基于超导系统的三比特杂化托佛利门一步实现方案

doi:10.3969/j.issn.1007-5461.2026.04.005

摘要/Abstract

摘要： 托佛利（Toffoli）门是量子态操控和量子算法所需的一类重要量子逻辑门。针对量子计算中托佛利门实现复杂度高的问题，提出了一种基于超导系统实现三比特托佛利门的新型理论方案。通过设计由两个微波腔与四能级超导人工原子组成的杂化系统，利用色散耦合与二阶微扰理论导出有效哈密顿量，构建了实现托佛利门操作的演化子空间。该方案通过单步操作即可完成三比特控制逻辑操作，显著提升运算效率。数值模拟表明，即使考虑微波腔与非目标能级耦合、腔间串扰及退相干效应，托佛利门操作仍能保持较高的平均保真度。此外，该设计思路可推广至其他量子系统，为三比特杂化托佛利门的高效实现提供了普适性框架。

关键词: 量子计算, 超导系统, 杂化量子门, 托佛利门

Abstract: The Toffoli gate is a crucial class of quantum logic gates essential for quantum state manipulation and quantum algorithms. Addressing the challenge of high implementation complexity for the Toffoli gate in quantum computation, we propose a novel theoretical scheme for realizing a three-qubit Toffoli gate based on a superconducting system. By designing a hybrid system comprising two microwave cavities and a four-level superconducting artificial atom, and utilizing dispersive coupling and second-order perturbation theory to derive an effective Hamiltonian, we successfully construct an evolution subspace enabling the Toffoli gate operation. This scheme accomplishes the three-qubit controlled logic operation in a single step, significantly enhancing computational efficiency. Numerical simulations demonstrate that the Toffoli gate operation maintains relatively high average fidelity even when considering microwave cavity coupling to non-target levels, inter-cavity crosstalk, and decoherence effects. Furthermore, this design is extendable to other quantum systems, providing a universal framework for the efficient implementation of three-qubit hybrid Toffoli gates.

Key words: quantum computation, superconducting system, hybrid quantum gate, Toffoli gate

中图分类号:

O431.2

康逸豪, 苏奇平, 杨垂平 . 基于超导系统的三比特杂化托佛利门一步实现方案[J]. 量子电子学报, 2026, 43(4): 549-563.

KANG Yihao, SU Qiping, YANG Chuiping . One⁃step implementation of a three⁃qubit hybrid Toffoli gate based on a superconducting system[J]. Chinese Journal of Quantum Electronics, 2026, 43(4): 549-563.

