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电路量子电动力学中基于超绝热捷径的控制相位门实现

王雪梅 张安琪 赵生妹

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电路量子电动力学中基于超绝热捷径的控制相位门实现

王雪梅, 张安琪, 赵生妹

Implementation of controlled phase gate based on superadiabatic shortcut in circuit quantum electrodynamics

Wang Xue-Mei, Zhang An-Qi, Zhao Sheng-Mei
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  • 针对绝热算法在系统演化过程中需要较长操作时间的问题, 本文提出了电路量子电动力学系统中基于超绝热捷径的两量子比特控制相位门的快速制备方案. 首先将量子比特的能级进行编码, 针对不同初始态分类讨论, 获得系统的有效哈密顿量. 通过反绝热驱动, 推导出系统有效哈密顿量的修正项, 以抑制不同本征态之间不必要的跃迁, 从而获得了高保真度的基于超绝热捷径控制相位门. 数值模拟验证了本方案的有效性, 最终保真度为0.991. 所提方案可以加速演化, 并且比绝热通道更有效. 此外, 本方案对谐振器的衰减和超导量子比特的退相干具有鲁棒性. 通过对谐振腔的泄漏、量子比特的自发辐射和退相位的影响分析, 得到的系统最终保真度始终保持在0.984以上.
    With high speed and big storage power, quantum computer has received increasing attention. The operation on the quantum computer can be composed of several single-bit and multi-bit quantum logic gates, among which the controlled phase gate is one of the essential two-qubit logic gates. Usually, the quantum gate is realized in a real physical system, and the circuit quantum electrodynamics system (QED) has become a promising candidate due to its long coherent time, easily coupled with other physical system and scaled up to large scale. In this work, we propose a scheme to fast implement a two-qubit controlled phase gate based on the circuit QED by using the superadiabatic-based shortcut, in order to solve the problem that the adiabatic algorithm needs a long time in the process of system evolution. Here, a coding strategy is first designed for the circuit QED system and the two transmon qubits, and the effective Hamiltonian of the system is then presented by dividing different initial states in the rotating-wave approximation. By using the superadiabatic-based shortcut algorithm for two iterations, a correction term in the same form as the system effective Hamiltonian is obtained through anti-diabatic driving, so that the effective Hamiltonian can suppress unwanted transitions between different instantaneous eigenstates. According to the evolution path, the appropriate boundary conditions are also obtained to complete the preparation of the controlled phase gate. The numerical simulation results show the availability of the proposed scheme, that is, the $ - \left| {11} \right\rangle $ state can be obtained by system evolution when the initial state is $ \left| {11} \right\rangle $, while the system does not change at all when the other initial states are prepared. Furthermore, the controlled phase gate with high-fidelity can be obtained . It is shown that the fidelity of the controlled phase gate is stable and greater than 0.991 when the evolution time is greater than $0.7{t \mathord{\left/ {\vphantom {t {{t_f}}}} \right. } {{t_{\rm f}}}}$. In addition, the proposed scheme can accelerate the evolution and is robust to decoherence. By the resonator decay and the spontaneous emission and dephasing of qubit, the final fidelity of the controlled phase gate is greater than 0.984. Since the controlled phase gate does not need additional parameters, the propsoed scheme is feasible in experiment.
      通信作者: 赵生妹, zhaosm@njupt.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 61871234)资助的课题.
      Corresponding author: Zhao Sheng-Mei, zhaosm@njupt.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61871234).
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    [2]

    Zhong H S, Wang H, Deng Y H, Chen M C, Peng L C, Luo Y H, Qin J, Wu D, Ding X, Hu Y, Hu P, Yang X Y, Zhang W J, Li H, Li Y X, Jiang X, Gan L, Yang G W, You L X, Wang Z, Li L, Liu N L, Lu C Y, Pan J W 2020 Science 370 1460Google Scholar

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    Preskill J 2018 Quantum 2 79Google Scholar

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    刘超, 邬云文 2018 物理学报 67 170302Google Scholar

    Liu C, Wu Y W 2018 Acta Phys. Sin. 67 170302Google Scholar

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    Yu D M, Wang H, Ma D D, Zhao X D, Qian J 2019 Opt. Express 27 23080Google Scholar

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    Shi H M, Yu Y F, Zhang Z M 2012 Chin. Phys. B 21 064205Google Scholar

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    Liu G Q, Pan X Y 2018 Chin. Phys. B 27 020304Google Scholar

    [8]

    Wan Y, Kienzler D, Erickson S D, Mayer K H, Tan T R, Wu J J, Vasconcelos H M, Glancy S, Knill E, Wineland D J, Wilson A C, Leibfried D 2019 Science 364 875Google Scholar

    [9]

    Qiu C D, Ne X F, Lu D W 2021 Chin. Phys. B 30 048201Google Scholar

    [10]

    Zhang M, Wei L F 2012 Chin. Phys. Lett. 29 080301Google Scholar

    [11]

    Wen J W, Kong X Y, Wei S J, Wang B X, Xin T, Long G L 2019 Phys. Rev. A 99 012320Google Scholar

    [12]

    Zhang F Y, Pei P, Li C, Song H S 2011 Chin. Phys. Lett. 28 120304Google Scholar

    [13]

    Alqahtani M M 2020 Quantum Inf. Process. 19 12Google Scholar

    [14]

    Yang C P, Zheng Z F, Zhang Y 2018 Opt. Lett. 43 5765Google Scholar

    [15]

    Göppl M, Fragner A, Baur M, et al. 2008 J. Appl. Phys. 104 113904Google Scholar

    [16]

    Clarke J, Wilhelm F K 2008 Nature 453 1031Google Scholar

    [17]

    Wendin G 2017 Rep. Prog. Phys. 80 106001Google Scholar

    [18]

    Arute F, Arya K, Babbush R, et al. 2019 Nature 574 505Google Scholar

    [19]

    Huang H L, Wu D, Fan D, et al. 2020 Sci. China Inf. Sci. 63 180501

    [20]

    Yang C P, Chu S I, Han S Y 2004 Phys. Rev. Lett. 92 117902Google Scholar

    [21]

    Wang X M, Zhang A Q, Xu P, Zhao S M 2021 Chin. Phys. B 30 030307Google Scholar

    [22]

    Wallraff A, Schuster D I, Blais A, et al. 2007 Phys. Rev. A 76 042319Google Scholar

    [23]

    Sangouard N, Lacour X, Guerin S, Jauslin H R 2005 Phys. Rev. A 72 062309Google Scholar

    [24]

    Zheng S B 2005 Phys. Rev. Lett. 95 080502Google Scholar

    [25]

    Shao X Q, Chen L, Zhang S, Yeon K H 2009 J. Phys. B 42 165507Google Scholar

    [26]

    Rousseaux B, Guerin S, Vitanov N V 2013 Phys. Rev. A 87 032328Google Scholar

    [27]

    Liang Y, Wu Q C, Su S L, Ji X, Zhang S 2015 Phys. Rev. A 91 032304Google Scholar

    [28]

    Liang Z T, Yue X X, Lv Q X, Du Y X, Huang W, Yan H, Zhu S L 2016 Phys. Rev. A 93 040305Google Scholar

    [29]

    Wang T H, Zhang Z X, Xiang L, Jia Z L, Duan P, Cai W Z, Gong Z H, Zong Z W, Wu M M, Wu J L, Sun L Y, Yin Y, Guo G P 2018 New J. Phys. 20 065003Google Scholar

    [30]

    Chu J, Li D Y, Yang X P, Song S Q, Han Z K, Yang Z, Dong Y Q, Zheng W, Wang Z M, Yu X M, Lan D, Tan X S, Yu Y 2020 Phys. Rev. A 13 064012Google Scholar

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    Spillane S M, Kippenberg T J, Painter O J, Vahala K J 2003 Phys. Rev. Lett. 91 043902Google Scholar

    [32]

    Spillane S M, Kippenberg T J, Vahala K J, Goh K W, Wilcut E, Kimble H J 2005 Phys. Rev. A 71 013817Google Scholar

    [33]

    Baksic A, Ribeiro H, Clerk A A 2016 Phys. Rev. Lett. 116 230503Google Scholar

  • 图 1  量子比特能级结构. qubit A和qubit B被囚禁在一个传输线性谐振器中, 每个量子比特拥有四个能级: $ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $, $ {\left| R \right\rangle _l} $$ {\left| e \right\rangle _l} $, $ l = {\text{A, B}} $, 其中$ {\left| e \right\rangle _l} $是辅助激发态. 有效信息被编码在$ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $$ {\left| R \right\rangle _l} $三个不同能级上. qubit A为控制比特, qubit B为受控比特.

    Fig. 1.  Structure of qubit energy level, qubit A and qubit B are fabricated in a transmission line resonator, each has four-level i.e., $ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $, $ {\left| R \right\rangle _l} $ and $ {\left| e \right\rangle _l} $, $ l = {\text{A, B}} $ here, $ {\left| e \right\rangle _l} $ is an auxiliary excited state. The information is encoded on the states $ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $ and $ {\left| R \right\rangle _l} $. Qubit A is a controlling qubit, and qubit B is a controlled qubit.

