Grover量子搜索算法理论研究
|
摘要
量子计算是利用量子力学原理进行信息处理的新近发展起来的前沿学科,近20年来的量子计算理论研究表明,量子计算在很多方面比经典计算优越得多,特别是在量子系统的模拟、大数因子分解和无序数据库搜索等问题上尤为突出。对于在无序数据库中搜索若干特定目标时,Grover量子搜索算法可以对许多(虽不是全部)启发式搜索的经典算法起到实质性的二次加速作用。Grover量子搜索算法在搜索时忽略搜索元素的性质,而把注意力放在那些元素的指标上,因此具有很强的通用性。同时,它可以有效地破译DES密码体系,具有加速搜索密码系统密钥的潜在用途。本文主要围绕Grover量子搜索算法中的相位匹配问题、算法的物理实现以及应用等方面进行了研究,具体内容如下:
量子纠缠是量子计算和量子信息处理中的重要资源,是产生量子加速的根本原因。基于线性光学和腔QED技术,分别提出制备量子多体纠缠态的物理方案。实验装置由简单的线性光学元件、原子–腔相互作用系统、单光子态、最大和非最大的双光子纠缠态、以及常规光子探测器所组成。在探测过程中,常规光子探测器的使用极大地降低了实验上对高质量探测器探测效率的需求,简化了实验的实现。
利用数学计算方法,研究和讨论了Grover量子搜索算法中的相位匹配问题,并提出π/3.61相位搜索算法。在这个算法中,搜索成功的概率至少为94.11%。π/3.61相位搜索算法克服了Grover量子搜索算法中成功概率随目标项数目增多而急剧下降的弱点,同时也证明了Grover量子搜索算法能够鲁棒的反对噪声和一定的扰动。利用原子间偶极相互作用和原子–腔相互作用,提出在腔QED中实现两量子比特Grover量子搜索算法。在这个方案中,实现Grover量子搜索算法所需要的两量子比特条件相位门操作可以很容易地被实现,且不需要执行辅助的单量子比特操作;同时,由于使用了强经典场,这个方案对于腔场衰减和热场效应不敏感。
基于Grover量子搜索算法,提出宇称(奇偶)确定算法和二次剩余算法。在宇称确定算法中,通过对满足条件f(x) = -1的元素进行计数,函数f(x)的宇称可以被确定。同时,讨论了不同情况下这个算法计算复杂度的上界和下界。在二次剩余算法中,算法的计算消耗主要集中在计算模M的二次剩余以及所需的迭代次数上。经典计算机上,求解二次剩余方程需要进行M/2次计算,利用量子算法则可以在O(M~(1/2)/2)步内以接近100%的成功概率求出二次剩余方程的解,实现了相对经典计算的二次加速。
基于广义Grover量子搜索算法,提出应用量子计算机算法直接测量任意一个未知的两量子比特纯态系统的纠缠度。我们具体地构建了广义Grover迭代算子,通过对两体纯态系统的两个拷贝应用量子算法以及对辅助工作量子比特进行测量,可以得到系统纠缠度的一个较好的、近似的估计值。这个方案在实验上的实现对于更加复杂的、任意量子比特数目的、有限维数量子系统的纠缠测量将是一个重大的推进,同时能够展现出量子计算机强大的计算能力。
基于腔QED技术,提出实现量子离散Fourier变换的有效的量子线路和物理方案。利用单原子–腔相互作用,提出物理方案实现N-比特量子离散Fourier变换。在这个方案中,通过发送原子通过一系列的经典场和腔场以及适当改变腔场的频率,得到一个可调的两量子比特条件相位门。在消相干时间范围内,所有的单比特和两比特量子门操作都能够被完成,有利于实现多比特量子Fourier变换。利用双原子–腔相互作用,提出物理方案实现N-比特量子离散Fourier变换。在这个方案中,基于CNOT门和SWCZ门操作(取代了原量子Fourier变换线路中复杂的受控-R_k门和SWAP门操作)和单量子比特门操作,设计了一个新的量子线路实现量子离散Fourier变换,并提出具体的原子–腔相互作用和原子–微波共振相互作用过程来实现这个量子线路。同时,我们提出具体的实验步骤并分析和讨论了这两个方案在实验实现上的可行性。
Quantum computation, which processes information by using the principles of quan-tum mechanics, is a frontier discipline that is freshly developed. During the past twodecades, the study on the theory of quantum computation shows that quantum computa-tion can outperform classical computation in many aspects, especially for the problems ofquantum system simulation, large prime factorization, searching in a disorder database,etc.. For searching several marked items in a disorder database, Grover quantum search al-gorithm achieves quadratic speedup for several (although not all) heuristic classical searchalgorithm. Grover quantum search algorithm neglects the character of the searching ele-ments and concentrates on the indices of those elements, thus it has strong universalness.In the meantime, it can efficiently decode DES coding system and has potential purposefor speeding up the search of cryptographic key in coding system. In this paper, we focuson the phase matching problem, physical realization, and application of Grover quantumsearch algorithm. The main contents are presented as the followings:
Quantum entanglement, which is the fundamental cause that produces quantumspeedup, is an important quantum resource in quantum information processing and lies atthe heart of quantum computation. Based on the linear optics and cavity QED technolo-gies, we present physical schemes for preparing quantum multipartite entangled states.The experimental setup comprises simple linear optical elements, maximally and non-maximally entangled two-photon polarization states, and conventional photon detectors.In the detection process, the use of conventional photon detectors greatly decreases thehigh-quality requirements of detectors in realistic experiments.
We investigate the phase matching problem in Grover quantum search algorithm byusing the method of mathematical calculation, and presentπ/3.61 phase search algorithm.In this algorithm, the success probability for searching is at least 94.11%.π/3.61 phasesearch algorithm overcomes the weakness that the success probability of getting correctresults usually decreases ?eetly with the increase of marked items when Grover quantumsearch algorithm is applied to search an unordered database. In the meantime, this workalso indicates that Grover quantum search algorithm is considerably robust to certainkinds of perturbations and is robust against modest noise in the initialization procedure; we present a scheme for implementing two-qubit Grover quantum search based on theatom-atom dipole-dipole interaction and the atom-cavity interaction in cavity QED. In thisscheme, all the two-qubit conditional phase operations required to achieve the two-qubitGrover quantum search can be implemented directly without performing any auxiliarysingle-qubit rotation operations. Moreover, the scheme is insensitive to both the cavitydecay and the thermal field with the assistance of a strong classical field.
We present parity determination algorithm and quadratic residue algorithm basedon Grover quantum search algorithm. In the parity determination algorithm, the parity offunction f(x) can be determined by counting exactly the number of satisfying f(x) = -1.We discuss the up and low bounds of the computational complexity of this algorithm. Inthe quadratic residue algorithm, the cost of the algorithm mainly depends on the calcula-tions of quadratic residue modulo M and the number of iterations. In classical computer,it needs M/2 calculations for solving quadratic residue equation. While in quantum com-puter, it needs only O(M~(1/2)) steps with the success probability of 100%, which realizesquadratic speedup compared with classical computation.
We present a method to measure directly the concurrence of an arbitrary two-qubitpure state system based on generalized Grover quantum search algorithm. We constructthe explicit generalized Grover quantum iteration operator and the concurrence can be cal-culated by applying quantum algorithm to two available copies of the bipartite system, anda final measurement on the auxiliary working qubits gives a better estimation of the con-currence. The implementation of the scheme would be an important step toward quantuminformation processing and more complex entanglement measure of finite-dimensionalquantum system with an arbitrary number of qubits. In the meantime, the scheme woulddirectly show the power of quantum computer.
