陳 娟，郭琰博，姜年權(quán)

基于STQ系統(tǒng)的一個(gè)四比特簇態(tài)制備方案

*陳 娟，郭琰博，姜年權(quán)

(溫州大學(xué)物理與電子信息工程學(xué)院，浙江，溫州 325035)

Transmon 比特通過(guò)電容與一個(gè)超導(dǎo)TLR（transmission line resonator）耦合。本文采用Transmon 比特與TLR之間有更強(qiáng)的耦合常量，提出了一個(gè)在STQ（superconducting transmon qubit）系統(tǒng)中制備四比特糾纏簇態(tài)（cluster state）的簡(jiǎn)單方案。與已有的方案相比，此方案有更長(zhǎng)的消相干時(shí)間。又由于Transmon 比特和TLR有以上的屬性，此方案在實(shí)驗(yàn)上更可行。

cluster態(tài)；超導(dǎo)Transmon比特；傳輸線型諧振腔

量子糾纏因其獨(dú)特的性質(zhì)而成為量子信息處理的重要資源[1]。其中，三體或多體糾纏不僅可以用于檢驗(yàn)量子力學(xué)非局域理論[2]，而且被用于許多量子信息處理方案中。比如：量子隱形傳態(tài)[3-4]，量子糾纏交換[5]和量子稠密編碼等[6-7]。因此, 探究和制備多體糾纏態(tài)具有重要意義。在近年來(lái)的研究中，已有多種不同的多體糾纏態(tài)被提出，如：Greenberger-Horne-Zeilinger (GHZ)態(tài)[8-9]、W 態(tài)[10]和cluster 態(tài)[11]等。其中，Briegel 和 Raussendor[11]提出的cluster 態(tài)比GHZ 態(tài)對(duì)消相干更不敏感[12-13]，而且量子計(jì)算機(jī)能通過(guò)cluster 態(tài)來(lái)實(shí)現(xiàn)[12]，正因?yàn)槿绱?，cluster 態(tài)備受人們的關(guān)注。其在不同的物理系統(tǒng)（如：腔量子電動(dòng)力學(xué)系統(tǒng)（QED））中的制備方案已經(jīng)被提出[14-21]。在文獻(xiàn)[20]中，Yang 等人提出了一個(gè)在熱腔QED中制備四原子cluster 態(tài)的方案，此方案包括兩個(gè)單比特操作和四個(gè)兩比特操作；在文獻(xiàn)[14-19,21]中，提出了需要三步來(lái)制備一個(gè)四比特cluster態(tài)的方案；鄭曉娟等提出了一個(gè)在腔QED中通過(guò)兩個(gè)三比特操作制備四比特cluster態(tài)的方案[22]。

(a)

(b)

圖 1 (a)Transmon量子比特基本的線路圖；(b)N個(gè)Transmon量子比特與TLR耦合的電路圖，其中頻率為d的微波輸入到TLR

Fig. 1 (a) The basic circuit of transmon qubit; (b) The circuit of N transmon qubits coupled with the TLR, in which a microwave ?eld of frequencydis applied to the input wire of the TRL

本文將基于STQ系統(tǒng)提出了一個(gè)有效地制備四比特糾纏cluster態(tài)的方案，與上述已有方案相比，該方案僅包括兩個(gè)三比特操作，方案中超導(dǎo)TLR被選為量子數(shù)據(jù)總線（QDB）[25]，并且Transmon量子位和QDB之間的耦合常數(shù)非常大[26]。因此，方案具有更長(zhǎng)的消相干時(shí)間和更短的操作時(shí)間。本文將在第二部分給出系統(tǒng)的模型和它的哈密頓量，在第三部分介紹如何制備四比特糾纏cluster態(tài)，最后討論此方案在實(shí)驗(yàn)中的可行性。

1 模型和哈密頓量

因此，總系統(tǒng)的哈密頓量為

2 制備四比特糾纏cluster態(tài)

現(xiàn)將介紹如何在一個(gè)由四個(gè)STQ和一個(gè)TLR組成的系統(tǒng)中制備四比特糾纏cluster態(tài)。利用前面已經(jīng)寫(xiě)出的系統(tǒng)的哈密頓量，選取