    图 2  (a) 脉冲幅度取不同值时, 参数$ {\theta _0}\left( t \right) $随时间的变化情况; $ {\varOmega _0} = 0.1 t_{\text{f}}^{ - 1} $(黑色虚线), ${\varOmega _0} = 50 t_{\text{f}}^{{{ - 1}}}$(浅绿色实线); (b) 脉冲幅度取不同值时, 参数$ {\theta _1}(t) $随时间的变化情况; $ {\varOmega _0} = 0.1 t_f^{ - 1} $(蓝色), $ {\varOmega _0} = 1 t_{\text{f}}^{ - 1} $(红色), $ {\varOmega _0} = 10 t_{\text{f}}^{ - 1} $(粉色), $ {\varOmega _0} = 20 t_{\text{f}}^{ - 1} $(浅绿色), $ {\varOmega _0} = 40 t_{\text{f}}^{ - 1} $(绿色).

    Fig. 2.  (a) Variation of $ {\theta _0}\left( t \right) $ with time when the amplitude of pulse has different values. $ {\varOmega _0} = 0.1 t_{\text{f}}^{ - 1} $(black dotted line), ${\varOmega _0} = 50 t_{\text{f}}^{{{ - 1}}}$ (light green line); (b) variation of $ {\theta _1}(t) $ with time when the amplitude of pulse has different values. $ {\varOmega _0} = 0.1 t_f^{ - 1} $(blue), $ {\varOmega _0} = 1 t_{\text{f}}^{ - 1} $(red), $ {\varOmega _0} = 10 t_{\text{f}}^{ - 1} $(pink), $ {\varOmega _0} = 20 t_{\text{f}}^{ - 1} $(light green), $ {\varOmega _0} = 40 t_{\text{f}}^{ - 1} $(green).

    图 3  保真度随时间变化.

    Fig. 3.  Variation of fidelity with time.

    图 4  参数为${\varOmega _0} = 0.3 t_{\text{f}}^{{{ - 1}}}$$g = 100 t_{\text{f}}^{{{ - 1}}}$时, 量子态$\left| {{\phi _1}} \right\rangle - $$ \left| {{\phi _5}} \right\rangle$布居随时间变化

    Fig. 4.  Variations of quantum state $\left| {{\phi _1}} \right\rangle -\left| {{\phi _5}} \right\rangle$ with time when ${\varOmega _0} = 0.3 t_{\text{f}}^{{{ - 1}}}$ and $g = 100 t_{\text{f}}^{{{ - 1}}}$.

    图 5  $ \left\langle {\psi (t)} \right|\left. {{\phi _1}} \right\rangle $随时间的变化图

    Fig. 5.  Variation of $ \left\langle {\psi (t)} \right|\left. {{\phi _1}} \right\rangle $varies with time.

    图 6  不同的退相干因素$ \kappa $, $ \gamma $$ {\gamma _\phi } $对最终保真度的影响 (a) $ \gamma $$ \kappa $的影响; (b) $ \gamma $$ {\gamma _\phi } $的影响; (c) 退相干中不含$ \gamma $时的影响

    Fig. 6.  Influence of different decoherence factors $ \kappa $, $ \gamma $ and $ {\gamma _\phi } $ to the final fidelity: (a) The effect of $ \gamma $ and $ \kappa $ to fidelity; (b) the effect of $ \gamma $ and $ {\gamma _\phi } $ to fidelity; (c) the effect of $ \kappa $ and $ {\gamma _\phi } $ to fidelity.

  • [1]

    Gong M, Wang S Y, Zha C, Chen M C, H L, Wu Y L, Zhu Q L, Zhao Y W, Li S W, Guo S J, Qian H R, Ye Y S, Chen F S, Ying C, Yu J L, Fan D J, Wu D H, Su H, Deng H, Rong H, Zhang K L, Cao S R, Lin J, Xu Y, Sun L H, Guo C, Li N, Liang F T, Bastidas V M, Nemoto K, Munro W J, Huo Y H, Lu C Y, Peng C Z, Zhu X B, Pan J W 2021 Science 372 948Google Scholar

    [2]

    Zhong H S, Wang H, Deng Y H, Chen M C, Peng L C, Luo Y H, Qin J, Wu D, Ding X, Hu Y, Hu P, Yang X Y, Zhang W J, Li H, Li Y X, Jiang X, Gan L, Yang G W, You L X, Wang Z, Li L, Liu N L, Lu C Y, Pan J W 2020 Science 370 1460Google Scholar

    [3]

    Preskill J 2018 Quantum 2 79Google Scholar

    [4]

    刘超, 邬云文 2018 物理学报 67 170302Google Scholar

    Liu C, Wu Y W 2018 Acta Phys. Sin. 67 170302Google Scholar

    [5]

    Yu D M, Wang H, Ma D D, Zhao X D, Qian J 2019 Opt. Express 27 23080Google Scholar

    [6]

    Shi H M, Yu Y F, Zhang Z M 2012 Chin. Phys. B 21 064205Google Scholar

    [7]

    Liu G Q, Pan X Y 2018 Chin. Phys. B 27 020304Google Scholar

    [8]

    Wan Y, Kienzler D, Erickson S D, Mayer K H, Tan T R, Wu J J, Vasconcelos H M, Glancy S, Knill E, Wineland D J, Wilson A C, Leibfried D 2019 Science 364 875Google Scholar

    [9]

    Qiu C D, Ne X F, Lu D W 2021 Chin. Phys. B 30 048201Google Scholar

    [10]

    Zhang M, Wei L F 2012 Chin. Phys. Lett. 29 080301Google Scholar

    [11]

    Wen J W, Kong X Y, Wei S J, Wang B X, Xin T, Long G L 2019 Phys. Rev. A 99 012320Google Scholar

    [12]

    Zhang F Y, Pei P, Li C, Song H S 2011 Chin. Phys. Lett. 28 120304Google Scholar

    [13]

    Alqahtani M M 2020 Quantum Inf. Process. 19 12Google Scholar

    [14]

    Yang C P, Zheng Z F, Zhang Y 2018 Opt. Lett. 43 5765Google Scholar

    [15]

    Göppl M, Fragner A, Baur M, et al. 2008 J. Appl. Phys. 104 113904Google Scholar

    [16]

    Clarke J, Wilhelm F K 2008 Nature 453 1031Google Scholar

    [17]

    Wendin G 2017 Rep. Prog. Phys. 80 106001Google Scholar

    [18]

    Arute F, Arya K, Babbush R, et al. 2019 Nature 574 505Google Scholar

    [19]

    Huang H L, Wu D, Fan D, et al. 2020 Sci. China Inf. Sci. 63 180501

    [20]

    Yang C P, Chu S I, Han S Y 2004 Phys. Rev. Lett. 92 117902Google Scholar

    [21]

    Wang X M, Zhang A Q, Xu P, Zhao S M 2021 Chin. Phys. B 30 030307Google Scholar

    [22]

    Wallraff A, Schuster D I, Blais A, et al. 2007 Phys. Rev. A 76 042319Google Scholar

    [23]

    Sangouard N, Lacour X, Guerin S, Jauslin H R 2005 Phys. Rev. A 72 062309Google Scholar

    [24]

    Zheng S B 2005 Phys. Rev. Lett. 95 080502Google Scholar

    [25]

    Shao X Q, Chen L, Zhang S, Yeon K H 2009 J. Phys. B 42 165507Google Scholar

    [26]

    Rousseaux B, Guerin S, Vitanov N V 2013 Phys. Rev. A 87 032328Google Scholar

    [27]

    Liang Y, Wu Q C, Su S L, Ji X, Zhang S 2015 Phys. Rev. A 91 032304Google Scholar

    [28]

    Liang Z T, Yue X X, Lv Q X, Du Y X, Huang W, Yan H, Zhu S L 2016 Phys. Rev. A 93 040305Google Scholar

    [29]

    Wang T H, Zhang Z X, Xiang L, Jia Z L, Duan P, Cai W Z, Gong Z H, Zong Z W, Wu M M, Wu J L, Sun L Y, Yin Y, Guo G P 2018 New J. Phys. 20 065003Google Scholar

    [30]

    Chu J, Li D Y, Yang X P, Song S Q, Han Z K, Yang Z, Dong Y Q, Zheng W, Wang Z M, Yu X M, Lan D, Tan X S, Yu Y 2020 Phys. Rev. A 13 064012Google Scholar

    [31]

    Spillane S M, Kippenberg T J, Painter O J, Vahala K J 2003 Phys. Rev. Lett. 91 043902Google Scholar

    [32]

    Spillane S M, Kippenberg T J, Vahala K J, Goh K W, Wilcut E, Kimble H J 2005 Phys. Rev. A 71 013817Google Scholar

    [33]

    Baksic A, Ribeiro H, Clerk A A 2016 Phys. Rev. Lett. 116 230503Google Scholar

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出版历程
  • 收稿日期:  2022-02-08
  • 修回日期:  2022-03-13
  • 上网日期:  2022-07-25
  • 刊出日期:  2022-08-05

电路量子电动力学中基于超绝热捷径的控制相位门实现

  • 南京邮电大学, 信号处理与传输研究院, 南京 210003
  • 通信作者: 赵生妹, zhaosm@njupt.edu.cn
    基金项目: 国家自然科学基金(批准号: 61871234)资助的课题.

