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Yu Lin, Xu Jing, Wu Jin-Lei, Ji Xin. Fast generating W state of three superconducting qubits via Lewis–Riesenfeld invariants. Chinese Physics B, 2017, 26(6): 060306  
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Fast generating W state of three superconducting qubits via Lewis–Riesenfeld invariants
Yu Lin, Xu Jing, Wu Jin-Lei, Ji Xin
Department of Physics, College of Science, Yanbian University, Yanji133002, China

 

† Corresponding author. E-mail: jixin@ybu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11464046).

Abstract

We propose a scheme for a fast generating three-qubit W state in a superconducting system by using a technique of shortcuts to adiabaticity, Lewis–Riesenfeld invariants. Three identical superconducting qubits (SQs) are connected by two coplanar waveguide resonators (CPWRs) capacitively. Under a certain limit condition, we convert the complicated SQ system into a simple three-state system. By designing experimentally accessible harmonic pulses, a three-SQ W state is implemented with quite short operation time and high fidelity. Numerical simulations prove that the scheme is robust against the parameter deviation. In addition, we also give detailed discussion about the scheme robustness against decoherence.

PACS: 03.67.Bg;42.50.Dv;42.50.Ex
Keyword:W state;superconducting qubits;shortcuts to adiabaticity
1. Introduction

Entanglement plays a crucial role in quantum information processing.[1–4] As two kinds of three (multi)-qubit entangled states, W states[5] and Greenberger–Horne–Zeilinger (GHZ) states[6] have received a great deal of attention. Comparing them, W states receive more attention than GHZ states because the former may keep bipartite entanglement when one of three qubits is collapsed, which will have advantages in quantum teleportation.[7] Therefore, a number of schemes have been proposed to investigate W states in different systems via different techniques for different purposes.[8–21]

As one of the most promising candidates for achieving quantum computers, the superconducting system has many advantages over other systems.[22–25] It has a natural superiority of scalability for achieving quantum computers. The superconducting device used in a quantum computer can be created more easily for large integration and scale with the mature microelectronic technique. In addition, coupling strengths among SQs can be controlled better with desired values by modulating electromagnetic signals and designing level configurations. By using superconducting systems, many schemes have been brought forward to generate W states.[26–29]

Among these references mentioned above for generating W states, a technique, stimulated Raman adiabatic passage (STIRAP), is widely used because of its robustness in quantum state transfer.[30–34] However, STIRAP usually requires a very long operation time, which will increase the probabilities of decoherence and errors and lead to useless dynamics finally. Therefore, a set of techniques called “shortcuts to adiabaticity (STA)” arise for speeding up adiabatic evolution processes.[35–44] Besides, some experimental realizations have been completed.[45,46] By using STA, some schemes were presented for the fast generation of W states in cavity quantum electrodynamics.[47–49] Very recently, in superconducting systems, Kang et al. proposed two schemes for fast generating W states.[50,51]

In this work, we implement the fast generation of three-SQ W states in a superconducting system by using a technique of STA, Lewis–Riesenfeld invariants. Unlike the scheme in the Refs. [50] and [51], on the one hand, the scheme we propose does not need four qubits which include an auxiliary qubit but only three qubits for the generation of three-SQ W state. On the other hand, the scheme needs to neither interpolate the system Hamiltonian nor add counterdiabatic driving pulses, which will reduce experimental difficulties, but only design accessible harmonic pulses to construct a Hermitian invariant of the system.

2. Model and effective dynamics

Figure 1 gives the fundamental model for generating a three-SQ W state. Three SQs are connected by two single-mode CPWRs capacitively. All SQs have identical Λ-type level configuration possessing one upper level and two lower levels and . For the j-th SQ, transition is resonantly driven by a time-dependent laser field with Rabi frequency . In addition, for SQ1, is resonantly coupled to CPWR-A and CPWR-B, synchronously, with coupling strength g1. For SQ2(3), is coupled to CPWR-A(B) with coupling strength . The interaction Hamiltonian of the superconducting system is ( ):

where is the annihilation operator of CPWR-A(B).

Fig. 1. (color online) (a) Diagrammatic sketch of the superconducting system. (b) Level configurations and relevant transitions of three SQs.

We suppose that the superconducting system is initially in state which represents three SQs in states , , and , respectively, and two CPWRs both are in the vacuum state. Then Hamiltonian (1) can be rewritten as

with the orthogonal complete basis states
where denotes a single photon in CPWR-A(B). When we introduce a set of orthogonal states
Hamiltonian (2) is written as
in which we have set and for convenience. Then using Eq. (5), we can easily find that will not be involved in the whole evolution when the initial state is . Thus Hamiltonian (5) can be written by

Next, we introduce a second set of orthogonal states as follows:

then Hamiltonian (6) becomes
By performing the unitary transformation and neglecting high oscillating terms under the limit condition , Hamiltonian (8) can be simplified into an effective Hamiltonian
which can take the place of the whole Hamiltonian (1) to approximatively govern the system evolution under the limit condition .