We present quantum circuits and physical protocols for implementing quantum dis-crete Fourier transformation based on cavity QED. Firstly, we present a physical protocolfor the implementation of N-bit quantum discrete Fourier transformation by using sin-gle atom-cavity interaction. In the protocol we design a tunable two-qubit conditionalphase gate by sending the atoms through a series of classical fields and cavities and bychanging the frequency of the cavity appropriately. All the one-bit and two-bit quantumgate operations can be achieved before the decoherence sets in, which are very powerfulfor the implementation of N-bit quantum Fourier transformation. Secondly, we present a physical protocol for implementing N-bit quantum discrete Fourier transformation byusing single two-atom-cavity interaction. In this protocol, we give a new quantum cir-cuit for implementing the quantum discrete Fourier transformation by using CNOT andSWCZ gate operations instead of the controlled-R_k and SWAP gate operations, togetherwith several one-qubit operatons. We also present the explicit atom-cavity interaction andatom-microwave interaction processes for the implementation of the quantum circuit. Inthe meantime, we present the detailed experimental procedure and analyze the experi-mental feasibility of the two protocols.
量子纠缠是量子计算和量子信息处理中的重要资源,是产生量子加速的根本原因。基于线性光学和腔QED技术,分别提出制备量子多体纠缠态的物理方案。实验装置由简单的线性光学元件、原子–腔相互作用系统、单光子态、最大和非最大的双光子纠缠态、以及常规光子探测器所组成。在探测过程中,常规光子探测器的使用极大地降低了实验上对高质量探测器探测效率的需求,简化了实验的实现。
利用数学计算方法,研究和讨论了Grover量子搜索算法中的相位匹配问题,并提出π/3.61相位搜索算法。在这个算法中,搜索成功的概率至少为94.11%。π/3.61相位搜索算法克服了Grover量子搜索算法中成功概率随目标项数目增多而急剧下降的弱点,同时也证明了Grover量子搜索算法能够鲁棒的反对噪声和一定的扰动。利用原子间偶极相互作用和原子–腔相互作用,提出在腔QED中实现两量子比特Grover量子搜索算法。在这个方案中,实现Grover量子搜索算法所需要的两量子比特条件相位门操作可以很容易地被实现,且不需要执行辅助的单量子比特操作;同时,由于使用了强经典场,这个方案对于腔场衰减和热场效应不敏感。
基于Grover量子搜索算法,提出宇称(奇偶)确定算法和二次剩余算法。在宇称确定算法中,通过对满足条件f(x) = -1的元素进行计数,函数f(x)的宇称可以被确定。同时,讨论了不同情况下这个算法计算复杂度的上界和下界。在二次剩余算法中,算法的计算消耗主要集中在计算模M的二次剩余以及所需的迭代次数上。经典计算机上,求解二次剩余方程需要进行M/2次计算,利用量子算法则可以在O(M~(1/2)/2)步内以接近100%的成功概率求出二次剩余方程的解,实现了相对经典计算的二次加速。
基于广义Grover量子搜索算法,提出应用量子计算机算法直接测量任意一个未知的两量子比特纯态系统的纠缠度。我们具体地构建了广义Grover迭代算子,通过对两体纯态系统的两个拷贝应用量子算法以及对辅助工作量子比特进行测量,可以得到系统纠缠度的一个较好的、近似的估计值。这个方案在实验上的实现对于更加复杂的、任意量子比特数目的、有限维数量子系统的纠缠测量将是一个重大的推进,同时能够展现出量子计算机强大的计算能力。
基于腔QED技术,提出实现量子离散Fourier变换的有效的量子线路和物理方案。利用单原子–腔相互作用,提出物理方案实现N-比特量子离散Fourier变换。在这个方案中,通过发送原子通过一系列的经典场和腔场以及适当改变腔场的频率,得到一个可调的两量子比特条件相位门。在消相干时间范围内,所有的单比特和两比特量子门操作都能够被完成,有利于实现多比特量子Fourier变换。利用双原子–腔相互作用,提出物理方案实现N-比特量子离散Fourier变换。在这个方案中,基于CNOT门和SWCZ门操作(取代了原量子Fourier变换线路中复杂的受控-R_k门和SWAP门操作)和单量子比特门操作,设计了一个新的量子线路实现量子离散Fourier变换,并提出具体的原子–腔相互作用和原子–微波共振相互作用过程来实现这个量子线路。同时,我们提出具体的实验步骤并分析和讨论了这两个方案在实验实现上的可行性。
Quantum computation, which processes information by using the principles of quan-tum mechanics, is a frontier discipline that is freshly developed. During the past twodecades, the study on the theory of quantum computation shows that quantum computa-tion can outperform classical computation in many aspects, especially for the problems ofquantum system simulation, large prime factorization, searching in a disorder database,etc.. For searching several marked items in a disorder database, Grover quantum search al-gorithm achieves quadratic speedup for several (although not all) heuristic classical searchalgorithm. Grover quantum search algorithm neglects the character of the searching ele-ments and concentrates on the indices of those elements, thus it has strong universalness.In the meantime, it can efficiently decode DES coding system and has potential purposefor speeding up the search of cryptographic key in coding system. In this paper, we focuson the phase matching problem, physical realization, and application of Grover quantumsearch algorithm. The main contents are presented as the followings:
Quantum entanglement, which is the fundamental cause that produces quantumspeedup, is an important quantum resource in quantum information processing and lies atthe heart of quantum computation. Based on the linear optics and cavity QED technolo-gies, we present physical schemes for preparing quantum multipartite entangled states.The experimental setup comprises simple linear optical elements, maximally and non-maximally entangled two-photon polarization states, and conventional photon detectors.In the detection process, the use of conventional photon detectors greatly decreases thehigh-quality requirements of detectors in realistic experiments.
We investigate the phase matching problem in Grover quantum search algorithm byusing the method of mathematical calculation, and presentπ/3.61 phase search algorithm.In this algorithm, the success probability for searching is at least 94.11%.π/3.61 phasesearch algorithm overcomes the weakness that the success probability of getting correctresults usually decreases ?eetly with the increase of marked items when Grover quantumsearch algorithm is applied to search an unordered database. In the meantime, this workalso indicates that Grover quantum search algorithm is considerably robust to certainkinds of perturbations and is robust against modest noise in the initialization procedure; we present a scheme for implementing two-qubit Grover quantum search based on theatom-atom dipole-dipole interaction and the atom-cavity interaction in cavity QED. In thisscheme, all the two-qubit conditional phase operations required to achieve the two-qubitGrover quantum search can be implemented directly without performing any auxiliarysingle-qubit rotation operations. Moreover, the scheme is insensitive to both the cavitydecay and the thermal field with the assistance of a strong classical field.
We present parity determination algorithm and quadratic residue algorithm basedon Grover quantum search algorithm. In the parity determination algorithm, the parity offunction f(x) can be determined by counting exactly the number of satisfying f(x) = -1.We discuss the up and low bounds of the computational complexity of this algorithm. Inthe quadratic residue algorithm, the cost of the algorithm mainly depends on the calcula-tions of quadratic residue modulo M and the number of iterations. In classical computer,it needs M/2 calculations for solving quadratic residue equation. While in quantum com-puter, it needs only O(M~(1/2)) steps with the success probability of 100%, which realizesquadratic speedup compared with classical computation.
We present a method to measure directly the concurrence of an arbitrary two-qubitpure state system based on generalized Grover quantum search algorithm. We constructthe explicit generalized Grover quantum iteration operator and the concurrence can be cal-culated by applying quantum algorithm to two available copies of the bipartite system, anda final measurement on the auxiliary working qubits gives a better estimation of the con-currence. The implementation of the scheme would be an important step toward quantuminformation processing and more complex entanglement measure of finite-dimensionalquantum system with an arbitrary number of qubits. In the meantime, the scheme woulddirectly show the power of quantum computer.