参考文献

[1] A. Sarkar and T. K. Bhattacharyya. Universal Toffoli gate in ballistic nanowires [J]. Appl. Phys. Lett., 2007, 90(17): 173101. [2] E. Zahedinejad, J. Ghosh, and B. C. Sanders. High-fidelity single-shot Toffoli gate via quantum control [J]. Phys. Rev. Lett., 2011, 114(2): 200502. [3] A. Fedorov, L. Steffen, M. Baur, et al. Implementation of a Toffoli gate with superconducting circuits [J]. Nature (London), 2011, 481(7380): 170–172. [4] E. Zahedinejad, J. Ghosh, and B. C. Sanders. High-fidelity single-shot Toffoli gate via quantum control [J]. Phys. Rev. Lett., 2015, 114(20): 200502. [5] C. Song, S.-B. Zheng, P. Zhang, et al., Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit [J]. Nat. Commun., 2017, 8(1): 1061. [6] H.-T. Liu, B.-J. Chen, J.-C. Zhang, et al. Direct implementation of high-fidelity three-qubit gates for superconducting processor with tunable couplers [J]. Phys. Rev. Lett., 2025, 135(5): 050602. [7] X.-M. Lin, Z.-W. Zhou, M.-Y. Ye, et al. One-step implementation of a multiqubit controlled-phase-flip gate [J]. Phys. Rev. A, 2006, 73(1): 012323. [8] T. Yamamoto, M. Neeley, E. Lucero, et al. Quantum process tomography of two-qubit controlled-Z and controlled-NOT gates using superconducting phase qubits [J]. Phys. Rev. B, 2010, 82(18): 184515. [9] S.-B. Zheng. Simplified construction and physical realization of ??-qubit controlled phase gates [J]. Phys. Rev. A, 2012, 86(1): 012326. [10] K. Lemr, K. Bartkiewicz, A. Cernoch, et al. Experimental implementation of optimal linear-optical controlled-unitary gates [J]. Phys. Rev. Lett, 2015, 114(15): 153602. [11] J. Fiurá?ek. Linear optical Fredkin gate based on partial-SWAP gate [J]. Phys. Rev. A, 2008, 78(3): 032317. [12] W.-Q. Liu, H.-R. Wei, and L.-C. Kwek. Low-cost Fredkin gate with auxiliary space [J]. Phys. Rev. Applied, 2020, 14(5): 054057. [13] Y. Li, L. Wan, H. Zhang, et al. Quantum Fredkin and Toffoli gates on a versatile programmable silicon photonic chip [J]. npj Quantum Inf., 2022, 8(1):112. [14] P. De, S. Ranwa, and S. Mukhopadhyay. Alternative scheme for implementation of 3 qubit Fredkin gate with photonic bandgap crystal [J]. Opt. Laser Technol., 2023, 167(12): 109804. [15] A. S?rensen and K. M?lmer. Quantum computation with ions in thermal motion [J]. Phys. Rev. Lett., 1999, 82(9): 1971-1974. [16] A. S?rensen and K. M?lmer. Entanglement and quantum computation with ions in thermal motion [J]. Phys. Rev. A, 2000, 62(2): 022311. [17] N. Navon, N. Akerman, S. Kotler, et al. Quantum process tomography of a M?lmer-S?rensen interaction [J]. Phys. Rev. A, 2014, 90(1): 010103(R). [18] A. Mitra, M. J. Martin, G. W. Biedermann, et al. Robust M?lmer-S?rensen gate for neutral atoms using rapid adiabatic Rydberg dressing [J]. Phys. Rev. A, 2020, 101(3): 030301(R). [19] Y.-H. Chen, R. Stassi, W. Qin, et al. Fault-tolerant multiqubit geometric entangling gates using photonic cat-state qubits [J]. Phys. Rev. Applied, 2022, 18(2): 024076. [20] M. D. Reed, L. DiCarlo, S. E. Nigg, et al. Realization of three-qubit quantum error correction with superconducting circuits [J]. Nature (London), 2012, 482(7385): 382–385. [21] S. E. Rasmussen, K. Groenland, R. Gerritsma, et al. Single-step implementation of high-fidelity n-bit Toffoli gates [J]. Phys. Rev. A, 2020, 101(2): 022308. [22] Y. Kim, A. Morvan, L. B. Nguyen, et al. High-fidelity three-qubit iToffoli gate for fixed-frequency superconducting qubits [J]. Nat. Phys., 2022, 18(7): 783–788. [23] J. T. Barreiro, M. Müller, P. Schindler, et al. An open-system quantum simulator with trapped ions [J]. Nature (London), 2011, 470(7335): 486–491 [24] Y, Dong, K. B. Whaley, and L. Lin. A quantum hamiltonian simulation benchmark [J]. npj Quantum Inf., 2022, 8(1):131. [25] S.-N. Sun, B. Marinelli, J. M. Koh, et al. Quantum computation of frequency-domain molecular response properties using a three-qubit iToffoli gate [J]. npj Quantum Inf., 2024, 10(1): 55. [26] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer [J]. SIAM J. Comput., 1997, 26(5): 1484–1509. [27] L. K. Grover. Quantum mechanics helps in searching for a needle in a haystack [J]. Phys. Rev. Lett., 1997, 79(2): 325-328. [28] E. A. Martinez, T. Monz, D. Nigg, et al. Compiling quantum algorithms for architectures with multi-qubit gates [J]. New J. Phys., 2016, 18(6): 063029. [29] A. J. Baker, G. B. P. Huber, N. J. Glaser, et al. Single shot i-Toffoli gate in dispersively coupled superconducting qubits [J]. Appl. Phys. Lett., 2022, 120(5): 054002. [30] C. Jones. Low-overhead constructions for the fault-tolerant Toffoli gate [J]. Phys. Rev. A, 2013, 87(2): 022328. [31] D. Maslov. Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization [J]. Phys. Rev. A, 2016, 93(2): 022311. [32] J. Heo and S. G. Choi. Toffoli gate with photonic qubits based on weak cross-Kerr nonlinearities [J]. Quantum Inf. Process., 2021, 20(10): 345. [33] J. Leng, F. Yang, X.-B. Wang. Decomposing (n+1)-qubit Toffoli gate with shallow circuit depth and no ancilla [J]. Adv. Quantum Technol., 2024, 7(2): 2300370. [34] E. Knill. Quantum computing with realistically noisy devices [J]. Nature (London), 2005, 434(7029): 39–44. [35] R. Raussendorf and J. Harrington. Fault-tolerant quantum computation with high threshold in two dimensions [J]. Phys. Rev. Lett., 2007, 98(19): 19050. [36] A. G. Fowler, M. Mariantoni, J. M. Martinis, et al. Surface codes: Towards practical large-scale quantum computation [J]. Phys. Rev. A, 2012, 86(3): 0323244. [37] Z.-L. Xiang, S. Ashhab, J.-Q. You, et al. Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems [J]. Rev. Mod. Phys., 2013, 85(2): 623-653. [38] R. Li and J.-Q. You. Anisotropic exchange coupling in a nanowire double quantum dot with strong spin-orbit coupling [J]. Phys. Rev. B, 2014, 90(3): 035303. [39] G. Massé, T. Coudreau, A. Keller, et al. Implementable hybrid entanglement witness [J]. Phys. Rev. A, 2020, 102(6): 062406. [40] S. Tiwary, H. Sharma, H. S. Dhar. Protecting information in a parametrically driven hybrid quantum system [J]. Quantum, 2025, 9(5): 1754. [41] Z.-L. Xiang, X.-Y. Lü, T.-F. Li, et al. Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator [J]. Phys. Rev. B, 2013, 87(14): 144516. [42] J. D. Pritchard, J. A. Isaacs, M. A. Beck, et al. Hybrid atom-photon quantum gate in a superconducting microwave resonator [J]. Phys. Rev. A, 2014, 89(1): 010301. [43] D. O. Krimer, B. Hartl, and S. Rotter. Hybrid quantum systems with collectively coupled spin states: suppression of decoherence through spectral hole burning [J]. Phys. Rev. Lett., 2015, 115(3): 033601. [44] A. A. Clerk, K. W. Lehnert, P. Bertet, et al. Hybrid quantum systems with circuit quantum electrodynamics [J]. Nat. Phys., 2020, 16(3): 257–267. [45] Q.-P. Su, Y. Zhang, and C.-P. Yang. Single-step implementation of a hybrid controlled-not gate with one superconducting qubit simultaneously controlling multiple target cat-state qubits [J]. Phys. Rev. A, 2022, 105(6): 062436. [46] C.-P. Yang, S.-I. Chu, and S. Han. Possible realization of entanglement, logical gates, and quantum information transfer with superconducting-quantum-interference-device qubits in cavity QED [J]. Phys. Rev. A, 2003, 67(4): 042311. [47] J.-Q. You and F. Nori. Quantum information processing with superconducting qubits in a microwave field [J]. Phys. Rev. B, 2003, 68(6): 064509. [48] A. Blais, R.-S. Huang, A. Wallra, et al. Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation [J]. Phys. Rev. A, 2004, 69(6): 062320. [49] J. Clarke and F. K. Wilhelm. Superconducting quantum bits [J]. Nature (London), 2008, 453(7198):1031–1042. [50] J.-Q. You and F. Nori. Atomic physics and quantum optics using superconducting circuits [J]. Nature (London), 2011, 474(7353): 589-597. [51] X. Gu, S.-N. Huai, F. Nori, et al. Polariton states in circuit QED for electromagnetically induced transparency [J]. Phys. Rev. A, 2016, 93(6): 063827. [52] X. Gu, A. F. Kockum, A. Miranowicz, et al. Microwave photonics with superconducting quantum circuits [J]. Phys. Rep., 2017, 718(1): 1-102. [53] W. Woods, G. Calusine, A. Melville, et al. Determining interface dielectric losses in superconducting coplanar waveguide resonators [J]. Phys. Rev. Applied, 2019, 12(1): 014012. [54] A. Melville, G. Calusine, W. Woods, et al. Comparison of dielectric loss in titanium nitride and aluminum superconducting resonators [J]. Appl. Phys. Lett., 2020, 117(12): 124004. [55] C. Wang, Y. Y. Gao, P. Reinhold, et al. A Schr?dinger cat living in two boxes [J]. Science, 2016, 352(6289): 1087-1091. [56] M. Reagor, W. Pfaff, C. Axline, et al. Quantum memory with millisecond coherence in circuit QED [J]. Phys. Rev. B, 2016, 94(1): 014506. [57] M. Hofheinz, H. Wang, M. Ansmann, et al. Synthesizing arbitrary quantum states in a superconducting resonator [J]. Nature (London), 2009, 459(7246): 546–549. [58] H. Wang, M. Hofheinz, J. Wenner, et al. Improving the coherence time of superconducting coplanar