] Z. Y. Xue, J. Zhou, Y. M. Chu, and Y. Hu, Nonadiabatic holonomic quantum computation with all-resonant control, Phys. Rev. A 94(2), 022331 (2016)

[41] Z. Y. Xue, F. L. Gu, Z. P. Hong, Z. H. Yang, D. W. Zhang, Y. Hu, and J. Q. You, Nonadiabatic holonomic quantum computation with dressed-state qubits, Phys. Rev. Appl. 7(5), 054022 (2017)

[42] P. Z. Zhao, G. F. Xu, Q. M. Ding, E. Sjöqvist, and D. M. Tong, Single-shot realization of nonadiabatic holonomic quantum gates in decoherence-free subspaces, Phys. Rev. A 95(6), 062310 (2017)

[43] L. N. Ji, T. Chen, and Z. Y. Xue, Scalable nonadiabatic holonomic quantum computation on a superconducting qubit lattice, Phys. Rev. A 100(6), 062312 (2019)

[44] Y. Wang, Y. Su, X. Chen, and C. Wu, Dephasingprotected scalable holonomic quantum computation on a Rabi lattice, Phys. Rev. Appl. 14(4), 044043 (2020)

[45] G. F. Xu, P. Z. Zhao, T. H. Xing, E. Sjöqvist, and D. M. Tong, Composite nonadiabatic holonomic quantum computation, Phys. Rev. A 95(3), 032311 (2017)

[46] Z. Zhu, T. Chen, X. Yang, J. Bian, Z. Y. Xue, and X. Peng, Single-loop and composite-loop realization of nonadiabatic holonomic quantum gates in a decoherence-free subspace, Phys. Rev. Appl. 12(2), 024024 (2019)

[47] Y. Sekiguchi, Y. Komura, and H. Kosaka, Dynamical decoupling of a geometric qubit, Phys. Rev. Appl. 12(5), 051001 (2019)

[48] B. J. Liu, Z. H. Huang, Z. Y. Xue, and X. D. Zhang, Superadiabatic holonomic quantum computation in cavity QED, Phys. Rev. A 95(6), 062308 (2017)

[49] B. J. Liu, X. K. Song, Z. Y. Xue, X. Wang, and M. H. Yung, Plug-and-play approach to nonadiabatic geometric quantum gates, Phys. Rev. Lett. 123(10), 100501 (2019)

[50] T. Yan, B. J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M. H. Yung, Y. Chen, and D. Yu, Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates, Phys. Rev. Lett. 122(8), 080501 (2019)

[51] S. Li, T. Chen, and Z. Y. Xue, Fast holonomic quantum computation on superconducting circuits with optimal control, Adv. Quantum Technol. 3(3), 2000001 (2020)

[52] M. Z. Ai, S. Li, Z. Hou, Z. Y. Xue, J. M. Cui, Y. F. Huang, C. F. Li, and G. C. Guo, Experimental realization of nonadiabatic holonomic single-qubit quantum gates with optimal control in a trapped ion, Phys. Rev. Appl. 14(5), 054062 (2020)

[53] M. Z. Ai, S. Li, Z. Y. Xue, J. M. Cui, Y. F. Huang, C. F. Li, and G. C. Guo, Experimental realization of nonadiabatic holonomic single-qubit quantum gates with two dark paths in a trapped ion, arXiv: 2101.07483 (2021)

[54] B. J. Liu, Y. S. Wang, and M. H. Yung, Global property condition-based non-adiabatic geometric quantum control, arXiv: 2008.02176(2020)

[55] S. Li and Z. Y. Xue, Dynamically corrected nonadiabatic holonomic quantum gates, arXiv: 2012.09034 (2020)

[56] G. F. Xu, D. M. Tong, and E. Sjöqvist, Path-shortening realizations of nonadiabatic holonomic gates, Phys. Rev. A 98(5), 052315 (2018)

[57] F. Zhang, J. Zhang, P. Gao, and G. Long, Searching nonadiabatic holonomic quantum gates via an optimization algorithm, Phys. Rev. A 100(1), 012329 (2019)

[58] T. Chen, P. Shen, and Z. Y. Xue, Robust and fast holonomic quantum gates with encoding on superconducting circuits, Phys. Rev. Appl. 14(3), 034038 (2020)

[59] B. J. Liu, Z. Y. Xue, and M. H. Yung, Brachistochrone non-Adiabatic holonomic quantum control, arXiv: 2001.05182 (2020)

[60] Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M. H. Yung, and Y. Yu, Experimental realization of universal time-optimal non-Abelian geometric gates, arXiv: 2004.10364 (2020)

[61] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Gu’ery-Odelin, and J. G. Muga, Fast optimal frictionless atom cooling in harmonic traps: Shortcut to adiabaticity, Phys. Rev. Lett. 104(6), 063002 (2010)

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

[63] H. R. Jr Lewis and W. B. Riesenfeld, An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field, J. Math. Phys. 10(8), 1458 (1969)

[64] X. Chen and J. G. Muga, Engineering of fast population transfer in three-level systems, Phys. Rev. A 86(3), 033405 (2012)

[65] Y. H. Chen, Z. C. Shi, J. Song, and Y. Xia, Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system, Phys. Rev. A 97(2), 023841 (2018)

[66] J. Zhou, S. Li, T. Chen, and Z. Y. Xue, Fast hybrid quantum state transfer and entanglement generation via no transition passage, Ann. Phys. (Berlin) 531(4), 1800402 (2019)

[67] Y. Yan, Y. Li, A. Kinos, A. Walther, C. Y. Shi, L. Rippe, J. Moser, S. Kröll, and X. Chen, Inverse engineering of shortcut pulses for high fidelity initialization on qubits closely spaced in frequency, Opt. Express 27(6), 8267 (2019)

[68] T. Chen and Z. Y. Xue, High-fidelity and robust geometric quantum gates that outperform dynamical ones, Phys. Rev. Appl. 14(6), 064009 (2020)

[69] Y. Zheng, S. Campbell, G. De Chiara, and D. Poletti, Cost of counterdiabatic driving and work output, Phys. Rev. A 94(4), 042132 (2016)

[70] O. Abah, R. Puebla, A. Kiely, G. De Chiara, M. Paternostro, and S. Campbell, Energetic cost of quantum control protocols, New J. Phys. 21(10), 103048 (2019)

[71] J. F. Poyatos, J. I. Cirac, and P. Zoller, Complete characterization of a quantum process: The two-bit quantum gate, Phys. Rev. Lett. 78(2), 390 (1997)

[72] M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, Coherence and decay of higher energy levels of a superconducting transmon qubit, Phys. Rev. Lett. 114(1), 010501 (2015)

RIGHTS & PERMISSIONS

Higher Education Press