We present quantum circuits and physical protocols for implementing quantum dis-crete Fourier transformation based on cavity QED. Firstly, we present a physical protocolfor the implementation of N-bit quantum discrete Fourier transformation by using sin-gle atom-cavity interaction. In the protocol we design a tunable two-qubit conditionalphase gate by sending the atoms through a series of classical fields and cavities and bychanging the frequency of the cavity appropriately. All the one-bit and two-bit quantumgate operations can be achieved before the decoherence sets in, which are very powerfulfor the implementation of N-bit quantum Fourier transformation. Secondly, we present a physical protocol for implementing N-bit quantum discrete Fourier transformation byusing single two-atom-cavity interaction. In this protocol, we give a new quantum cir-cuit for implementing the quantum discrete Fourier transformation by using CNOT andSWCZ gate operations instead of the controlled-R_k and SWAP gate operations, togetherwith several one-qubit operatons. We also present the explicit atom-cavity interaction andatom-microwave interaction processes for the implementation of the quantum circuit. Inthe meantime, we present the detailed experimental procedure and analyze the experi-mental feasibility of the two protocols.
引文
1 M. A. Nielsen, I. L. Chuang原著,赵千川译.量子计算和量子信息–量子信息部分[M].第一版.,北京:清华大学出版社, 2004:3–250.
2陈汉武.量子计算和量子信息[M].第一版.,南京:东南大学出版社, 2006:6–52.
3李承祖,黄明球,陈平形,等.量子通信和量子计算[M].第一版.,长沙:国防科技大学出版社, 2000:101–182.
4 C. H. Bennett, G. Brassard, C. Cre′peau, et al. Teleporting an Unknown QuantumState via Dual Classical and Einstein-podolsky-rosen Channels[J]. Phys. Rev. Lett.,1993, 70(13):1895–1899.
5 W. L. Li, C. F. Li, G. C. Guo. Probabilistic Teleportation and Entanglement Match-ing[J]. Phys. Rev. A, 2000, 61(3):034301.
6 S. B. Zheng. Scheme for Approximate Conditional Teleportation of an Un-known Atomic State without the Bell-state Measurement[J]. Phys. Rev. A, 2004,69(6):064302.
7 C. H. Bennett, G. Brassard. Quantum Cryptography: Public Key Distribution andCoin Tossing[C]//Proceedings of IEEE International Conference on Computers,System and Signal Proceessing, Bangalore, IEEE, (New York). l984:175–179.
8 C. H. Bennett, G. Brassard, N. D. Mermin. Quantum Cryptography without Bell’sTheorem[J]. Phys. Rev. Lett., 1992, 68(5):557–559.
9 J. C. Boileau, K. Tamaki, J. Batuwantudawe, et al. Unconditional Security ofa Three State Quantum Key Distribution Protocol[J]. Phys. Rev. Lett., 2005,94(4):040503.
10 A. Cabello. Quantum Key Distribution in the Holevo Limit[J]. Phys. Rev. Lett.,2000, 85(26):5635–5638.
11 H. K. Lo, X. F. Ma, K. Chen. Decoy State Quantum Key Distribution[J]. Phys.Rev. Lett., 2005, 94(23):230504.
12 C. H. Bennett, S. J. Wiesner. Communication via One- and Two-particle Operatorson Einstein-podolsky-rosen States[J]. Phys. Rev. Lett., 1992, 69(20):2881–2884.
13 K. Mattle, H. Weinfurter, P. G. Kwiat, et al. Dense Coding in Experimental Quan-tum Communication[J]. Phys. Rev. Lett., 1996, 76(25):4656–4659.
14 X. S. Liu, G. L. Long, D. M. Tong, et al. General Scheme for Superdense Codingbetween Multiparties[J]. Phys. Rev. A, 2002, 65(2):022304.
15 A. Karlsson, M. Koashi, N. Imoto. Quantum Entanglement for Secret Sharing andSecret Splitting[J]. Phys. Rev. A, 1999, 59(1):162–168.
16 M. Hillery, V. Buz′ek, A. Berthiaume. Quantum Secret Sharing[J]. Phys. Rev. A,1999, 59(3):1829–1834.
17 R. Cleve, D. Gottesman, H. K. Lo. How to Share a Quantum Secret[J]. Phys. Rev.Lett., 1999, 83(19):648–651.
18 H. F. Wang, X. Ji, S. Zhang. Improving the Security of Multiparty Quantum SecretSplitting and Quantum State Sharing[J]. Phys. Lett. A, 2006, 358(1):11–14.
19 S. Bandyopadhyay. Teleportation and Secret Sharing with Pure Entangled States[J].Phys. Rev. A, 2000, 62(1):012308.
20 V. Karimipour, A. Bahraminasab, S. Bagherinezhad. Entanglement Swapping ofGeneralized Cat States and Secret Sharing[J]. Phys. Rev. A, 2002, 65(4):042320.
21 S. Bandyopadhyay, V. Karimipour. Quantum Secret Sharing Based on ReusableGreenberger-horne-zeilinger States as Secure Carriers[J]. Phys. Rev. A, 2003,67(4):044302.
22 H. F. Wang, S. Zhang, Y. K. Hwang. Multiparty Quantum Secret Sharing via Intro-ducing Auxiliary Particles[J]. J. Korean Phys. Soc., 2006, 49(2):472–476.
23 K. Bostro¨m, T. Felbinger. Deterministic Secure Direct Communication Using En-tanglement[J]. Phys. Rev. Lett., 2002, 89(18):187902.
24 A. Beige, B. G. Englert, C. Kurtsiefer, et al. Secure Communication with a PubliclyKnown Key[J]. Acta. Phys. Pol. A, 2002, 101(3):357–368.
25 F. G. Deng, G. L. Long, X. S. Liu. A Two-step Quantum Direct Communica-tion Pro- Tocol Using Einstein-podolsky-rosen Pair Block[J]. Phys. Rev. A, 2003,68(4):042317.
26 H. F. Wang, S. Zhang, Y. K. Hwang, et al. Quantum Secure Direct Communicationby Using a Ghz State[J]. J. Korean Phys. Soc., 2006, 49(2):459–463.
27 B. A. Nguyen. Quantum Doalogue[J]. Phys. Lett. A, 2004, 328(1):6–10.
28 M. Christophe, P. D. Townsend. Quantum Key Distribution Over Distances as Longas 30 Km[J]. Opt. Lett., 1995, 20(16):1695.
29 A. Muller, J. Brejuet, N. Gisin. Experimental Demonstration of Quantum Cryptog-raphy Using Polarized Photons in Optical Fibre Over more Than 1 Km[J]. Euro-physics Letters, 1993, 23(6):383–388.
30 J. D. Franson, H. Ilves. Quantum Cryptography Using Polarization Feedback[J]. J.Mod. Opt., 1994, 41(12):2391–2396.
31 C. Liang, D. H. F. Liang, et al. Quantum Key Distribution Over 1.1 Km in an 850Nm Experimental All-fiber System[J]. Acta Phys. Sin., 2001, 50(8):1429–1433.
32 A. Muller, T. Herzog, B. Huttner, et al. Plug and Play Systems for Quantum Cryp-tography[J]. Appl. Phys. Lett., 1997, 70(7):793–795.
33 M. A. Rowe, D. Kielpinski, V. Meyer, et al. Experimental Violation of a Bell’sInequality with Efficient Detection[J]. Nature, 2001, 409(2):791–794.
34 G. Ribordy, J. D. Gautier. Automated Plug & Play Quantum Key Distribution[J].Electronics letters, 1998, 34(22):2116–2117.
35 S. J. D. Phoenix, S. M. Barnett, P. D. Townsend, et al. Multi-user Quantum Cryp-tography on Optical Networks[J]. J. Mod. Opt., 1995, 42(6):1155–1163.
36 W. T. Buttler, R. J. Hughes, P. G. Kwiat, et al. Practical Free-space Quantum KeyDistribution Over 1 Km[J]. Phys. Rev. Lett., 1998, 81(15):3283–3286.