摘要: 针对绝热算法在系统演化过程中需要较长操作时间的问题, 本文提出了电路量子电动力学系统中基于超绝热捷径的两量子比特控制相位门的快速制备方案. 首先将量子比特的能级进行编码, 针对不同初始态分类讨论, 获得系统的有效哈密顿量. 通过反绝热驱动, 推导出系统有效哈密顿量的修正项, 以抑制不同本征态之间不必要的跃迁, 从而获得了高保真度的基于超绝热捷径控制相位门. 数值模拟验证了本方案的有效性, 最终保真度为0.991. 所提方案可以加速演化, 并且比绝热通道更有效. 此外, 本方案对谐振器的衰减和超导量子比特的退相干具有鲁棒性. 通过对谐振腔的泄漏、量子比特的自发辐射和退相位的影响分析, 得到的系统最终保真度始终保持在0.984以上.

English Abstract

    • 量子计算机拥有传统计算机无法获取的速度和强大的存储能力, 受到了越来越广泛的关注[1,2]. 理论上, 量子计算机上操作可以由数个单比特和多比特的量子逻辑门组合而成. 将所需的多个量子门按照设计方案排列组合起来, 通过量子线路实现量子计算[3], 其中不可或缺的两比特量子逻辑门之一是控制相位门[4], 其由控制比特和受控比特组成. 当控制比特为$ \left| 1 \right\rangle $且受控比特为$ \left| 1 \right\rangle $时, 量子态$ \left| {11} \right\rangle $会产生180°的相位翻转, 而其他量子态$ \left| {00} \right\rangle $, $ \left| {01} \right\rangle $, $ \left| {10} \right\rangle $均不发生相位变化.

      量子门的制备依赖于物理实现系统. 常用的物理系统有离子阱系统、核磁共振系统、腔量子电动力学系统和电路量子电动力学(QED)系统[5-14]. 电路QED系统是由超导量子比特和超导谐振腔组成[15,16], 其中超导谐振腔主要包括中心导体和两边的地(中心导体的长度决定了谐振腔的模式频率), 超导量子比特由约瑟夫森结构成, 超导量子比特作为人工原子与超导谐振腔电磁耦合在一起[17]. 2019年, 具有量子优越性的量子计算模型首次在超导量子系统得到证明[18]. 基于其可设计性、易扩展性、长相干性, 以及与其他物理系统易耦合等特性, 电路QED系统现已成为实现量子计算最有前途的系统之一[19]. 在电路QED系统中, 广泛使用的超导量子比特为Transmon. 与超导电荷量子比特相比, Transmon具有对电荷噪声敏感度低、退相干时间长等优点[20-22].

      早在2005年, Sangouard等[23]提出了一个基于绝热算法制备SWAP门的方案. 同年, Zheng[24]提出了一个基于绝热算法构建$ {\text{π }} $相位门的方案. 在2009年, Shao等[25]基于量子芝诺动力学提出了一个构建控制非门(CNOT门)的方案. 到2013年时, Rousseaux等[26]则实现了任意比特门的制备. 与此同时, 2015年, Liang等[27]在腔量子电动力学系统中提出了快速构建控制相位门的方案, 利用的是LR不变量捷径方法. 该方案不仅可以实现一个量子比特的相位门, 还可以实现两个量子比特的控制相位门, 接着进一步推广到n个量子比特的控制相位门. 2016年, Liang等[28]提出了一种基于超绝热捷径方法实现了通用量子门的方案, 并基于金刚石氮空中心系统给出了详细说明. 2018年, Wang等[29]运用超绝热捷径方法构建了在超导Xmon量子系统中的高保真度的单比特量子门. 2020年, Chu等[30]提出了一种基于超导电路的两比特量子门的构建方案, 结合参数调制和超绝热捷径方法, 并以SWAP门和控制Z门为例进行了演示. 与其他捷径方法相比, 超绝热捷径方法能够实现高保真度, 并且操作简单可行. 但是, 在电路QED系统中基于超绝热捷径方法的相位控制门还未被讨论.

      因此, 本文基于超绝热捷径方法, 提出一种在电路量子电动力学系统中快速制备两比特控制相位门的方案. 利用反向导热场思想, 获得哈密顿量的修正项, 抑制了本征态间不需要的跃迁. 使用第2次迭代的本征态为演化路径, 由于不需要初始态和最终目标态之间的直接耦合, 可降低了实验难度. 通过数值模拟, 对本方案进行验证和性能分析. 本方案有以下3个优点: 1)与绝热算法相比, 本方案不需要严格遵守绝热近似条件; 2)本方案只保留有用的本征态间的跃迁, 节省了演化时间; 3)本方案给出了初态与终态间的直接耦合方案, 提高了方案的可行性. 总之, 本方案缩短了演化时间、降低了实验实现难度, 其最终保真度可接近于1, 因而具有重要的参考价值.

    • 首先给出电路量子电动力学系统, 系统中两个Transmon囚禁在一个传输线谐振器中. 图1是电路量子电动力学系统中量子比特的能级结构图, 每个量子比特拥有4个能级, 分别为$ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $, $ {\left| R \right\rangle _l} $$ {\left| e \right\rangle _l} $, 其中$ {\left| e \right\rangle _l} $是辅助激发态, 下标$ l = {\text{A, B}} $是区分不同的量子比特. 将有效信息编码在$ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $$ {\left| R \right\rangle _l} $这3个不同能级上. 量子比特$ l = {\text{A}} $为控制比特, 其中$ {\left| R \right\rangle _{\text{A}}} $$ {\left| e \right\rangle _{\text{A}}} $之间的跃迁是与耦合强度为$ {g_{\text{A}}} $的谐振器发生的共振耦合, 其他能级之间的跃迁是被禁止的. 量子比特$ l = {\text{B}} $为受控比特, $ {\left| R \right\rangle _{\text{B}}} $$ {\left| e \right\rangle _{\text{B}}} $之间的跃迁是与耦合强度为$ {g_{\text{B}}} $的谐振器发生的共振耦合, $ {\left| {{L_1}} \right\rangle _{\text{B}}} \leftrightarrow {\left| e \right\rangle _{\text{B}}} $, $ {\left| {{L_2}} \right\rangle _{\text{B}}} \leftrightarrow {\left| e \right\rangle _{\text{B}}} $的跃迁分别是由时间独立的拉比频率为$ {\varOmega _{{\text{ }}1}}(t) $, $ {\varOmega _{{\text{ }}2}}(t) $的经典脉冲来驱动的, 同时其他能级之间的跃迁也是被禁止的.

      图  1  量子比特能级结构. qubit A和qubit B被囚禁在一个传输线性谐振器中, 每个量子比特拥有四个能级: $ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $, $ {\left| R \right\rangle _l} $$ {\left| e \right\rangle _l} $, $ l = {\text{A, B}} $, 其中$ {\left| e \right\rangle _l} $是辅助激发态. 有效信息被编码在$ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $$ {\left| R \right\rangle _l} $三个不同能级上. qubit A为控制比特, qubit B为受控比特.

      Figure 1.  Structure of qubit energy level, qubit A and qubit B are fabricated in a transmission line resonator, each has four-level i.e., $ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $, $ {\left| R \right\rangle _l} $ and $ {\left| e \right\rangle _l} $, $ l = {\text{A, B}} $ here, $ {\left| e \right\rangle _l} $ is an auxiliary excited state. The information is encoded on the states $ {\left| {{L_1}} \right\rangle _l} $, $ {\left| {{L_2}} \right\rangle _l} $ and $ {\left| R \right\rangle _l} $. Qubit A is a controlling qubit, and qubit B is a controlled qubit.

      $ {g_{\text{A}}} = {g_{\text{B}}} = g $, 利用超绝热捷径方法快速获得两比特控制相位量子门. 在旋转波近似条件下[31], 电路QED系统的总哈密顿量可写为

      $ \begin{split} {H}_{\text{total}}=&{\displaystyle \sum _{l=\text{A,B}}\left[{g}_{l}a{|e\rangle }_{l}\langle R|\right]}+{\varOmega }_{1}\left(t\right){|e\rangle }_{\text{B}}\langle {L}_{1}|\\&+{\varOmega }_{2}\left(t\right){|e\rangle }_{\text{B}}\langle {L}_{2}|+{\rm h.c.}\text{, } \end{split} $

      其中, $ \hbar = 1 $, $ a\left( {{a^\dagger }} \right) $是谐振器的湮灭(产生)算子. 在该系统中, 量子信息被编码为${\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}} = $$\left| {00} \right\rangle$, $ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| R \right\rangle _{\text{B}}} = \left| {01} \right\rangle $, $ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}} = \left| {10} \right\rangle $, $ {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} = \left| {11} \right\rangle $. $ {\left| 0 \right\rangle _{\text{C}}} $表示谐振器处于真空态, $ {\left| 1 \right\rangle _{\text{C}}} $表示单光子态. 要制备控制相位门, 需实现以下逻辑:

      $ \begin{split} {|0\rangle }_{\text{C}}{|{L}_{1}\rangle }_{\text{A}}{|{L}_{1}\rangle }_{\text{B}}=|00\rangle \to \text{ }{|0\rangle }_{\text{C}}{|{L}_{1}\rangle }_{\text{A}}{|{L}_{1}\rangle }_{\text{B}}=\text{ }|00\rangle \text{, }\\ \end{split} \tag{2a} $