3. Fast generating three-SQ state via Lewis–Riesenfeld invariants

In this section, according to effective Hamiltonian (9), we implement the fast generation of three-SQ W state via Lewis–Riesenfeld invariants. First of all, it is necessary to give a brief introduction of Lewis–Riesenfeld invariants theory.[52] A quantum system is governed by a time-dependent Hamiltonian H(t), and its time-dependent Hermitian invariant I(t) satisfies

The solution of the time-dependent Schrödinger equation can be written as a superposition of I(t) eigenvectors
for which is time-independent amplitude and αn is the Lewis–Riesenfeld phase. The Lewis–Riesenfeld phase is defined by

We choose one Hermitian invariant of the effective Hamiltonian (9) as

which has the dimension of energy. ν and β are two parameters which will be determined later. Eigenvectors of I(t) with eigenvalues and , respectively, are
Inserting Eqs. (9) and (13) into Eq. (10), we obtain two Rabi frequencies as
in which the dot represents a time derivative.

For generating the W state, the system should evolve along the invariant dark mode from the initial state to the target state

which demands
where is the final operation time, and the scheme starts at t = 0 and ends at . Besides, must have the same eigenvectors as I(t) at two boundary times.[53] It requires and , which give two boundary conditions with respect to two Rabi frequencies
Substituting Eqs. (16) and (17) into Eq. (15), we find two additional boundary conditions
Now all boundary conditions have been given. However, the conditions imply infinite and , and hence we must weaken the conditions and can set
with a time-independent small value ε.

According to the boundary conditions we discuss the above points, we choose ν and β as

Correspondingly, two Rabi frequencies are
which are experimentally accessible harmonic pulses with the amplitude
Now we can use Eq. (11) to calculate the final fidelity for generating the W state
with corresponding Lewis–Riesenfeld phases
Then, for appropriate Rabi frequencies and the final fidelity , we choose , i.e.,
Up to now, we have implemented the construction of STA for a fast generating three-SQ W state via Lewis–Riesenfeld invariants.

4. Numerical simulations and discussion

In this section, we discuss the availability and robustness of the scheme by numerical simulations. First of all, we need to ensure that ε = 0.1526 is right and pick an appropriate operation time. In Fig. 2, we plot a contour image of the final fidelity versus ε and . We can clearly see that guarantees a high-fidelity generation of the three-SQ W state with a suitable final time. is inversely proportional to , which requires a large enough to meet the limit condition , so the final fidelity increases with the increase of .

Fig. 2. (color online) Contour image for the final fidelity versus ε and .

Next, in order to show that our proposed STA scheme is fast, we contrast the STA scheme and the STIRAP scheme for generating three-SQ W state in Fig. 3. In the STIRAP scheme, we adopt the following Gaussian Rabi frequencies, respectively corresponding to and :

with two parameters and . Since the STIRAP scheme needs either a very long operation time or very large differences between system eigen energies to weaken non-adiabatic couplings[54] as shown in Fig. 3, for the STIRAP scheme a larger , which gives higher system eigen energies, will shorten the operation time; however, that will consume more resources. Besides, the limit condition requires a smaller for high-fidelity generation of three-SQ W state. Therefore, for the STIRAP scheme in Fig. 3, when the final fidelity impossibly increases up to near unity even with an infinitely large operation time, which indicates that shortening the operation time by increasing the system energy is limited by the condition . For the STA scheme, when the operation time is the final fidelity is over 0.94 which gives the harmonic pulse amplitude . Correspondingly, however, in the STIRAP scheme when it requires , which is 4 times that for the STA scheme to reach the same final fidelity. Therefore, the STA scheme is fast for generating the three-SQ W state. In the following discussion, we pick as the final operation time.

Fig. 3. (color online) Contrast between the STA scheme and the STIRAP scheme.

To see the STA scheme availability, it is necessary to show the time dependence of the fidelity and populations for states investigated. So in Fig. 4, we plot the time dependence of the fidelity for generating a three-SQ W state and populations P1 for , for , and for other states. Apparently, Figure 4(a) shows that the fidelity is nearly unity at the final time . Figure 4(b) further proves that the STA scheme implements three-SQ W nearly perfectly, because all populations of are about 1/3 at the final time but the sum of populations for other states not involved in the W state is near zero.

Fig. 4. (color online) (a) Time dependence of the fidelity; (b) time dependence of the populations for , and other states.