37 G. Ribordy, J. Brendel, J. D. Fautier, et al. Long-distance Entanglement-basedQuantum Key Distribution[J]. Phys. Rev. A, 2001, 63(1):012309.
38 W. Tittel, J. Brendel, H. Zbinden, et al. Quantum Cryptography Using EntangledPhotons in Energy-time Bell States[J]. Phys. Rev. Lett., 2000, 84(20):4737–4740.
39 W. Tittel, H. Zbinden, N. Gisin. Experimental Demonstration of Quantum SecretSharing[J]. Phys. Rev. A, 2001, 63(4):042301.
40 Y. H. Kim, S. P. Kulik, Y. Shih. Quantum Teleportation of a Polarization State witha Com- Plete Bell State Measurement[J]. Phys. Rev. Lett., 2001, 86(7):1370.
41 A. Furusawa, J. L. Sorensen, S. L. Braunstein, et al. Unconditional Quantum Tele-portation[J]. Science, 1998, 282(5389):706–709.
42 D. Bouwmeester, J. W. Pan, K. Mattle, et al. Experimental Quantum Teleporta-tion[J]. Nature, 1997, 390(6660):575–579.
43 J. W. Pan, M. Daniell, S. Gasparoni, et al. Experimental Demonstration of Four-photon Entanglement and High-fidelity Teleportation[J]. Phys. Rev. Lett., 2001,86(20):4435–4438.
44 M. A. Nielsen, E. Knill, R. La?amme. Complete Quantum Teleportation UsingNuclear Magnetic Resonance[J]. Nature, 1998, 396(6706):52–55.
45 Y. A. Chen, S. Chen, Z. S. Yuan, et al. Memory-built-in Quantum Teleportationwith Photonic and Atomic Qubits[J]. Nature Physics, 2008, 4(2):103–107.
46 P. W. Shor. Algorithms for Quantum Computer Computation: Discrete Logarithmsand Factoring[C]//Proc. 35th Annu. Sym. Found. Comput. Science. l994:124.
47 L. K. Grover. Quantum Mechanics Helps in Searching for a Needle in aHaystack[J]. Phys. Rev. Lett, 1997, 79(23):325–328.
48 R. L. Rivest, A. Shamir, L. M. Adleman. A Method of Obtaining Digital Signaturesand Public-key Cryptosystems[J]. Comm. ACM, 1978, 21(2):120–126.
49 D. E. Knuth. The Art of Computer Programming[C]//Vol. 2: Seminumerical Algo-rithms, Seconded., Addison-Wesley. l981:25–60.
50 C. H. Hardy, E. M. Wright. An Introduction to the Theory of Numbers[C]//NewYork: Oxford University, 1979:1–52.
51 A. Ekert, R. Jozsa. Quantum Computation and Shor’s Factoring Algorithm[J]. Rev.Mod. Phys., 1996, 68(3):733–753.
52 C. Zalka. Grover’s Quantum Searching Algorithm Is Optimal[J]. Phys. Rev. A,1999, 60(4):2746–2751.
53 C. H. Bennett, E. Bernstein, G. Brassard, et al. Strengths and Weaknesses of Quan-tum Computing[J]. SIAM J. Comput., 1997, 60(5):1510–1523.
54 M. Boyer, G. Brassard, P. Hoyer, et al. Tight Bounds on Quantum Searching[J].Fortsch. Phys. Prog. Phys., 1998, 46(4 & 5):493–505.
55 G. Brassard, P. Hoyer, A. Tapp. Quantum Counting[J]. quantum-ph/9805082.
56 M. Mosca. Quantum Searching, Counting and Amplitude Amplification by Eigen-vector Analysis[C]//Proceedings of International Workshop on Randomized Algo-rithms. l998:90–100.
57 A. Einstein, B. Podolsky, N. Rosen. Can Quantum Mechanical Description of Phys-ical Reality Be Considered Complete[J]. Phys. Rev., 1935, 47(10):777–780.
58 E. Schro¨dinger. Die Genenwartige Situation in Der Quantenme Chanik[J]. Natur-wissenschafter, 1935, 23(8):807–812.
59 D. Gottesman, I. L. Chuang. Demonstrating the Viability of Universal QuantumComputation Using Teleportation and Single-qubit Operations[J]. Nature, 1999,402(11):390–393.
60 Y. Yeo, W. K. Chua. Teleportation and Dense Coding with Genuine MultipartiteEntanglement[J]. Phys. Rev. Lett., 2006, 96(6):060502.
61 X. W. Wang, G. J. Yang. Generation and Discrimination of a Type of Four-partiteEntangled State[J]. Phys. Rev. A, 2008, 78(2):024301.
62 J. Kempe, D. Bacon, D. A. Lidar, et al. Theory of Decoherence-free Fault-tolerantUniversal Quantum Computation[J]. Phys. Rev. A, 2001, 63(4):042307.
63 H. W. P. G. Kwiat, K. Mattle, et al. New High-intensity Source of Polarization-entangled Photon Pairs[J]. Phys. Rev. Lett., 1995, 75(24):4337–4341.
64 X. B. Zou, K. Pahlke, W. Mathis. Generation of an Entangled Four-photon WState[J]. Phys. Rev. A, 2002, 66(4):044302.
65 H. F. Wang, S. Zhang. Linear Optical Generation of Multipartite Entanglement withConventional Photon Detectors[J]. Phys. Rev. A, 2009, 79(4):042336.
66 Y. Tokunaga, T. Yamamoto, M. Koashi, et al. Simple Experimental Scheme ofPreparing a Four-photon Entangled State for the Teleportation-based Realization ofa Linear Optical Controlled-not Gate[J]. Phys. Rev. A, 2005, 71(3):030301.
67 D. Bouwmeester, J. W. Pan, M. Daniell, et al. Observation of Three-photonGreenberger-horne-zeilinger Entanglement[J]. Phys. Rev. Lett., 1999, 82(7):1345–1349.
68 E. Knill, R. La?amme, G. Milburn. A Scheme for Efficient Quantum Computationwith Linear Optics[J]. Nature, 2001, 409(1):46–52.
69 T. C. Ralph, N. K. Langford, T. B. Bell. Linear Optical Controlled-not Gate in theCoincidence Basis[J]. Phys. Rev. A, 2002, 65(6):062324.
70 H. F. Hofmann, S. Takeuchi. Quantum Phase Gate for Photonic Qubits Using onlyBeam Splitters and Postselection[J]. Phys. Rev. A, 2002, 66(2):024308.
71 J. L. O’Brien, G. J. Pryde, A. G. White, et al. Demonstration of an All-opticalQuantum Controlled-not Gate[J]. Nature, 2003, 426(11):264–267.
72 T. B. Pittman, M. J. Fitch, B. C. Jacobs, et al. Experimental Controlled-notLogic Gate for Single Photons in the Coincidence Basis[J]. Phys. Rev. A, 2003,68(3):032316.
73 T. B. Pittman, B. C. Jacobs, J. D. Franson. Demonstration of Feed-forward Controlfor Linear Optics Quantum Computation[J]. Phys. Rev. A, 2002, 66(5):052305.
74 T. B. Pittman, B. C. Jacobs, J. D. Franson. Probabilistic Quantum Logic OperationsUsing Polarizing Beam Splitters[J]. Phys. Rev. A, 2002, 64(6):062311.
75 X. B. Zou, S. L. Zhang, K. Li, et al. Linear Optical Implementation of the Two-qubit Controlled Phase Gate with Conventional Photon Detectors[J]. Phys. Rev. A,2007, 75(3):034302.
76 X. B. Zou, K. Li, G. C. Guo. Linear Optical Scheme for Direct Implementa-tion of a Nondestructive N-qubit Controlled Phase Gate[J]. Phys. Rev. A, 2006,74(4):044305.