這些條件在實(shí)驗(yàn)上能很容易實(shí)現(xiàn)。因此，哈密頓量 (4) 變?yōu)?/p>

應(yīng)用上面相同的理論，得到演化算符為

因此，任意初態(tài)演化為

這是一個(gè)四比特糾纏cluster態(tài)。

3 討論和結(jié)論

很明顯非常小。所以說(shuō)，在第一步中只有三個(gè)Transmon比特演化。第一個(gè)Transmon比特一直處于基態(tài)。同理可得第二步中Transmon比特被激發(fā)的可能性

所以方案是可行的。

總之，本文提出了一個(gè)在STQ系統(tǒng)中制備四比特糾纏cluster態(tài)的方案。因?yàn)樗玫牧孔颖忍赜懈L(zhǎng)的消相干時(shí)間和更大的耦合量，所以該方案在實(shí)驗(yàn)上更加可行。同時(shí)也計(jì)算了無(wú)關(guān)Transmon比特被激發(fā)的可能性，發(fā)現(xiàn)對(duì)于態(tài)的制備來(lái)說(shuō)，影響非常小。

[1] Einstein A, Podolsky B, Rosen N. Can quantum-mechanical description of physical reality be considered complete?[J]. Physical review, 1935, 47(10): 777-780.

[2] Zheng S B. One-step synthesis of multiatom Greenberger-Horne-Zeilinger states[J]. Physical review letters, 2001, 87(23): 230404-1.

[3] Vitali D, Fortunato M, Tombesi P. Complete quantum teleportation with a Kerr nonlinearity[J]. arXiv preprint quant-ph/0003082, 2000.

[4] Zheng X J, Fang M F, Cai J W, et al. Teleportation of atomic entangled states with a thermal cavity[J]. Chinese Physics, 2006, 15(12): 2840-2846.

[5] Yang M, Zhao Y, Song W, et al. Entanglement concentration for unknown atomic entangled states via entanglement swapping[J]. arXiv preprint quant-ph/0411157, 2004.

[6] Bennett C H, Wiesner S J. Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states[J]. Physical review letters, 1992, 69(20): 2881.

[7] Xiao-Juan Z, Shuai C, Mao-Fa F, et al. Scheme for implementing quantum dense coding with four-particle decoherence-free states in an ion trap[J]. Chinese Physics B, 2008, 17(2): 431.

[8] Greenberger D M, Horne M A, Zeilinger A. Going beyond Bell’s theorem[M]//Bell’s theorem, quantum theory and conceptions of the universe. Springer Netherlands, 1989: 69-72.

[9] Greenberger D M, Horne M A, Shimony A, et al. Bell’s theorem without inequalities[J]. Am. J. Phys, 1990, 58(12): 1131-1143.

[10] Dür W, Vidal G, Cirac J I. Three qubits can be entangled in two inequivalent ways[J]. Physical Review A, 2000, 62(6): 062314.

[11] Briegel H J, Raussendorf R. Persistent entanglement in arrays of interacting particles[J]. Physical Review Letters, 2001, 86(5): 910.

[12] Walther P, Aspelmeyer M, Resch K J, et al. Experimental violation of a cluster state Bell inequality[J]. Physical review letters, 2005, 95(2): 020403.

[13] Scarani V, Acín A, Schenck E, et al. Nonlocality of cluster states of qubits[J]. Physical Review A, 2005, 71(4): 042325.

[14] Zou X B, Mathis W. Generating a four-photon polarization-entangled cluster state[J]. Physical Review A, 2005, 71(3): 032308.

[15] Zheng S B. Generation of cluster states in ion-trap systems[J]. Physical Review A, 2006, 73(6): 65802.

[16] Yang W X, Zhan Z M, Li J H. Efficient scheme for multipartite entanglement and quantum information processing with trapped ions[J]. Physical Review A, 2005, 72(6): 062108.

[17] Zhang X L, Gao K L, Feng M. Preparation of cluster states and W states with superconducting quantum-interference-device qubits in cavity QED[J]. Physical Review A, 2006, 74(2): 024303.

[18] Zou X B, Mathis W. Schemes for generating the cluster states in microwave cavity QED[J]. Physical Review A, 2005, 72(1): 013809.

[19] Ye L, Yu L B, Guo G C. Generation of entangled states in cavity QED[J]. Physical review a, 2005, 72(3): 034304.