In a real experiment, perfect controls of parameters are scarcely possible. Hence it is necessary to investigate the sensitivities of the STA scheme to variations in control parameters. Here we define as the deviation of a parameter with x denoting the ideal value and denoting the actual value. In Fig. 5, we mainly consider the sensitivities of the STA scheme to variations in , , and g. Through comparing Figs. 5(a)–5(c), we can find that the STA scheme is insensitive to variations in and g but somewhat sensitive to those in . Numerically speaking, however, the STA scheme is insensitive to variations in all three parameters because the final fidelity is always beyond 0.98 in each of the three figures.

Fig. 5. (color online) Effects of the variations in the control parameters on final fidelity. (a) Final fidelity versus and . (b) Final fidelity versus and . (c) Final fidelity versus and .

For more convenient simplification, we set and . However, perfectly obtaining the two groups of equal parameters is scarcely possible in experiment either. Therefore, it is necessary to investigate the effects of the variations in the equal parameters on the final fidelity. In Fig. 6, we plot the effects of the variations in each pair of equal parameters on the final fidelity, in which and are for and , respectively. Because of the symmetry of SQ2 and SQ3, Figures 6(a) and 6(b) are identical. Generally speaking, the variations in each pair of equal parameters will destroy the final fidelity to a certain extent. However, as shown in Figs. 6(a)–6(d), the lowest final fidelity is still over 0.975, which indicates that the scheme is robust against the variations in the equal parameters.

Fig. 6. (color online) Effects of the variations in the equal parameters on the final fidelity. (a) Final fidelity versus and . (b) Final fidelity versus and . (c) Final fidelity versus and . (d) Final fidelity versus and .

Finally, we consider the STA scheme robustness against decoherence caused by SQ energy relaxations, CPWR photon decay and SQ dephasing. Taking decoherence into account, the whole evolution of the superconducting system will be governed by the master equation

in which is Hamiltonian (expression (1)).
κl is the photon decay rate of CPWR-l, and ( ) is the dephasing rate (energy relaxation rate) of the j-th SQ for the decay path . For simplicity, we set , , and .

By solving the master equation numerically, we can plot three-dimensional images of the final fidelities versus the two corresponding decoherence factors from among SQ energy relaxation, CPWR photon decay and SQ dephasing. Through comparing Figs. 7(a)–7(c), one can clearly find that the STA scheme for generating the three-SQ W state is insensitive to SQ energy relaxation and CPWR photon decay, especially the latter. Figure 7(a) shows that the final fidelity can keep over 0.98 even when in the absence of SQ dephasing. However, in the presence of SQ dephasing even when , SQ dephasing will dominate the effects of decoherence on the final fidelities. But in Fig. 7(b) the final fidelity reaches to very high when and , so the STA scheme is still robust against decoherence.

Fig. 7. (color online) (a) Final fidelity versus and with . (b) Final fidelity versus and with . (c) Final fidelity versus and with .

By using a set of recent experimental parameters { , , , } with ,[55] we can obtain the following parameter relations { , , }. Adopting the parameter relations above, we can achieve the generation of the three-SQ W state with the relatively high fidelity over .

5. Conclusions

In this paper, we achieve the fast generation of the W state of three SQs via Lewis–Riesenfeld invariants. Based on the skilled simplification of the superconducting system, the STA scheme is used for shortening the operation time greatly. There are no additions of auxiliary driving pulses but experimentally accessible harmonic pulses adopted in the scheme. Adequate numerical simulations show that the STA scheme for the fast generation of the three-SQ W state is robust against variations in parameter and decoherence induced by SQ energy relaxation and CPWR photon decay. The scheme is somewhat sensitive to SQ dephasing, but the high-fidelity three-SQ W state may still be implemented under the current experimental parameters.