77 H. F. Wang, S. Zhang, K. H. Yeon. Linear Optical Scheme for Entanglement Con-centration of Two Partially Entangled Three-photon W States[J]. Eur. Phys. J. D,2010, 56(2):271–275.
78 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Linear Optical Implementation of anAncilla-free Quantum Swap Gate[J]. Phys. Scr., 2010, 81(1):015011.
79 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Scheme for Implementing Linear OpticalQuantum Iswap Gate with Conventional Photon Detectors[J]. J. Opt. Soc. Am. B,2010, 27(1):27–31.
80 Y. Xia, J. Song, H. S. Song. Linear Optical Protocol for Preparation of N-photonGreenberger-horne-zeilinger State with Conventional Photon Detectors[J]. Appl.Phys. Lett., 2008, 92(12):021127.
81 Y. Xia, J. Song, H. S. Song, et al. Controlled Generation of Four-photonPolarization-entangled Decoherence-free States with Conventional Photon Detec-tors[J]. J. Opt. Soc. Am. B, 2009, 26(1):129–132.
82 S. Glancy, J. M. LoSecco, H. M. Vasconcelos, et al. Imperfect Detectors in LinearOptical Quantum Computers[J]. Phys. Rev. A, 2002, 65(6):062317.
83 Z. Y. Ou, L. Mandel. Observation of Spatial Quantum Beating with SeparatedPhotodetectors[J]. Phys. Rev. Lett., 1998, 61(1):54–57.
84 F. DeMartini, G. DiGiuseppe, M. Marrocco. Single-mode Generation of QuantumPhoton States by Excited Single Molecules in a Microcavity Trap[J]. Phys. Rev.Lett., 1996, 76(6):900–903.
85 H. J. Kimble, M. Dagenais, L. Mandel. Photon Antibunching in Resonance Fluo-rescence[J]. Phys. Rev. Lett., 1977, 39(11):691–695.
86 S. Brattke, B. T. H. Varcoe, H. Walther. Generation of Photon Number Stateson Demand via Cavity Quantum Electrodynamics[J]. Phys. Rev. Lett., 2001,86(16):3534–3537.
87 P. Michler, A. Imamoglu, M. D. Mason, et al. Quantum Correlation Among Photonsfrom a Single Quantum Dot at Room Temperature[J]. Nature, 2000, 406(8):968–970.
88 J. Kim, O. Benson, H. Kan, et al. A Single-photon Turnstile Device[J]. Nature,1999, 397(2):500–503.
89 N. Akopian, N. H. Lindner, E. Poem, et al. Entangled Photon Pairs from Semicon-ductor Quantum Dots[J]. Phys. Rev. Lett., 2006, 96(13):130501.
90 O. Benson, C. Santori, M. Pelton, et al. Regulated and Entangled Photons from aSingle Quantum Dot[J]. Phys. Rev. Lett., 2000, 84(11):2513–2516.
91 A. Barenco, D. Deutsch, A. Ekert. Conditional Quantum Dynamics and LogicGates[J]. Phys. Rev. Lett., 1995, 74(5):4083–4086.
92 T. Sleator, H. Weinfurter. Realizable Universal Quantum Logic Gates[J]. Phys.Rev. Lett., 1995, 74(5):4087–4090.
93 X. B. Zou, K. Pahlke, W. Mathis. Quantum Entanglement of Four Distant AtomsTrapped in Different Optical Cavities[J]. Phys. Rev. A, 2004, 69(5):052314.
94 Y. X. Gong, X. B. Zou, X. L. Niu, et al. Generation of Arbitrary Four-photon Polarization-entangled Decoherence-free States[J]. Phys. Rev. A, 2008,77(4):042317.
95 G. C. Guo, Y. S. Zhang. Scheme for Preparation of the W State via Cavity QuantumElectrodynamics[J]. Phys. Rev. A, 2002, 65(5):054302.
96 X. B. Zou, K. Pahlke, W. Mathis. Conditional Generation of the Greenberger-horne-zeilinger State of Four Distant Atoms via Cavity Decay[J]. Phys. Rev. A,2003, 68(7):024302.
97 Z. J. Deng, M. Feng, K. L. Gao. Preparation of Entangled States of Four Re-mote Atomic Qubits in Decoherence-free Subspace[J]. Phys. Rev. A, 2007,75(2):024302.
98 S. Bose, P. L. Knight, M. B. Plenio, et al. Proposal for Teleportation of an AtomicState via Cavity Decay[J]. Phys. Rev. Lett., 1999, 83(24):5158–5161.
99 H. F. Wang, S. Zhang. Scheme for Linear Optical Preparation of a Type of Four-photon Entangled State with Conventional Photon Detectors[J]. Eur. Phys. J. D,2009, 53(3):359–363.
100 L. M. Duan, M. D. Lukin, J. Cirac, et al. Long-distance Quantum Communicationwith Atomic Ensembles and Linear Optics[J]. Nature, 2001, 414(9):413–418.
101 C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez, et al. Creation of Entangled States ofDistant Atoms by Interference[J]. Phys. Rev. A, 1999, 59(2):1025–1033.
102 X. L. Feng, Z. M. Zhang, X. D. Li, et al. Entangling Distant Atoms by Interferenceof Polarized Photons[J]. Phys. Rev. Lett., 2003, 90(21):217902.
103 C. Simon, W. T. M. Irvine. Robust Long-distance Entanglement and a Loophole-free Bell Test with Ions and Photons[J]. Phys. Rev. Lett., 2003, 91(11):110405.
104 J. Hong, H. W. Lee. Quasideterministic Generation of Entangled Atoms in a Cav-ity[J]. Phys. Rev. Lett., 2002, 89(23):237901.
105 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Schemes for Generation of MultipartiteEntanglement of Atoms Trapped in Separate Optical Cavities[J]. J. Phys. B: At.Mol. Phys., 2009, 42(17):175506.
106 J. Song, Y. Xia, H. S. Song, et al. Quantum Computation and Entangled-stateGeneration Through Adiabatic Evolution in Two Distant Cavities[J]. Eur. Phys.Lett., 2007, 80(12):60001.
107 J. Song, Y. Xia, H. S. Song, et al. Four-dimensional Entangled State Generation inRemote Cavities[J]. Eur. Phys. J. D, 2008, 50(1):91–96.
108 J. Song, Y. Xia, H. S. Song. Quantum Gate Operations Using Atomic QubitsThrough Cavity Input-output Process[J]. Eur. Phys. Lett., 2009, 87(10):50005.
109 J. Shu, X. B. Zou, Y. F. Xiao, et al. Generating Four-mode Multiphoton EntangledStates in Cavity Qed[J]. Phys. Rev. A, 2006, 74(4):044301.
110 G. S. Jin, S. S. Li, S. L. Feng, et al. Generation of a Supersinglet of Three Three-level Atoms in Cavity Qed[J]. Phys. Rev. A, 2005, 71(3):034307.
111 X. B. Zou, K. Pahlke, W. Mathis. Generation of an Entangled State of Two Three-level Atoms in Cavity Qed[J]. Phys. Rev. A, 2003, 67(4):044301.
112 Y. F. Xiao, X. B. Zou, G. C. Guo. Generation of Atomic Entangled States withSelective Resonant Interaction in Cavity Quantum Electrodynamics[J]. Phys. Rev.A, 2007, 75(1):012310.
113 P. Dong, Z. Y. Xue, M. Yang, et al. Generation of Cluster States[J]. Phys. Rev. A,2006, 73(3):033818.
114 L. Ye, L. B. Yu, G. C. Guo. Generation of Entangled States in Cavity Qed[J]. Phys.Rev. A, 2005, 72(3):034304.
115 G. W. Lin, M. Y. Ye, L. B. Chen, et al. Generation of the Singlet State for ThreeAtoms in Cavity Qed[J]. Phys. Rev. A, 2007, 76(1):014308.