[20] Rong-Can Y, Hong-Cai L, Mei-Xiang C, et al. Generation of four-atom cluster states in thermal cavity and implementing remote controlled not gate[J]. Chinese Physics, 2006, 15(10): 2315.

[21] Shao-Hua X, Ke-Hui S. Generation of two-atom cluster state via cavity QED[J]. Chinese Physics Letters, 2006, 23(6): 1466.

[22] Xiao-Juan Z, Hui X, Mao-Fa F, et al. Preparation of the four-qubit cluster states in cavity QED and the trapped-ion system[J]. Chinese Physics B, 2010, 19(3): 034207.

[23] Koch J, Terri M Y, Gambetta J, et al. Charge-insensitive qubit design derived from the Cooper pair box[J]. Physical Review A, 2007, 76(4): 042319.

[24] Schreier J A, Houck A A, Koch J, et al. Suppressing charge noise decoherence in superconducting charge qubits[J]. Physical Review B, 2008, 77(18): 180502.

[25] Blais A, Huang R S, Wallraff A, et al. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation[J]. Physical Review A, 2004, 69(6): 062320.

[26] You-Bang Z, Qun-Yong Z, Yu-Wu W. Schemes for splitting quantum information with four-particle genuine entangled states[J]. Communications in Theoretical Physics, 2010, 53(5): 847.

[27] Blais A, Gambetta J, Wallraff A, et al. Quantum-information processing with circuit quantum electrodynamics[J]. Physical Review A, 2007, 75(3): 032329.

[28] Astafiev O, Pashkin Y A, Nakamura Y, et al. Quantum noise in the Josephson charge qubit[J]. arXiv preprint cond-mat/0411216, 2004.

[29] Kowal M, Jachimowicz P, Sobiczewski A. Fission barriers for even-even superheavy nuclei[J]. Physical Review C, 2010, 82(1): 014303.

[30] Makhlin Y, Sch?n G, Shnirman A. Quantum-state engineering with Josephson-junction devices[J]. Reviews of modern physics, 2001, 73(2): 357.

[31] Clarke J, Wilhelm F K. Superconducting quantum bits[J]. Nature, 2008, 453(7198): 1031-1042.

[32] S. Haroche, Fundamental Systems in Qunantum Optics, edited by J. Dalibard and J.Zinn-Just in (Elsevier, New York, 1992), p. 767.

[33] Zheng S B. Generation of entangled states for many multilevel atoms in a thermal cavity and ions in thermal motion[J]. Physical Review A, 2003, 68(3): 035801.

[34] Yang C P, Liu Y, Nori F. Phase gate of one qubit simultaneously controlling n qubits in a cavity[J]. Physical Review A, 2010, 81(6): 062323.

[35] DiCarlo L, Reed M D, Sun L, et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit[J]. Nature, 2010, 467(7315): 574-578.

[36] Yang C P. Quantum information transfer with superconducting flux qubits coupled to a resonator[J]. arXiv preprint arXiv:1012.2030, 2010.

Scheme To PreparE four-qubit cluster states Based On STQ sYstem

*CHEN Juan，GUO Yan-bo，JIANG Nian-quan

( College of Physics and Electronic Information Engineering of Wenzhou University, Wenzhou, Zhejiang 325035,China)

Transmon qubits capacitively coupled to a superconducting transmission line resonator (TLR). We adopt transmon qubits which have much stronger coupling constant with TLR and propose a simple scheme to prepare a four-qubit entangled cluster state in superconducting transmon qubit (STQ) system. Compared with the scheme firstly introduced by Zheng Xiao-Juan et al, our schemes have longer dephasing time. Based on the favorable properties of transmons and TLR, our method is more feasible in experiment.

cluster states; superconducting Transmon qubit; transmission line resonator

O413.1

A

10.3969/j.issn.1674-8085.2014.03.006

1674-8085(2014)03-0027-05

2013-12-09；

2014-01-11

國(guó)家自然科學(xué)基金項(xiàng)目(10947017/A05)

*陳 娟(1989-)，女，江西撫州人，碩士生，主要從事量子信息研究(E-mail: 214386945@qq.com);

郭琰博(1990-)，男，甘肅天水人，碩士生，主要從事量子信息研究(E-mail:344516047@qq.com);

姜年權(quán)(1966-)，男，安徽安慶人，教授，博士，主要從事量子信息和太陽(yáng)能電池研究(E-mail:jiangnq@wzu.edu.cn).