      $ \begin{split} {|0\rangle }_{\text{C}}{|{L}_{1}\rangle }_{\text{A}}{|{L}_{2}\rangle }_{\text{B}}=|01\rangle \to \text{ }{|0\rangle }_{\text{C}}{|{L}_{1}\rangle }_{\text{A}}{|{L}_{2}\rangle }_{\text{B}}=\text{ }|01\rangle \text{, }\\ \end{split}\tag{2b} $

      $ \begin{split} {|0\rangle }_{\text{C}}{|{L}_{2}\rangle }_{\text{A}}{|{L}_{1}\rangle }_{\text{B}}=|10\rangle \to \text{ }{|0\rangle }_{\text{C}}{|{L}_{2}\rangle }_{\text{A}}{|{L}_{1}\rangle }_{\text{B}}=\text{ }|10\rangle \text{, }\\ \end{split}\tag{2c} $

      $ \begin{split} {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} = \left| {11} \right\rangle \to - {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} = - \left| {11} \right\rangle . \\ \end{split}\tag{2d}$

      假设系统的初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $, 此时谐振器处于真空$ {\left| 0 \right\rangle _{\text{C}}} $态, 量子比特A处于$ \left| R \right\rangle $态, 量子比特B处于$ \left| {{L_2}} \right\rangle $态. 根据总哈密顿量, 系统将会在由$ {\phi _1} $, $ {\phi _2} $, ···, $ {\phi _5} $张成的子空间中进行演化, 其中,

      $ \left| {{\phi _1}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}}, \tag{3a}$

      $ \left| {{\phi _2}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| e \right\rangle _{\text{B}}}, \tag{3b}$

      $ \left| {{\phi _3}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}}, \tag{3c}$

      $ \left| {{\phi _4}} \right\rangle = {\left| 1 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| R \right\rangle _{\text{B}}}, \tag{3d}$

      $ \left| {{\phi _5}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| e \right\rangle _{\text{A}}}{\left| R \right\rangle _{\text{B}}}. \tag{3e}$

      系统总哈密顿量可写为

      $ {H_{{\text{total}}}} = {H_{\text{C}}} + {H_{\text{L}}}\left( t \right), \tag{4a}$

      $ {H_{\text{C}}} = g\left| {{\phi _5}} \right\rangle \left\langle {{\phi _4}} \right| + g\left| {{\phi _2}} \right\rangle \left\langle {{\phi _4}} \right| +{\rm h.c.}{\text{ }} , \tag{4b}$

      $ {H_{\text{L}}}\left( t \right) = {\varOmega _1}\left( t \right)\left| {{\phi _2}} \right\rangle \left\langle {{\phi _3}} \right| + {\varOmega _2}\left( t \right)\left| {{\phi _2}} \right\rangle \left\langle {{\phi _1}} \right| + {\rm h.c.} \tag{4c}$

      求解得到, $ {H_{\text{C}}} $的本征值为$ {E_0} = 0 $, $ {E_1} = \sqrt 2 g $$ {E_2} = - \sqrt 2 g $, 同时相对应的本征态为

      $ \begin{split} \left| {{\psi _0}} \right\rangle & = \dfrac{1}{{\sqrt 2 }}\left[ {\left| {{\phi _2}} \right\rangle - \left| {{\phi _5}} \right\rangle } \right],\\ \left| {{\psi _1}} \right\rangle &= \dfrac{1}{2}\left[ {\left| {{\phi _2}} \right\rangle + \sqrt 2 \left| {{\phi _4}} \right\rangle + \left| {{\phi _5}} \right\rangle } \right], \\ \left| {{\psi _2}} \right\rangle & = \dfrac{1}{2}\left[ {\left| {{\phi _2}} \right\rangle - \sqrt 2 \left| {{\phi _4}} \right\rangle + \left| {{\phi _5}} \right\rangle } \right]. \end{split} $

      因此, (4)式中的哈密顿量可以用$ {H_{\text{C}}} $的本征值、本征态来表达, 系统哈密顿量$ {H_{{\text{total}}}} $可以改写为

      $ {\text{ }}{H_{{\text{total}}}} = {H_{\text{C}}} + {H_{\text{L}}}\left( t \right), \tag{5a}$

      $ {\text{ }}{H_{\text{C}}} = \sqrt 2 g\left| {{\psi _1}} \right\rangle \left\langle {{\psi _1}} \right| - \sqrt 2 g\left| {{\psi _2}} \right\rangle \left\langle {{\psi _2}} \right| + {\rm h.c.}, \tag{5b}$

      $ \begin{split} {H_{\text{L}}}\left( t \right) =& \frac{{{\varOmega _1}\left( t \right)}}{2}\left[ {\sqrt 2 \left| {{\psi _0}} \right\rangle + \left| {{\psi _1}} \right\rangle + \left| {{\psi _2}} \right\rangle } \right]\left\langle {{\phi _3}} \right| \\ & + \frac{{{\varOmega _2}\left( t \right)}}{2}\left[ {\sqrt 2 \left| {{\psi _0}} \right\rangle + \left| {{\psi _1}} \right\rangle + \left| {{\psi _2}} \right\rangle } \right]\left\langle {{\phi _1}} \right| +{ \rm h.c.} \\ \end{split}\tag{5c}$

      对系统哈密顿量通过幺正算符$ {U_{\text{C}}} = \exp \left[ { - {\text{i}}{H_{\text{C}}}t} \right] $进行绘景变换. 在${\varOmega _1}\left( t \right), {\varOmega _2}\left( t \right) \ll g$的条件下, 忽略高频振荡项, 利用此绘景变换容易算出系统的有效哈密顿量为

      $ \begin{split} {H_{{\text{eff}}}}\left( t \right) &= \frac{{{\varOmega _1}\left( t \right)}}{{\sqrt 2 }}\left| {{\psi _0}} \right\rangle \left\langle {{\phi _3}} \right| + \frac{{{\varOmega _2}\left( t \right)}}{{\sqrt 2 }}\left| {{\psi _0}} \right\rangle \left\langle {{\phi _1}} \right| + {\rm h.c.} \\ \quad \quad \quad &= {\varOmega _{\text{p}}}\left( t \right)\left| {{\psi _0}} \right\rangle \left\langle {{\phi _3}} \right| \,+\, {\varOmega _{\text{s}}}\left( t \right)\left| {{\psi _0}} \right\rangle \left\langle {{\phi _1}} \right|\, +\, {\rm h.c.} {\text{,}} \end{split} $

      其中, $ {\varOmega _{\text{p}}}\left( t \right) = {{{\varOmega _1}\left( t \right)} \mathord{\left/ {\vphantom {{{\varOmega _1}\left( t \right)} {\sqrt 2 }}} \right. } {\sqrt 2 }} $$ {\varOmega _{\text{s}}}\left( t \right) = {{{\varOmega _2}\left( t \right)} \mathord{\left/ {\vphantom {{{\varOmega _2}\left( t \right)} {\sqrt 2 }}} \right. } {\sqrt 2 }} $. (6)式可以认为是一个由两个基态$ \left| {{\phi _1}} \right\rangle $, $ \left| {{\phi _3}} \right\rangle $和激发态$ \left| {{\psi _0}} \right\rangle $组成的三能级量子系统.

      系统有效哈密顿量${H_{\rm eff}}\left( t \right)$的瞬时本征值, 分别为$ {\varepsilon _0} = 0 $, $ {\varepsilon _1} = \varOmega \left( t \right) $$ {\varepsilon _2} = - \varOmega \left( t \right) $, 相对应的瞬时本征态为

      $ \left| {{\varphi _0}\left( t \right)} \right\rangle = - \sin {\theta _0}\left( t \right)\left| {{\phi _1}} \right\rangle + \cos {\theta _0}\left( t \right)\left| {{\phi _3}} \right\rangle \text{, } \tag{7a}$

      $ \left| {{\varphi _ + }\left( t \right)} \right\rangle = \frac{1}{{\sqrt 2 }}\left[ {\cos {\theta _0}\left( t \right)\left| {{\phi _1}} \right\rangle + \left| {{\psi _0}} \right\rangle + \sin {\theta _0}\left( t \right)\left| {{\phi _3}} \right\rangle } \right] \text{, } \tag{7b}$

      $ \left| {{\varphi _ - }\left( t \right)} \right\rangle = \frac{1}{{\sqrt 2 }}\left[ {\cos {\theta _0}\left( t \right)\left| {{\phi _1}} \right\rangle - \left| {{\psi _0}} \right\rangle + \sin {\theta _0}\left( t \right)\left| {{\phi _3}} \right\rangle } \right] \text{, } \tag{7c}$

      其中$ \varOmega (t) {=} \sqrt {\varOmega _{\text{p}}^2 (t) {+} \varOmega _{\text{s}}^{\text{2}}(t)} $$\tan {\theta _0}(t) = \varOmega _{\text{p}} (t)/ { {\varOmega _{\text{s}}}(t)}$.