116 X. B. Zou, J. Kim, H. W. Lee. Generation of Two-mode Nonclassical MotionalStates and a Fredkin Gate Operation in a Two-dimensional Ion Trap[J]. Phys. Rev.A, 2001, 63(6):065801.
117 X. B. Zou, K. Pahlke, W. Mathis. Generation of Two-mode Nonclassical Statesand a Quantum-phase-gate Operation in Trapped-ion Cavity Qed[J]. Phys. Rev. A,2002, 65(6):064303.
118 N. Gisin, G. Ribordy, W. Tittel, et al. Quantum Cryptography[J]. Rev. Mod. Phys.,2002, 74(1):145–195.
119 S. Tanzilli, W. Tittel, M. Halder, et al. A Photonic Quantum Information Inter-face[J]. Nature, 2005, 437(9):116–120.
120 T. Yamamoto, K. Tamaki, M. Koashi, et al. Polarization-entangled W State UsingParametric Down-conversion[J]. Phys. Rev. A, 2002, 66(6):064301.
121 G. L. Long, Y. S. Li, W. L. Zhang, et al. Phase Matching in Quantum Searching[J].Phys. Lett. A, 1999, 262(1):27–34.
122 G. L. Long, W. L. Zhang, Y. S. Li, et al. Arbitrary Phase Rotation of the MarkedState Can Not Be Used for Grover’s Quantum Search Algorithm[J]. quantum-ph/9904077v1.
123 P. H?yer. Arbitrary Phases in Quantum Amplitude Amplification[J]. Phys. Rev. A,2000, 62(5):052304.
124 G. L. Long. Grover Algorithm with Zero Theoretical Failure Rate[J]. Phys. Rev.A, 2001, 64(2):022307.
125 G. L. Long, S. Li, Y. Sun. Phase Matching Condition for Quantum Search with aGeneralized Initial State[J]. Phys. Lett. A, 2002, 294(3-4):143–152.
126 L. K. Grover. Fixed-point Quantum Search[J]. Phys. Rev. Lett., 2005,95(15):150501.
127 L. K. Grover. Quantum Computers Can Search Rapidly by Using Almost anyTransformation[J]. Phys. Rev. Lett., 1998, 80(19):4329–4332.
128 P. C. Li, S. Y. Li. Phase Matching in Grover’s Algorithm[J]. Phys. Lett. A, 2007,366(1-2):42–46.
129 F. Yamaguchi, P. Milman, M. Brune, et al. Quantum Search with Two-atom Colli-sions in Cavity Qed[J]. Phys. Rev. A, 2002, 66(1):010302.
130 Z. J. Deng, M. Feng, K. L. Gao. Simple Scheme for the Two-qubit Grover Searchin Cavity Qed[J]. Phys. Rev. A, 2005, 72(3):034306.
131 W. L. Yang, C. Y. Chen, M. Feng. Implementation of Three-qubit Grover Search inCavity Quantum Electrodynamics[J]. Phys. Rev. A, 2007, 76(5):054301.
132 H. F. Wang, S. Zhang, K. H. Yeon. Implementation of Grover Quantum Searchvia Cavity Quantum Electrodynamics[J]. J. Korean Phys. Soc., 2008, 53(6):3144–3150.
133 N. Bhattacharya, H. B. van Linden van den Heuvell, R. J. C. Spreeuw. Implemen-tation of Quantum Search Algorithm Using Classical Fourier Optics[J]. Phys. Rev.Lett., 2002, 88(13):137901.
134 I. L. Chuang, N. Gershenfeld, M. Kubinec. Experimental Implementation of FastQuantum Searching[J]. Phys. Rev. Lett., 1998, 80(15):3408–3411.
135 M. Feng. Grover Search with Pairs of Trapped Ions[J]. Phys. Rev. A, 2001,63(5):052308.
136 S. Fujiwara, S. Hasegawa. General Method for Realizing the Conditional Phase-shift Gate and a Simulation of Grover’s Algorithm in an Ion-trap System[J]. Phys.Rev. A, 2005, 71(1):012337.
137 B. Deb, G. Kurizk. Formation of Giant Quasibound Cold Diatoms by Strong Atom-cavity Coupling[J]. Phys. Rev. Lett., 1999, 83(4):714–717.
138 A. S?rensen, K. M?Imer. Entanglement and Quantum Computation with Ions inThermal Motion[J]. Phys. Rev. A, 2000, 62(2):022311.
139 A. Rauschenbeutel, G. Nogues, S. Osnaghi, et al. Coherent Operation of a TunableQuantum Phase Gate in Cavity Qed[J]. Phys. Rev. Lett., 1999, 83(24):5166–5169.
140 A. Rauschenbeutel, G. Nogues, S. Osnaghi, et al. Step-by-step Engineered Multi-particle Entanglement[J]. Science, 2000, 288(6):2024–2028.
141 A. Rauschenbeutel, P. Bertet, S. Osnaghi, et al. Controlled Entanglement of TwoField Modes in a Cavity Quantum Electrodynamics Experiment[J]. Phys. Rev. A,2001, 64(5):050301.
142 P. Bertet, A. Auffeves, P. Maioli, et al. Direct Measurement of the Wigner Functionof a One-photon Fock State in a Cavity[J]. Phys. Rev. Lett., 2002, 89(20):200402.
143 G. Lenz, P. Meystre. Resonance Fluorescence from Two Identical Atoms in aStanding-wave Field[J]. Phys. Rev. A, 1993, 48(4):3365–3374.
144 R. G. Brewer. Two-ion Superradiance Theory[J]. Phys. Rev. A, 1995, 52(4):2965–2970.
145 E. V. Goldstein, P. Pax, P. Meystre. Dipole-dipole Interaction in Three-dimensionalOptical Lattices[J]. Phys. Rev. A, 1996, 53(4):2604–2615.
146 A. Beige, G. C. Hegerfeldt. Transition from Antibunching to Bunching for TwoDipole-interacting Atoms[J]. Phys. Rev. A, 1998, 58(5):4133–4139.
147 G. K. Brennen, I. H. Deutsch, P. S. Jessen. Entangling Dipole-dipole Interactionsfor Quantum Logic with Neutral Atoms[J]. Phys. Rev. A, 2000, 61(6):062309.
148 J. Guo. Effects of Resonant Dipole Interaction between Atoms on Laser Cooling:The Case of Two Identical Atoms[J]. Phys. Rev. A, 1994, 50(4):R2830–R2833.
149 A. Beige, S. F. Huelga, P. L. Knight, et al. Coherent Manipulation of Two Dipole―dipole Interacting Ions[J]. J. Mod. Opt., 2000, 47(1-2):401–414.
150 S. B. Zheng. Quantum-information Processing and Multiatom-entanglement Engi-neering with a Thermal Cavity[J]. Phys. Rev. A, 2002, 66(6):060303.
151 E. Farhi, J. Goldstone, S. Gutmann, et al. Limit on the Speed of Quantum Compu-tation in Determining Parity[J]. Phys. Rev. Lett., 1998, 81(24):5442–5444.
152 R. Stadelhofer, D. Suter, W. Banzhaf. Quantum and Classical Parallelism inParity Algorithms for Ensemble Quantum Computers[J]. Phys. Rev. A, 2005,71(3):032345.
153 M. O. Rabin. Digitalized Signatures and Public-key Functions as Intractable as Fac-torizations[C]//Technical Report, MIT/LCS/TR212, MIT Lab., Computer Science,Cambridge, Mass. l979.
154 R. L. Rivest, A. Shamir, L. Adleman. Public Key Cryptography[J]. Comm. Assoc.Comput. Mach., 1978, 21(6):120–125.
155 L. Harn, T. Kiesler. Improved Rabin’s Scheme with High Efficiency[J]. ElectronicsLetter, 1989, 25(11):726–728.
156 D. H. Nyang, J. S. Song. Fast Digital Signature Scheme Based on the QuadraticResidue Problem[J]. Electronics Letter, 1997, 33(3):205–206.