resonators [J]. Appl. Phys. Lett., 2009, 95(23): 233508. [59] M. H. Devoret and R. J. Schoelkopf. Superconducting circuits for quantum information: an outlook [J]. Science, 2013, 339(6124): 1169-1174. [60] M. H. Michael, M. Silveri, R. T. Brierley, et al. New class of quantum error-correcting codes for a bosonic mode[J]. Phys. Rev. X, 2016, 6(3): 031006 [61] H. Putterman, J. Iverson, Q. Xu, et al. Stabilizing a bosonic qubit using colored dissipation [J]. Phys. Rev. Lett., 2022, 128(11): 110502. [62] A. Hajr, B. Qing, K. Wang, et al. High-coherence Kerr-cat qubit in 2D architecture [J]. Phys. Rev. X, 2024, 14(4): 041049. [63] M. Hofheinz, E. M. Weig, M. Ansmann, et al. Generation of Fock states in a superconducting quantum circuit [J]. Nature (London), 2008, 454(7202): 310–314. [64] H. Wang, M. Mariantoni, R. C. Bialczak, et al. Deterministic entanglement of photons in two superconducting microwave resonators [J]. Phys. Rev. Lett., 2011, 106(6): 060401. [65] M. Kounalakis, G. E. W. Bauer, and Y. M. Blanter. Analog quantum control of magnonic cat states on a chip by a superconducting qubit [J]. Phys. Rev. Lett., 2022, 129(3): 037205. [66] T. Hillmann and F. Quijandria. Quantum error correction with dissipatively stabilized squeezed-cat qubits [J]. Phys. Rev. A, 2023, 107(3): 032423. [67] K. S. Dubovitskii, D. M. Basko, J. S. Meyer, et al. Theory of quasiparticle-induced errors in driven-dissipative Schr?dinger cat qubits [J]. Phys. Rev. B, 2024, 110(2): 024505. [68] X. Wang, T. Liu, A. F. Kockum, et al. Tunable chiral bound states with giant atoms [J]. Phys. Rev. Lett., 2021, 126(3): 043602. [69] K. Nie, A. Bista, K. Chow, et al. Parametrically controlled microwave-photonic interface for the fluxonium [J]. Phys. Rev. Applied, 2024, 22(5): 054021. [70] S. Li, Z. Ni, L. Zhang, et al. Autonomous stabilization of Fock states in an oscillator against multiphoton losses [J]. Phys. Rev. Lett., 2024, 132(20): 203602 [71] D. F. James and J. Jerke. Effective Hamiltonian theory and its applications in quantum information [J]. Can. J. Phys., 2007, 85(6): 625-632. [72] Q. Guo, S.-B. Zheng, J. Wang, et al. Dephasing-insensitive quantum information storage and processing with superconducting qubits [J]. Phys. Rev. Lett., 2018, 121(13): 130501. [73] W. Ning, X.-J. Huang, P.-R. Han, et al. Deterministic entanglement swapping in a superconducting circuit [J]. Phys. Rev. Lett., 2019, 123(6): 060502. [74] Y. Xu, Y. Ma, W. Cai, et al. Demonstration of controlled-phase gates between two error-correctable photonic qubits [J]. Phys. Rev. Lett., 2020, 124(1): 120501. [75] M. A. Nielsen. A simple formula for the average gate fidelity of a quantum dynamical operation [J]. Phys. Lett. A, 2002, 303(4): 249-252. [76] Y.-x. Liu, J.-Q. You, L.-F. Wei, et al. Optical selection rules and phase-dependent adiabatic state control in a superconducting quantum circuit [J]. Phys. Rev. Lett., 2005, 95(8): 087001. [77] A. Vezvaee, E. Takou, P. Hilaire, et al. Avoiding leakage and errors caused by unwanted transitions in lambda systems [J]. PRX Quantum, 2023, 4(3): 030312. [78] C.-P. Yang, J.-H. Ni, L. Bin, et al. Preparation of maximally-entangled states with multiple cat-state qutrits in circuit QED [J]. Front. Phys. 2024, 19(3): 31201. [79] S. Ashhab and F. Nori. Switchable coupling for superconducting qubits using double resonance in the presence of crosstalk [J]. Phys. Rev. B, 2007, 76(13): 132513. [80] C.-P. Yang, Q.-P. Su, S.-B. Zheng, et al. Crosstalk-insensitive method for simultaneously coupling multiple pairs of resonators [J]. Phys. Rev. A, 2016, 93(4): 042307. [81] P. Mundada, G. Zhang, T. Hazard, et al. Suppression of qubit crosstalk in a tunable coupling superconducting circuit [J]. Phys. Rev. Applied, 2019, 12(5), 054023. [82] Z. Leghtas, S. Touzard, I. M. Pop, et al. Confining the state of light to a quantum manifold by engineered two-photon loss [J]. Science, 2015, 347(6224): 853-857. [83] S. Puri, S. Boutin, and A. Blais. Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving [J]. npj Quantum Inf., 2017, 3(1): 18. [84] Q.-P. Su, Y. Zhang, L. Bin, et al. Efficient scheme for realizing a multiplex-controlled phase gate with photonic qubits in circuit quantum electrodynamics [J]. Front. Phys., 2022, 17(5): 53505. [85] Y. Xu, W. Cai, Y. Ma, et al. Single-loop realization of arbitrary nonadiabatic holonomic single-qubit quantum gates in a superconducting circuit [J]. Phys. Rev. Lett., 2018, 121(11): 110501.