      进行第1次迭代. 对在绝热框架下的有效哈密顿量$ {H_{{\text{eff}}}}\left( t \right) $进行酉变换${U_0}\left( t \right) = \sum\nolimits_{k = 0, \pm } {\left| {{\varphi _k}} \right\rangle } \left\langle {{\varphi _k}\left( t \right)} \right|$. 变换后的哈密顿量变为

      $ \begin{split} {H_1}\left( t \right) = \;& {U_0}\left( t \right){H_{{\text{eff}}}}\left( t \right)U_0^\dagger \left( t \right) + {\rm i}{{\dot U}_0}\left( t \right)U_0^\dagger \left( t \right) \\ \quad \quad \;\, =\;& \varOmega \left( t \right)\left( {\left| {{\varphi _ + }} \right\rangle \left\langle {{\varphi _ + }} \right| - \left| {{\varphi _ - }} \right\rangle \left\langle {{\varphi _ - }} \right|} \right) \\ + \;&\frac{{{{\dot \theta }_0}\left( t \right)}}{{\sqrt 2 }}\left( {{\rm i}\left| {{\varphi _ + }} \right\rangle \left\langle {{\varphi _0}} \right| + {\rm i}\left| {{\varphi _ - }} \right\rangle \left\langle {{\varphi _0}} \right| + {\rm h.c.}} \right). \end{split} $

      由(8)式可得, 在绝热近似条件$\left| {\dot \theta \left( t \right)} \right| \ll \sqrt 2 \varOmega \left( t \right)$下, 系统的演化遵循绝热态$ \left| {{\varphi _{0, \pm }}\left( t \right)} \right\rangle $. (8)式的最后一项代表不同本征态之间的跃迁. 因此, 随着哈密顿量参数的缓慢变化, 系统的演化只会遵循一个本征态, 但需要很长的时间. 否则, 随着系统哈密顿量的更快变化将引起不同本征态之间的跃迁. 然而, 参考文献[32, 33]提出了一种反向导热的哈密顿量修正项来抑制不同本征态之间的跃迁. 现对(8)式引入反向导热修正项$ - {\text{i}}{\dot U_0}\left( t \right)U_0^\dagger \left( t \right) $. 因此, 在绝热框架下, 修正项可以写为:

      $ \begin{split} H_{\rm CD}^1\left( t \right) &= - {{\rm i}}{{\dot U}_0}\left( t \right)U_0^ + \left( t \right) \\ \quad \quad \quad &= {{\rm i}}{{\dot \theta }_0}\left( t \right)\left( {\left| {{\phi _3}} \right\rangle \left\langle {{\phi _1}} \right| - \left| {{\phi _1}} \right\rangle \left\langle {{\phi _3}} \right|} \right). \end{split} $

      与(6)式的哈密顿量相比, 反向导热修正项哈密顿量需要$ \left| {{\phi _1}} \right\rangle $$ \left| {{\phi _3}} \right\rangle $之间的直接耦合, 但是完成此过程十分困难. 所以本文采用参考文献[33]中提出的方法, 通过修改脉冲的形式来获得修正项.

      进行第2次迭代. 首先, 可获得$ {H_1}\left( t \right) $的本征值为$ {\varepsilon '_0} = 0 $, $ {\varepsilon '_ + } = \varOmega '\left( t \right) $$ {\varepsilon '_ - } = - \varOmega '\left( t \right) $, 相对应的瞬时本征态为

      $\begin{split} \left| {{\varphi '_0}\left( t \right)} \right\rangle =\; & \frac{1}{{\sqrt 2 }}\left[ - {\text{i}}\sin {\theta _1}\left( t \right)\left| {{\varphi _ + }} \right\rangle \right.\\ &+ \sqrt 2 \cos {\theta _1}\left( t \right)\left| {{\varphi _0}} \right\rangle \\ & +\left. {\text{i}}\sin {\theta _1}\left( t \right)\left| {{\varphi _ - }} \right\rangle \right], \end{split}\tag{10a} $

      $ \begin{split} \left| {{\varphi '_ + }\left( t \right)} \right\rangle =& \frac{1}{2}{\text{i}}\left[ {1 + \cos {\theta _1}\left( t \right)} \right]\left| {{\varphi _ + }} \right\rangle + \frac{1}{{\sqrt 2 }}\sin {\theta _1}\left( t \right)\left| {{\varphi _0}} \right\rangle \\ &+ \frac{1}{2}{\text{i}}\left[ {1 - \cos {\theta _1}\left( t \right)} \right]\left| {{\varphi _ - }} \right\rangle , \end{split} \tag{10b} $

      $ \begin{split} \left| {{\varphi '_ - }\left( t \right)} \right\rangle =& \frac{1}{2}{\text{i}}\left[ {1 - \cos {\theta _1}\left( t \right)} \right]\left| {{\varphi _ + }} \right\rangle - \frac{1}{{\sqrt 2 }}\sin {\theta _1}\left( t \right)\left| {{\varphi _0}} \right\rangle \\& + \frac{1}{2}{\text{i}}\left[ {1 + \cos {\theta _1}\left( t \right)} \right]\left| {{\varphi _ - }} \right\rangle , \end{split}\tag{10c} $

      其中$ \varOmega' (t) = \sqrt {{{\dot \theta }_0}{{( t )}^2} + {\varOmega ^2} (t)} $$\tan {\theta _1} (t) = {\dot\theta}_0 (t)/\varOmega (t)$.

      对于哈密顿量$ {H_1}\left( t \right) $进行${U_1}(t) = \displaystyle\sum\limits_{k = 0, \pm } |\varphi'_k \rangle \langle {{ \varphi '_k} (t)} |$变换, 得到

      $ \begin{split} \;& H_2(t) = {U_1}(t){H_1}(t)U_1^\dagger (t) + {{\rm i}}{{\dot U}_1}(t) U_1^\dagger ( t) \\ = \;& \varOmega ' (t) \big( {| {{{\varphi '}_ + }} \rangle \langle {{{\varphi }'_ + }} | - | {{{\varphi }'_-}} \rangle \langle {{{\varphi}'_ - }} |} \big) \\&+ \frac{{{{\dot \theta }_1} (t)}}{{\sqrt 2 }} \big( {{\rm{i}} | {{{\varphi '}_ + }} \rangle \langle {{{\varphi '}_0}} | + {{\rm i}} | {{{\varphi '}_ - }} \rangle \langle {{{\varphi '}_0}} | + {\rm h.c.}} \big). \end{split} $

      为了与(9)式相等, 以$ \left\{ {\left| {{\phi _1}} \right\rangle , \left| {{\phi _3}} \right\rangle , \left| {{\psi _0}} \right\rangle } \right\} $为基, 新的反向导热哈密顿量修正项为

      $ \begin{split} H_{{\text{CD}}}^2\left( t \right) = \; & - {\text{i}}U_0^ + \left( t \right)\dot U_1^ + \left( t \right){U_1}\left( t \right){U_0}\left( t \right) \\ =\;& {{\dot \theta }_1}\left( t \right)\left( - \cos {\theta _0}\left( t \right)\left| {{\psi _0}} \right\rangle \left\langle {{\phi _3}} \right| \right. \\ & \left.+ \sin {\theta _0}\left( t \right)\left| {{\psi _0}} \right\rangle \left\langle {{\phi _1}} \right| \right) + {\rm h.c.} \end{split} $

      经过两次迭代获得修正项$ H_{{\text{CD}}}^2\left( t \right) $, 用来抑制不需要的本征态之间的跃迁, 从而花费更少的演化操作时间. 系统有效哈密顿量$ {H_{{\text{eff}}}}\left( t \right) $加上修正项$ H_{{\text{CD}}}^{\text{2}}\left( t \right) $后为

      $ \begin{split} & {H_{\bmod }}\left( t \right) = {H_{{\text{eff}}}}\left( t \right) + H_{{\text{CD}}}^2\left( t \right) \\ =\;& \left[ {{\varOmega _{\text{p}}}\left( t \right) - {{\dot \theta }_1}\left( t \right)\cos {\theta _0}\left( t \right)} \right]\left| {{\psi _0}} \right\rangle \left\langle {{\phi _3}} \right| \\ & +\left[ {{\varOmega _{\text{s}}}\left( t \right) + {{\dot \theta }_1}\left( t \right)\sin {\theta _0}\left( t \right)} \right]\left| {{\psi _0}} \right\rangle \left\langle {{\phi _1}} \right| + {\rm h.c.} \\ =\;& \varOmega _{\text{p}}^{\text{A}}\left( t \right)\left| {{\psi _0}} \right\rangle \left\langle {{\phi _3}} \right| + \varOmega _{\text{s}}^{\text{A}}\left( t \right)\left| {{\psi _0}} \right\rangle \left\langle {{\phi _1}} \right| + {\rm h.c.} , \end{split} $

      其中, $ \varOmega _{\text{p}}^{\text{A}}\left( t \right) = {\varOmega _{\text{p}}}\left( t \right) - {\dot \theta _1}\left( t \right)\cos {\theta _0}\left( t \right) $$\varOmega _{\text{s}}^{\text{A}}\left( t \right) =$${\varOmega _{\text{s}}}\left( t \right)+ {\dot \theta _1}\left( t \right)\sin {\theta _0}\left( t \right)$.