157 Y. Lecerf. Reversible Turing Machines: Recursive Insolubility[J]. C. R Acad.Francaise Sci., 1963, 257(7):2597–2600.
158 C. H. Bennett. Logical Reversibility of Computation[J]. IBM J. Res. Develop,1973, 17(6):525–532.
159 A. Scho¨nhage, V. Strassen. Schnelle Multiplikation GroSSer Zahlen[J]. Comput-ing, 1971, 7(3-4):281–292.
160 G. Chen, S. A. Fulling, J. Chen. Generalization of Grover’s Algorithm to Multiob-ject Search in Quantum Computing, Part I: Continuous Time and Discrete Time[J].quantum-ph/0007123.
161 H. F. Wang, S. Zhang. Implementation of N-qubit Deutsch–jozsa Algorithm UsingResonant Interaction in Cavity Qed[J]. Chin. Phys. B, 2008, 17(4):1165–1173.
162 C. P. Yang, S. Han. N-qubit-controlled Phase Gate with Superconducting Quantum-interference Devices Coupled to a Resonator[J]. Phys. Rev. A, 2005, 72(3):032311.
163 H. F. Wang, S. Zhang, K. H. Yeon. Implementing Quantum Discrete Fourier Trans-form by Using Cavity Quantum Electrodynamics[J]. J Korean Phys. Soc., 2008,53(4):1787–1790.
164 A. Gabris, G. S. Agarwal. Vacuum-induced Stark Shifts for Quantum Logic Usinga Collective System in a High-quality Dispersive Cavity[J]. Phys. Rev. A, 2005,71(5):052316.
165 H. F. Wang, S. Zhang, K. H. Yeon. Implementation of Grover Quantum Search viaCavity Quantum Electrodynamics[J]. J Korean Phys. Soc., 2008, 53(6):3144–3150.
166 Y. F. Xiao, X. B. Zou, G. C. Guo. Implementing a Conditional N-qubit Phase Gatein a Largely Detuned Optical Cavity[J]. Phys. Rev. A, 2007, 75(1):014302.
167 C. Y. Chen, M. Feng, K. L. Gao. Toffoli Gate Originating from a Single ResonantInteraction with Cavity Qed[J]. Phys. Rev. A, 2006, 73(6):064304.
168 A. O. Niskanen, J. J. Vartiainen, M. M. Salomaa. Optimal Multiqubit Operationsfor Josephson Charge Qubits[J]. Phys. Rev. Lett., 2003, 90(19):197901.
169 J. I. Cirac, P. Zoller. Quantum Computations with Cold Trapped Ions[J]. Phys. Rev.Lett., 1995, 74(20):4091–4094.
170 M. Sˇasˇura, V. Buzˇck. Multiparticle Entanglement with Quantum Logic Networks:Application to Cold Trapped Ions[J]. Phys. Rev. A, 2001, 64(1):012305.
171 V. Vedral, M. A. Rippin, P. L. Knight. Quantifying Entanglement[J]. Phys. Rev.Lett., 1997, 78(12):2275–2279.
172 O. Gu¨hne, P. Hyllus, A. Ekert, et al. Detection of Entanglement with Few LocalMeasurements[J]. Phys. Rev. A, 2002, 66(6):062305.
173 A. Salles, F. de MELO, J. C. Retamal, et al. Single Observable Voncurrence Mea-surement without Simultaneous Copies[J]. Phys. Rev. A, 2006, 74(6):060303.
174 P. Horodecki. Measuring Quantum Entanglement without Prior State Reconstruc-tion[J]. Phys. Rev. Lett., 2003, 90(16):167901.
175 R. T. Thew, K. Nemoto, A. G. White, et al. Qudit Quantum-state Tomography[J].Phys. Rev. A, 2002, 66(1):012303.
176 D. F. V. James, P. G. Kwiat, W. J. Munro, et al. Measurement of Qubits[J]. Phys.Rev. A, 2001, 64(5):052312.
177 N. Kiesel, C. Schmid, G. To′th, et al. Experimental Observation of Four-photonEntangled Dicke State with High Fidelity[J]. Phys. Rev. Lett., 2007, 98(6):063604.
178 A. G. White, D. F. V. James, P. H. Eberhard, et al. Nonmaximally EntangledStates: Production, Characterization, and Utilization[J]. Phys. Rev. Lett., 1999,83(16):3103–3107.
179 J. Rehacek, B. G. Englert, D. Kaszlikowski. Minimal Qubit Tomography[J]. Phys.Rev. A, 2004, 70(5):052321.
180 A. Ling, K. P. Soh, A. Lamas-Linares, et al. Experimental Polarization State To-mography Using Optimal Polarimeters[J]. Phys. Rev. A, 2006, 74(2):022309.
181 G. To′thl, O. Gu¨hne. Detecting Genuine Multipartite Entanglement with Two LocalMeasurements[J]. Phys. Rev. Lett., 2005, 94(6):060501.
182 F. A. Bovino, G. Castagnoli, A. Ekert, et al. Direct Measurement of NonlinearProperties of Bipartite Quantum States[J]. Phys. Rev. Lett., 2005, 95(24):240407.
183 M. Lewenstein, B. Kraus, J. I. Cirac, et al. Optimization of Entanglement Wit-nesses[J]. Phys. Rev. A, 2000, 62(5):052310.
184 S. P. Walborn, P. H. S. Ribeiro, L. Davidovich, et al. Experimental Determinationof Entanglement with a Single Measurement[J]. Nature, 2006, 440(4):1022–1024.
185 G. Romero, C. E. Lo′pez, F. Lastra, et al. Direct Measurement of Concurrence forAtomic Two-qubit Pure States[J]. Phys. Rev. A, 2007, 75(3):032303.
186 J. M. Cai, Z. W. Zhou, G. C. Guo. Entanglement Measurement Based on Two-particle Interference[J]. Phys. Rev. A, 2006, 73(2):024301.
187 S. M. Lee, S. W. Ji, H. W. Lee, et al. Proposal for Direct Measurement of Concur-rence via Visibility in a Cavity Qed System[J]. Phys. Rev. A, 2008, 77(4):040301.
188 H. F. Wang, S. Zhang. Application of Quantum Algorithms to Direct Measurementof Concurrence of a Two-qubit Pure State[J]. Chin. Phys. B, 2009, 18(7):2642–2648.
189 O. Biham, M. A. Nielsen, T. J. Osborne. Entanglement Monotone Derived fromGrover’s Algorithm[J]. Phys. Rev. A, 2002, 65(6):062312.
190 D. Shapira, Y. Shimoni, O. Biham. Groverian Measure of Entanglement for MixedStates[J]. Phys. Rev. A, 2006, 73(4):044301.
191 P. Benioff. Quantum Mechanical Hamiltonian Models of Turing Machines[J]. J.Stat. Phys., 1982, 29(6):515–518.
192 P. Benioff. Quantum Mechanical Hamiltonian Models of Turing Machines ThatDissipates No Energy[J]. Phys. Rev. Lett., 1982, 48(23):1581–1585.
193 D. Deutsch. Quantum Computational Networks[J]. Proc. Roy. Soc. lond. A, 1989,425(13):73–78.
194 S. Lloyd. A Potentially Realizable Quantum Computer[J]. Science, 1993,261(9):1569–1571.
195 J. Watrous. On One Dimensional Quantum Cellular Automata[C]//Proceeding ofthe 36th IEEE Symposium on Foundations of Computer Science. l995:528.
196 S. L. Braunstein, C. M. Caves, R. Jozsa, et al. Separability of Very Noisy MixedStates and Implications for Nmr Quantum Computing[J]. Phys. Rev. Lett., 1999,83(5):1054–1057.