Reference
[1] Ekert A K 1991 Phys. Rev. Lett. 67 661
[2] Bennett C H Wiesner S J 1992 Phys. Rev. Lett. 69 2881
[3] Zheng S B Guo G C 2000 Phys. Rev. Lett. 85 2392
[4] Vidal G 2003 Phys. Rev. Lett. 91 147902
[5] Dür W Vidal G Cirac J I 2000 Phys. Rev. 62 062314
[6] Greenberger D M Horne M A Zeilinger A 1989 Going beyond Bell’s theorem. In Bell’s theorem, quantum theory, and conceptions of the universe Dordrecht Kluwer 69
[7] Bennett C H Brassard G Crepeau C Jozsa R Peres A Wootters W K 1993 Phys. Rev. Lett. 70 1895
[8] Bastin T Thiel C von Zanthier J Lamata L Solano E Agarwal G S 2009 Phys. Rev. Lett. 102 053601
[9] Wang X W Yang G J Su Y H Xie M 2009 Quantum Inf. Process. 8 431
[10] Chen R X Shen L T 2011 Phys. Lett. 375 3840
[11] Song P J Lü X Y Si L G Yang X X 2011 Chin. Phys. 20 050308
[12] Lu M Xia Y Song J An N B 2013 J. Opt. Soc. Am. 30 2142
[13] Chen Y H Xia Y Song J 2013 Quantum Inf. Process. 12 3771
[14] Zhang C L Li W Z Chen M F 2013 Opt. Commun. 311 301
[15] Zhang C L Li W Z Chen M F 2014 Opt. Commun. 312 269
[16] Sun W M Su S L Jin Z Liang Y Zhu A D Wang H F Zhang S 2015 J. Opt. Soc. Am. 32 1873
[17] Su S L Chen L Guo Q Wang H F Zhu A D Zhang S 2015 Chin. Phys. 24 020305
[18] Fan R H Guo B H Guo J J 2015 Acta Phys. Sin. 64 0140301 in Chinese
[19] Geetha P J Yashodamma K O Sudha 2015 Chin. Phys. 24 110302
[20] Gu Z Y Qian S W 2015 Chin. Phys. 24 040301
[21] Zhan H T Yu X T Xiong P Y Zhang Z C 2016 Chin. Phys. 25 050305
[22] Makhlin Y Schön G Shnirman A 2001 Rev. Mod. Phys. 73 357
[23] Yu Y Han S Chu X Chu S I Wang Z 2002 Science 296 889
[24] Vion D Aassime A Cottet A Joyez P Pothier H Urbina C Esteve D Devoret M H 2002 Science 296 886
[25] Pashkin Y A Yamamoto T Astafiev O Nakamura Y Averin D V Tsai J S 2003 Nature 421 823
[26] Deng Z J Gao K L Feng M 2006 Phys. Rev. 74 064303
[27] Song K H Xiang S H Liu Q Lu D H 2007 Phys. Rev. 75 032347
[28] Chen M F Zhang C L Ma S S 2013 Opt. Commun. 306 21
[29] Wei X Chen M F 2015 Quantum Inf. Process. 14 2419
[30] Bergmann K Theuer H Shore B W 1998 Rev. Mod. Phys. 70 1003
[31] Demirplak M Rice S A 2003 J. Phys. Chem. 107 9937
[32] Král P Thanopulos I Shapiro M 2007 Rev. Mod. Phys. 79 53
[33] Zhang C L Chen M F 2013 Chin. Phys. 22 050307
[34] Zhang C L Chen M F 2015 Chin. Phys. 24 070310
[35] Chen X Lizuain I Ruschhaupt A Guéry-Odelin D Muga J G 2010 Phys. Rev. Lett. 105 123003
[36] del Campo A 2013 Phys. Rev. Lett. 111 100502
[37] Torrontegui E Ibáñez S Martínez-Garaot S Modugno M del Campo A Guéry-Odelin D Ruschhaupt A Chen X Muga J G 2013 Adv. At. Mol. Opt. Phys. 62 117
[38] Martínez-Garaot S Torrontegui E Chen X Muga J G 2014 Phys. Rev. 89 053408
[39] Ibáñez S Chen X Torrontegui E Muga J G Ruschhaupt A 2012 Phys. Rev. Lett. 109 100403
[40] Chen Y H Xia Y Chen Q Q Song J 2014 Phys. Rev. 89 033856
[41] Chen Y H Xia Y Chen Q Q Song J 2015 Phys. Rev. 91 012325
[42] Song X K Ai Q Qiu J Deng F G 2016 Phys. Rev. 93 052324
[43] Chen Y H Xia Y Wu Q C Huang B H Song J 2016 Phys. Rev. 93 052109
[44] Baksic A Ribeiro H Clerk A A 2016 Phys. Rev. Lett. 116 230503
[45] Du Y X Liang Z T Li Y C Yue X X Lv Q X Huang W Chen X Yan H Zhu S L 2016 Nat. Commun. 7 12479
[46] An S Lv D delCampo A Kim K 2016 Nat. Commun. 7 12999
[47] Chen Y H Xia Y Chen Q Q Song J 2014 Laser Phys. Lett. 11 115201
[48] Huang X B Zhong Z R Chen Y H 2015 Quantum Inf. Process. 14 4475
[49] Chen Y H Huang B H Song J Xia Y 2016 Opt. Commun. 380 140
[50] Kang Y H Chen Y H Wu Q C Huang B H Song J Xia Y 2016 Sci. Rep. 6 36737
[51] Kang Y H Chen Y H Shi Z C Song J Xia Y 2016 Phys. Rev. 94 052311
[52] Lewis H R Riesenfeld W B 1969 J. Math. Phys. 10 1458
[53] Chen X Muga J G 2012 Phys. Rev. 86 033405
[54] Kuklinski J R Gaubatz U Hioe F T Bergmann K 1989 Phys. Rev. 40 6741
[55] Xiang Z L Ashhab S You J Q Nori F 2013 Rev. Mod. Phys. 85 623