      从以上讨论中可得出, 哈密顿量$ H_{{\text{CD}}}^2\left( t \right) $能够抑制非必须的跃迁. 同时为获得控制相位门, 参数$ {\theta _{0, 1}}\left( t \right) $应取不同值. 当初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $时, 经过演化后想要得到${\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} = \left| {11} \right\rangle \to - {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}$${\left| {{L_2}} \right\rangle _{\text{B}}} = - \left| {11} \right\rangle$, 此时参数$ {\theta _{0, 1}}\left( t \right) $应满足边界条件$ {\theta _1}\left( 0 \right) = {\theta _1}\left( {{t_{\text{f}}}} \right) = 0 $${\theta _0}\left( 0 \right) = - \dfrac{{\text{π }}}{2}$${\theta _0}\left( {{t_{\text{f}}}} \right) = \dfrac{{\text{π }}}{2}$, 其中$ {t_{\text{f}}} $是动态演化的最终时间. 原因如下, 当${\theta _1}\left( 0 \right) = $${\theta _1}\left( {{t_{\text{f}}}} \right) = 0$, 然后有$ \left| {{{\varphi '}_0}\left( 0 \right)} \right\rangle = \left| {{{\varphi '}_0}\left( {{t_{\text{f}}}} \right)} \right\rangle = \left| {{\varphi _0}} \right\rangle $, 即$ {H_1}\left( t \right) $的瞬时本征态和$ {H_{{\text{eff}}}}\left( t \right) $的瞬时本征态是重合的. 所以当初始态为$ \left| {{\phi _1}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $、目标态为$- \left| {{\phi _1}} \right\rangle = - {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}}$时, 根据$ \left| {{\varphi _0}\left( t \right)} \right\rangle $应满足边界条件${\theta _1}\left( 0 \right) = {\theta _1}\left( {{t_{\text{f}}}} \right) = 0$${\theta _0}\left( 0 \right) = - {{\text{π }}}/{2}$${\theta _0}\left( {{t_{\text{f}}}} \right) = $ ${{\text{π }}}/{2}$.

    • 对基于超绝热捷径的控制相位门进行分析. 设系统的初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}} $, 此时谐振器处于真空态$ {\left| 0 \right\rangle _{\text{C}}} $, 量子比特A处于$ \left| {{L_1}} \right\rangle $态, 量子比特B处于$ \left| {{L_1}} \right\rangle $态. 根据总哈密顿量, 系统会在由张成的子空间中进行演化:

      $ \left| {{\phi _6}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}}, \tag{14a} $

      $ \left| {{\phi _7}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| e \right\rangle _{\text{B}}}, \tag{14b} $

      $ \left| {{\phi _8}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}}, \tag{14c} $

      $ \left| {{\phi _9}} \right\rangle = {\left| 1 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| R \right\rangle _{\text{B}}}.\tag{14d} $

      系统总哈密顿量可写为

      $ {H'_{{\text{total}}}} = {H'_{\text{C}}} + {H'_{\text{L}}}\left( t \right), \tag{15a} $

      /$ {H'_{\text{C}}} = {g_{\text{B}}}\left| {{\phi _7}} \right\rangle \left\langle {{\phi _9}} \right| + {\rm{h.c.}}, \tag{15b} $

      $ {\text{ }}{H'_{\text{L}}}\left( t \right) = {\varOmega _1}\left( t \right)\left| {{\phi _7}} \right\rangle \left\langle {{\phi _6}} \right| + {\varOmega _2}\left( t \right)\left| {{\phi _7}} \right\rangle \left\langle {{\phi _8}} \right| + {\rm h.c.}\tag{15c} $

      求得$ {H'_{\text{C}}} $的本征值为$ {E'_1} = {g_{\text{B}}} $$ {E'_2} = - {g_{\text{B}}} $, 同时相应的本征态为$\left| {{{\psi '}_1}} \right\rangle = \dfrac{1}{{\sqrt 2 }}\left[ {\left| {{\phi _7}} \right\rangle + \left| {{\phi _9}} \right\rangle } \right]$, $\left| {{{\psi '}_2}} \right\rangle = $$\dfrac{1}{{\sqrt 2 }}\left[ {\left| {{\phi _7}} \right\rangle - \left| {{\phi _9}} \right\rangle } \right]$. 利用$ {H'_{\text{C}}} $$ {E'_{1, 2}} $, $ \left| {{{\psi '}_{1, 2}}} \right\rangle $可以将系统哈密顿量$ {H'_{{\text{total}}}} $重新写为

      $ {H'_{{\text{total}}}} = {H'_{\text{C}}} + {H'_{\text{L}}}\left( t \right), \tag{16a} $

      $ {H'_{\text{C}}} = {g_{\text{B}}}\left| {{{\psi}'_1}} \right\rangle \left\langle {{{\psi }'_1}} \right| - {g_{\text{B}}}\left| {{{\psi}'_2}} \right\rangle \left\langle {{{\psi}'_2}} \right| + {\rm h.c.} {\text{,}} \tag{16b} $

      $ \begin{split} {H'_{\text{L}}}\left( t \right) =\,& \frac{{{\varOmega _1}\left( t \right)}}{{\sqrt 2 }}\left[ {\left| {{{\psi '}_1}} \right\rangle + \left| {{{\psi '}_2}} \right\rangle } \right]\left\langle {{\phi _6}} \right| \\& + \frac{{{\varOmega _2}\left( t \right)}}{{\sqrt 2 }}\big[ {| {{{\psi '}_1}} \rangle - | {{{\psi '}_2}} \rangle } \big]\langle {{\phi _8}} | + {\rm h.c.} \end{split}\tag{16c} $

      在此基础上, 对系统哈密顿量通过幺正算符${U'_{\text{C}}} = \exp \left[ { - {\text{i}}{{H}'_{\text{C}}}t} \right]$进行一个绘景变换. 利用此绘景变换可算出, 在${\varOmega _1}\left( t \right), {\varOmega _2}\left( t \right) \ll g$的条件下, 忽略高频振荡项, 系统的有效哈密顿量为$ {H'_{{\text{eff}}}}\left( t \right) = 0 $. 同理, 当系统的初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $时, 系统会在由$ {\phi _6}, {\phi _7}, \cdots , {\phi _9} $张成的子空间中进行演化, 所以经过推导系统的有效哈密顿量为$ {H'_{{\text{eff}}}}\left( t \right) = 0 $. 因此, 当系统的初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}} $$ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_1}} \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $时, 系统是不会随时间进行演化的, 是被禁止的.

      进一步设系统的初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}} $, 此时谐振器处于真空$ {\left| 0 \right\rangle _{\text{C}}} $态, 量子比特A处于$ \left| {{L_2}} \right\rangle $态, 量子比特B处于$ \left| {{L_1}} \right\rangle $态. 根据总哈密顿量, 系统会在由$ {\phi _{10}}, {\phi _{11}}, \cdots , {\phi _{13}} $张成的子空间中进行演化, 此时有

      $ \left| {{\phi _{10}}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}}, \tag{17a} $

      $ \left| {{\phi _{11}}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| e \right\rangle _{\text{B}}}, \tag{17b} $

      $ \left| {{\phi _{12}}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}}, \tag{17c} $

      $ \left| {{\phi _{13}}} \right\rangle = {\left| 1 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| R \right\rangle _{\text{B}}}. \tag{17d} $

      系统总哈密顿量可写为

      $ {H''_{{\text{total}}}} = {H''_{\text{C}}} + {H''_{\text{L}}}\left( t \right), \tag{18a} $

      $ {H''_{\text{C}}}={g}_{\text{B}}|{\phi }_{11}\rangle \langle {\phi }_{13}|+{\rm h.c.}\text{, } \tag{18b} $

      $ \begin{split} {\text{ }}{H''_{\text{L}}}\left( t \right) = {\varOmega _1}\left( t \right)\left| {{\phi _{11}}} \right\rangle \left\langle {{\phi _{10}}} \right| + {\varOmega _2}\left( t \right)\left| {{\phi _{11}}} \right\rangle \left\langle {{\phi _{12}}} \right| + {\rm h.c.}\\ \end{split} \tag{18c} $

      可以求得$ {H_{\text{C}}} $的本征值为$ {E''_1} = {g_{\text{B}}} $$ {E''_2} = - {g_{\text{B}}} $, 同时相对应的本征态为$\left| {{{\psi}''_1}} \right\rangle = \dfrac{1}{{\sqrt 2 }}\left[ {\left| {{\phi _7}} \right\rangle + \left| {{\phi _9}} \right\rangle } \right]$, $\left| {{{\psi} ''_2}} \right\rangle = \dfrac{1}{{\sqrt 2 }}\left[ {\left| {{\phi _7}} \right\rangle - \left| {{\phi _9}} \right\rangle } \right]$. 利用$ {H''_{\text{C}}} $$ {E''_{1, 2}} $, $\left| {{{\psi}''_{1, 2}}} \right\rangle$可以将系统哈密顿量$ {H''_{{\text{total}}}} $重新写为

      $ {H''_{{\text{total}}}} = {H''_{\text{C}}} + {H''_{\text{L}}}\left( t \right), \tag{19a} $

      $ {H''_{\text{C}}} = {g_{\text{B}}}\left| {{\psi ''_1}} \right\rangle \left\langle {{\psi ''_1}} \right| - {g_{\text{B}}}\left| {{\psi ''_2}} \right\rangle \left\langle {{\psi ''_2}} \right| + {\rm h.c.}{\text{ ,}} \tag{19b} $

      $ \begin{split} {H''_{\text{L}}}\left( t \right) =& \frac{{{\varOmega _1}\left( t \right)}}{{\sqrt 2 }}\left[ {\left| {{\psi ''_1}} \right\rangle + \left| {{\psi ''_2}} \right\rangle } \right]\left\langle {{\phi _{10}}} \right| \\& + \frac{{{\varOmega _2}\left( t \right)}}{{\sqrt 2 }}\left[ {\left| {{\psi ''_1}} \right\rangle - \left| {{\psi ''_2}} \right\rangle } \right]\left\langle {{\phi _{12}}} \right| + {\rm h.c.} \end{split}\tag{19c} $

      在此基础上, 对系统哈密顿量通过幺正算符${U''_{\text{C}}} = \exp \left[ { - {\text{i}}{H''_{\text{C}}}t} \right]$进行一个绘景变换. 利用此绘景变换容易算出, 在${\varOmega _1}\left( t \right), {\varOmega _2}\left( t \right) \ll g$的条件下, 忽略高频振荡项, 系统的有效哈密顿量变为${H''_{{\text{eff}}}}\left( t \right) =$ $0$. 因此, 当系统的初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| {{L_2}} \right\rangle _{\text{A}}}{\left| {{L_1}} \right\rangle _{\text{B}}} $时, 系统不随时间进行演化, 被禁止.

      综上所述, 在电路QED系统中, 利用超绝热捷径方法可以快速制备两比特的控制相位量子门.

    • 本节将对超绝热捷径快速制备两比特的控制相位量子门方案进行数值模拟. 经过之前的讨论可得, 当初态为$ {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $时, 经过演化后, 需获得$ {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} = \left| {11} \right\rangle \to - {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} = - \left| {11} \right\rangle $, 参数$ {\theta _{0, 1}}\left( t \right) $应满足边界条件$ {\theta _1}\left( 0 \right) = {\theta _1}\left( {{t_f}} \right) = 0 $${\theta _0}\left( 0 \right) = $$- {{\text{π }}}/{2}$${\theta _0}\left( {{t_{\text{f}}}} \right) = {{\text{π }}}/{2}$, 脉冲$ {\varOmega _1}\left( t \right) $$ {\varOmega _2}\left( t \right) $可以根据文献[14]选择高斯型脉冲:

      $ \begin{split} {\varOmega _1}\left( t \right) =& - 20{\varOmega _0}\exp \left[ {{{ - {{\left( {t - {\tau _{11}}} \right)}^2}}/{{T^2}}}} \right] \\ &+ 10{\varOmega _0}\exp \left[ {{{ - {{\left( {t - {\tau _{12}}} \right)}^2}} /{{T^2}}}} \right] \\ & - 0.1{\varOmega _0}\exp \left[ {{{ - {{\left( {t - {\tau _{13}}} \right)}^2}} /{{T^2}}}} \right], \end{split} \tag{20a} $

      $ {\varOmega _2}\left( t \right) = 0.3{\varOmega _0}\exp \left[ {{{ - {{\left( {t - {\tau _{21}}} \right)}^2}}/{{T^2}}}} \right] , \tag{20b}$

      其中, $ {\varOmega _{{\text{ }}0}} $是脉冲振幅, $ {t_{\text{f}}} $是总的演化时间, $ {\tau _{11}} = $$0.01{t_{\text{f}}}$, $ {\tau _{12}} = 0.8{t_{\text{f}}} $, $ {\tau _{13}} = 0.6{t_{\text{f}}} $$ {\tau _{21}} = 0.45{t_{\text{f}}} $是脉冲时延, $ T = 0.2{t_{\text{f}}} $是脉冲宽度. 因此, 驱动脉冲$ {\varOmega _1}\left( t \right) $$ {\varOmega _2}\left( t \right) $可以用来确定$ {\theta _0}\left( t \right) $$ {\theta _1}\left( t \right) $. 图2$ {\theta _0}\left( t \right) $$ {\theta _1}\left( t \right) $随演化时间变化的情况, 描述了当脉冲振幅$ {\varOmega _{{\text{ }}0}} $作为变量取不同值时对$ {\theta _0}\left( t \right) $$ {\theta _1}\left( t \right) $的影响. 通过图2(a)可以发现, 脉冲幅度$ {\varOmega _{{\text{ }}0}} $的取值大小对$ {\theta _0}\left( t \right) $几乎没有影响, 无论脉冲幅度$ {\varOmega _{{\text{ }}0}} $取什么值(分别取$ {\varOmega _0} = 0.1 t_{\text{f}}^{ - 1} $${\varOmega _0} = 50 t_{\text{f}}^{{{ - 1}}}$, 两条曲线重合), 在整个演化过程中均满足边界条件${\theta _0}\left( 0 \right) = $$- {{\text{π }}}/{2}$, ${\theta _0}\left( {{t_{\text{f}}}} \right) = {{\text{π }}}/{2}$. 而图2(b)是与时间相关的参数$ {\theta _1}\left( t \right) $, 可以得到, 当$ {\varOmega _{{\text{ }}0}} $的取值越大时两边边界更趋向于0. 但在整个超绝热捷径演化过程中, 应满足条件${\varOmega _k}\left( t \right) \ll g\left( {k = 1, 2} \right)$, 所以$ {\varOmega _{{\text{ }}0}} $的取值又不能太大. 综上, 在后面的讨论中选取${\varOmega _0} = 0.3 t_{\text{f}}^{{{ - 1}}}$$g = 100 t_{\text{f}}^{{{ - 1}}}$.

      图  2  (a) 脉冲幅度取不同值时, 参数$ {\theta _0}\left( t \right) $随时间的变化情况; $ {\varOmega _0} = 0.1 t_{\text{f}}^{ - 1} $(黑色虚线), ${\varOmega _0} = 50 t_{\text{f}}^{{{ - 1}}}$(浅绿色实线); (b) 脉冲幅度取不同值时, 参数$ {\theta _1}(t) $随时间的变化情况; $ {\varOmega _0} = 0.1 t_f^{ - 1} $(蓝色), $ {\varOmega _0} = 1 t_{\text{f}}^{ - 1} $(红色), $ {\varOmega _0} = 10 t_{\text{f}}^{ - 1} $(粉色), $ {\varOmega _0} = 20 t_{\text{f}}^{ - 1} $(浅绿色), $ {\varOmega _0} = 40 t_{\text{f}}^{ - 1} $(绿色).

      Figure 2.  (a) Variation of $ {\theta _0}\left( t \right) $ with time when the amplitude of pulse has different values. $ {\varOmega _0} = 0.1 t_{\text{f}}^{ - 1} $(black dotted line), ${\varOmega _0} = 50 t_{\text{f}}^{{{ - 1}}}$ (light green line); (b) variation of $ {\theta _1}(t) $ with time when the amplitude of pulse has different values. $ {\varOmega _0} = 0.1 t_f^{ - 1} $(blue), $ {\varOmega _0} = 1 t_{\text{f}}^{ - 1} $(red), $ {\varOmega _0} = 10 t_{\text{f}}^{ - 1} $(pink), $ {\varOmega _0} = 20 t_{\text{f}}^{ - 1} $(light green), $ {\varOmega _0} = 40 t_{\text{f}}^{ - 1} $(green).

      为验证本文方案的有效性, 图3给出了随时间变化的目标态保真度图. 定义态$\left| {\varPsi \left( t \right)} \right\rangle$为系统随时间演化的实际状态, 求解系统保真度$ F\left( t \right) = $${\left| {\left\langle {\varPsi \left( t \right)} \right|\left. {{\phi _1}} \right\rangle } \right|^2}$, 其中$ \left| {{\phi _1}} \right\rangle $为上文中提到的初始态$ {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}}{\left| {{L_2}} \right\rangle _{\text{B}}} $. 在整个演化过程中, 当时间超过$ 0.7{t \mathord{\left/ {\vphantom {t {{t_{\text{f}}}}}} \right. } {{t_{\text{f}}}}} $后, 保真度逐渐保持稳定, 最终保真度数值为0.9914. 由此可以得出, 本文方案是可行有效的, 能够制备出高保真度的控制相位门. 然而, 由于在绘景变换时, 忽略了高频振荡项, 得到系统的有效哈密顿量; 另外, 在控制参数的过程中, 量子系统演化时的期望量子态会演化到其他非必要的量子状态上. 因而最终保真度达不到1.

      图  3  保真度随时间变化.

      Figure 3.  Variation of fidelity with time.

      图4描绘了在整个演化过程中所有态的布居数图. 它表明在系统演化后可得到高保真度的控制相位门. 从图4可以得出, 当时间超过$ 0.7{t \mathord{\left/ {\vphantom {t {{t_{\text{f}}}}}} \right. } {{t_{\text{f}}}}} $以后所有态趋于稳定. 中间态$ \left| {{\phi _2}} \right\rangle $, $ \left| {{\phi _3}} \right\rangle $$ \left| {{\phi _5}} \right\rangle $只在$ 0.2{t \mathord{\left/ {\vphantom {t {{t_{\text{f}}}}}} \right. } {{t_{\text{f}}}}} $$ 0.7{t \mathord{\left/ {\vphantom {t {{t_{\text{f}}}}}} \right. } {{t_{\text{f}}}}} $时间间隔内出现且数值不超过0.5, $ \left| {{\phi _4}} \right\rangle $态的布居数在整个演化过程中几乎为0. 因此, 在所提出的方法中, 可以忽略谐振器衰减引起的退相干影响.

      图  4  参数为${\varOmega _0} = 0.3 t_{\text{f}}^{{{ - 1}}}$$g = 100 t_{\text{f}}^{{{ - 1}}}$时, 量子态$\left| {{\phi _1}} \right\rangle - $$ \left| {{\phi _5}} \right\rangle$布居随时间变化

      Figure 4.  Variations of quantum state $\left| {{\phi _1}} \right\rangle -\left| {{\phi _5}} \right\rangle$ with time when ${\varOmega _0} = 0.3 t_{\text{f}}^{{{ - 1}}}$ and $g = 100 t_{\text{f}}^{{{ - 1}}}$.

      已知态$\left| {\varPsi \left( t \right)} \right\rangle$为系统随时间演化的实际演化状态. 实际演化态$\left| {\varPsi \left( t \right)} \right\rangle$和初始态$ \left| {{\phi _1}} \right\rangle = {\left| 0 \right\rangle _{\text{C}}}{\left| R \right\rangle _{\text{A}}} $${\left| {{L_2}} \right\rangle _{\text{B}}}$内积, 可给出相位的变化情况. 图5显示了$\left\langle {\varPsi \left( t \right)} \right|\left. {{\phi _1}} \right\rangle$随时间变化情况. 当演化时间为$ 0{t \mathord{\left/ {\vphantom {t {{t_{\text{f}}}}}} \right. } {{t_{\text{f}}}}} $时, 内积值为+1, 说明实际态$\left| {\varPsi \left( t \right)} \right\rangle$和初始态$ \left| {{\phi _1}} \right\rangle $重合, 没有相位差; 随着时间的推移, 其内积$\left\langle {\varPsi \left( t \right)} \right|\left. {{\phi _1}} \right\rangle$值逐渐变小; 当演化时间为$ {t_{\text{f}}} $时, 内积值为–1, 说明此时实际态$\left| {\varPsi \left( t \right)} \right\rangle$和初始态$ \left| {{\phi _1}} \right\rangle $的相位已经相差180°, 即实际态变为$\left| {\varPsi \left( t \right)} \right\rangle = - \left| {{\phi _1}} \right\rangle$. 这验证了相位的变化, 实现了控制相位门的制备.

      图  5  $ \left\langle {\psi (t)} \right|\left. {{\phi _1}} \right\rangle $随时间的变化图

      Figure 5.  Variation of $ \left\langle {\psi (t)} \right|\left. {{\phi _1}} \right\rangle $varies with time.

      考虑谐振器衰减和量子比特退相位率, 并分析两者对控制相位门产生的影响. 退相干过程可以用主方程来描述, 可以写成:

      $ \begin{split}\dot{\rho }=-\text{i}\left[{H}_{\text{total}},\rho \right]+{\displaystyle {\sum }_{k}\left[{b}_{k}\rho {b}_{k}^{†}-\frac{1}{2}\left({b}_{k}^{†}\rho {b}_{k}+\rho {b}_{k}^{†}{b}_{k}\right)\right]}\text{ }\text{, }\\\end{split} $

      其中$ {b_k} $为耗散项, 表示为

      $ {b_1} = \sqrt {{\gamma _1}} {\left| R \right\rangle _{\text{A}}}\left\langle e \right| , \tag{22a} $

      $ {b_2} = \sqrt {{\gamma _2}} {\left| {{L_1}} \right\rangle _{\text{B}}}\left\langle e \right| \text{, } \tag{22b} $

      $ {b_3} = \sqrt {{\gamma _3}} {\left| {{L_2}} \right\rangle _{\text{B}}}\left\langle e \right| \text{, } \tag{22c} $

      $ {b_4} = \sqrt {{\gamma _4}} {\left| R \right\rangle _{\text{B}}}\left\langle e \right| \text{, } \tag{22d} $

      $ {b_5} = \sqrt \kappa a \text{, } \tag{22e} $

      $ {b}_{6}=\sqrt{{\gamma }_{{\phi }_{R\text{A}}}}\left({|e\rangle }_{\text{A}}\langle e|-{|R\rangle }_{\text{A}}\langle R|\right)\text{, } \tag{22f} $

      $ {b_7} = \sqrt {{\gamma _{{\phi _{{L_1}{\text{B}}}}}}} \left( {{{\left| e \right\rangle }_{\text{B}}}\left\langle e \right| - {{\left| {{L_1}} \right\rangle }_{\text{B}}}\left\langle {{L_1}} \right|} \right) \text{, } \tag{22g} $

      $ {b_8} = \sqrt {{\gamma _{{\phi _{{L_2}B}}}}} \left( {{{\left| e \right\rangle }_{\text{B}}}\left\langle e \right| - {{\left| {{L_2}} \right\rangle }_{\text{B}}}\left\langle {{L_2}} \right|} \right) \text{, } \tag{22h} $

      $ {b_9} = \sqrt {{\gamma _{{\phi _{R{\text{B}}}}}}} \left( {{{\left| e \right\rangle }_{\text{B}}}\left\langle e \right| - {{\left| R \right\rangle }_{\text{B}}}\left\langle R \right|} \right) \text{, } \tag{22i} $

      其中, $ {\gamma _n}\left( {n = 1, 2, \cdots , 4} \right) $表示第$ l $个量子比特从能级态$ {\left| R \right\rangle _l} $($ {\left| e \right\rangle _l} $)到能级态$ {\left| e \right\rangle _l} $($ {\left| {{L_1}} \right\rangle _l}, {\left| {{L_2}} \right\rangle _l} $)的自发辐射率; $ \kappa $表示谐振腔的衰减率; $ {\gamma _{{\phi _{il}}}} $($ i = R, {L_1}, {L_2} $; $ l = A, B $)表示第$ l $个量子比特的退相位率. 为简化计算过程, 令$ {\gamma _n} = \gamma $$ {\gamma _{{\phi _{il}}}} = {\gamma _\phi } $.

      为了量化谐振器衰减和量子比特退相干引起的最终保真度变化, 图6给出了不同的退相干因素对最终保真度的影响, 其中图6(a)绘制了最终保真度$ F\left( {{t_{\text{f}}}} \right) $与谐振器衰减$ \kappa $和量子比特自发辐射率$ {\gamma _n} $的变化关系图, 图6(b)图6(c)分别绘制了最终保真度$ F\left( {{t_{\text{f}}}} \right) $$ {\gamma _\phi } $$ \gamma $$ F\left( {{t_{\text{f}}}} \right) $$ {\gamma _\phi } $$ \kappa $的关系图. $ \kappa $的值从0到0.01$t_{\text{f}}^{{{ - 1}}}$, $ \gamma $的范围是从0到0.01$t_{\text{f}}^{{{ - 1}}}$. 从图6(a)图6(c)可以得到, 不同$ \kappa $值时, 最终保真度可以保持在0.985以上, 这表明了所提出的方案对谐振器衰减是具有鲁棒性的. 如图6(b)所示, 量子比特退相干对最终保真度的影响也比较小. 即使$\gamma = 0.01 t_{\text{f}}^{{{ - 1}}}$${\gamma _\phi } = 0.001 t_{\text{f}}^{{{ - 1}}}$, 最终保真度也超过0.984. 因此, 该控制相位门对谐振器衰减和量子比特退相干具有鲁棒性.

      图  6  不同的退相干因素$ \kappa $, $ \gamma $$ {\gamma _\phi } $对最终保真度的影响 (a) $ \gamma $$ \kappa $的影响; (b) $ \gamma $$ {\gamma _\phi } $的影响; (c) 退相干中不含$ \gamma $时的影响

      Figure 6.  Influence of different decoherence factors $ \kappa $, $ \gamma $ and $ {\gamma _\phi } $ to the final fidelity: (a) The effect of $ \gamma $ and $ \kappa $ to fidelity; (b) the effect of $ \gamma $ and $ {\gamma _\phi } $ to fidelity; (c) the effect of $ \kappa $ and $ {\gamma _\phi } $ to fidelity.

    • 本文提出了电路QED系统中利用超绝热捷径方法来快速制备两比特控制相位量子门的方案. 在两比特信息$ \left| 0 \right\rangle $, $ \left| 1 \right\rangle $编码的基础上, 获得系统的哈密顿量和有效哈密顿量. 运用超绝热捷径方法进行两次迭代, 获得与系统有效哈密顿量相同形式的修正项, 以抑制其他本征态之间的跃迁; 根据演化路径, 求得合适的边界条件, 完成控制相位门的制备. 数值模拟结果表明, 两比特控制相位量子门可高效制备. 当初态为$ \left| {11} \right\rangle $时, 通过系统演化得$ - \left| {11} \right\rangle $; 其他态为初态时, 经过演化系统是不会发生任何改变. 数值仿真结果显示控制相位门最终保真度为0.991. 最后对谐振腔的泄漏、量子比特的自发辐射和退相位的影响进行分析, 仿真结果表明, 最终保真度可保持在0.984以上, 所以由此方案制备的控制相位门是鲁棒且稳定的. 但是本方案还存在一定局限性, 如图4中量子态$ \left| {{\phi _2}} \right\rangle , \left| {{\phi _5}} \right\rangle $的布居数高于0.4, 比其他系统中相应的布居数大, 表明本方案对激发态比较敏感. 同时, 为了获得和系统有效哈密顿量具有相同形式的修正项, 本方案进行了两次迭代, 因而计算量比较大.

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