197 R. Schack, C. M. Caves. Explicit Product Ensembles for Separable QuantumStates[J]. quantum-ph/9904109.
198 R. Schack, C. M. Caves. Classical Model for Bulk-ensemble Nmr Quantum Com-putation[J]. quantum-ph/9903101.
199 M. Brune, P. Nussenzveig, F. Schmidt-Kaler, et al. From Lamb Shift to Light Shifts:Vacuum and Subphoton Cavity Fields Measured by Atomic Phase Sensitive Detec-tion[J]. Phys. Rev. Lett., 1994, 72(21):3339–3342.
200 Q. A. Turchette, C. J. Hood, W. Lange, et al. Measurement of Conditional PhaseShifts for Quantum Logic[J]. Phys. Rev. Lett., 1995, 75(25):4710–4713.
201 M. O. Scully, M. S. Zubairy. Cavity Qed Implementation of the Discrete QuantumFourier Transform[J]. Phys. Rev. A, 2002, 65(5):052324.
202 Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, et al. Implementation of the Quan-tum Fourier Transform[J]. Phys. Rev. Lett., 2001, 86(9):1889–1891.
203 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Protocol and Quantum Circuit for Imple-menting N-bit Discrete Quantum Fourier Transform in Cavity Qed[J]. J. Phys. B:At. Mol. Phys., 2010, 53(4):in press.
204 M. Brune, E. Hagley, J. Dreyer, et al. Observing the Progressive Decoherence ofthe“meter”in a Quantum Measurement[J]. Phys. Rev. Lett., 1996, 77(24):4887–4890.
205 S. B. Zheng, G. C. Guo. Generation of Schro¨dinger Cat States via the Jaynes-cummings Model with Large Detuning[J]. Phys. Lett. A, 1996, 223(5):332–4890.
206 M. J. Holland, D. F. Walls, P. Zoller. Quantum Nondemolition Measurementsof Photon Number by Atomic Beam De?ection[J]. Phys. Rev. Lett., 1991,67(13):1716–1719.
207 S. B. Zheng, G. C. Guo. Efficient Scheme for Two-atom Entanglement and Quan-tum Information Processing in Cavity Qed[J]. Phys. Rev. Lett., 2000, 85(11):2392–2395.
208 J. M. Raimond, M. Brune, S. Haroche. Manipulating Quantum Entanglement withAtoms and Photons in a Cavity[J]. Rev. Mod. Phys., 2001, 73(3):565–582.
209 A. Barenco, C. H. Bennett, R. Cleve, et al. Elementary Gates for Quantum Compu-tation[J]. Phys. Rev. A, 1995, 52(5):3457–3467.
210 J. F. Poyatos, J. I. Cirac, P. Zoller. Complete Characterization of a Quantum Process:The Two-bit Quantum Gate[J]. Phys. Rev. Lett., 1997, 78(2):390–393.
2陈汉武.量子计算和量子信息[M].第一版.,南京:东南大学出版社, 2006:6–52.
3李承祖,黄明球,陈平形,等.量子通信和量子计算[M].第一版.,长沙:国防科技大学出版社, 2000:101–182.
4 C. H. Bennett, G. Brassard, C. Cre′peau, et al. Teleporting an Unknown QuantumState via Dual Classical and Einstein-podolsky-rosen Channels[J]. Phys. Rev. Lett.,1993, 70(13):1895–1899.
5 W. L. Li, C. F. Li, G. C. Guo. Probabilistic Teleportation and Entanglement Match-ing[J]. Phys. Rev. A, 2000, 61(3):034301.
6 S. B. Zheng. Scheme for Approximate Conditional Teleportation of an Un-known Atomic State without the Bell-state Measurement[J]. Phys. Rev. A, 2004,69(6):064302.
7 C. H. Bennett, G. Brassard. Quantum Cryptography: Public Key Distribution andCoin Tossing[C]//Proceedings of IEEE International Conference on Computers,System and Signal Proceessing, Bangalore, IEEE, (New York). l984:175–179.
8 C. H. Bennett, G. Brassard, N. D. Mermin. Quantum Cryptography without Bell’sTheorem[J]. Phys. Rev. Lett., 1992, 68(5):557–559.
9 J. C. Boileau, K. Tamaki, J. Batuwantudawe, et al. Unconditional Security ofa Three State Quantum Key Distribution Protocol[J]. Phys. Rev. Lett., 2005,94(4):040503.
10 A. Cabello. Quantum Key Distribution in the Holevo Limit[J]. Phys. Rev. Lett.,2000, 85(26):5635–5638.
11 H. K. Lo, X. F. Ma, K. Chen. Decoy State Quantum Key Distribution[J]. Phys.Rev. Lett., 2005, 94(23):230504.
12 C. H. Bennett, S. J. Wiesner. Communication via One- and Two-particle Operatorson Einstein-podolsky-rosen States[J]. Phys. Rev. Lett., 1992, 69(20):2881–2884.
13 K. Mattle, H. Weinfurter, P. G. Kwiat, et al. Dense Coding in Experimental Quan-tum Communication[J]. Phys. Rev. Lett., 1996, 76(25):4656–4659.
14 X. S. Liu, G. L. Long, D. M. Tong, et al. General Scheme for Superdense Codingbetween Multiparties[J]. Phys. Rev. A, 2002, 65(2):022304.
15 A. Karlsson, M. Koashi, N. Imoto. Quantum Entanglement for Secret Sharing andSecret Splitting[J]. Phys. Rev. A, 1999, 59(1):162–168.
16 M. Hillery, V. Buz′ek, A. Berthiaume. Quantum Secret Sharing[J]. Phys. Rev. A,1999, 59(3):1829–1834.
17 R. Cleve, D. Gottesman, H. K. Lo. How to Share a Quantum Secret[J]. Phys. Rev.Lett., 1999, 83(19):648–651.
18 H. F. Wang, X. Ji, S. Zhang. Improving the Security of Multiparty Quantum SecretSplitting and Quantum State Sharing[J]. Phys. Lett. A, 2006, 358(1):11–14.
19 S. Bandyopadhyay. Teleportation and Secret Sharing with Pure Entangled States[J].Phys. Rev. A, 2000, 62(1):012308.
20 V. Karimipour, A. Bahraminasab, S. Bagherinezhad. Entanglement Swapping ofGeneralized Cat States and Secret Sharing[J]. Phys. Rev. A, 2002, 65(4):042320.
21 S. Bandyopadhyay, V. Karimipour. Quantum Secret Sharing Based on ReusableGreenberger-horne-zeilinger States as Secure Carriers[J]. Phys. Rev. A, 2003,67(4):044302.
22 H. F. Wang, S. Zhang, Y. K. Hwang. Multiparty Quantum Secret Sharing via Intro-ducing Auxiliary Particles[J]. J. Korean Phys. Soc., 2006, 49(2):472–476.
23 K. Bostro¨m, T. Felbinger. Deterministic Secure Direct Communication Using En-tanglement[J]. Phys. Rev. Lett., 2002, 89(18):187902.
24 A. Beige, B. G. Englert, C. Kurtsiefer, et al. Secure Communication with a PubliclyKnown Key[J]. Acta. Phys. Pol. A, 2002, 101(3):357–368.
25 F. G. Deng, G. L. Long, X. S. Liu. A Two-step Quantum Direct Communica-tion Pro- Tocol Using Einstein-podolsky-rosen Pair Block[J]. Phys. Rev. A, 2003,68(4):042317.
26 H. F. Wang, S. Zhang, Y. K. Hwang, et al. Quantum Secure Direct Communicationby Using a Ghz State[J]. J. Korean Phys. Soc., 2006, 49(2):459–463.
27 B. A. Nguyen. Quantum Doalogue[J]. Phys. Lett. A, 2004, 328(1):6–10.
28 M. Christophe, P. D. Townsend. Quantum Key Distribution Over Distances as Longas 30 Km[J]. Opt. Lett., 1995, 20(16):1695.
29 A. Muller, J. Brejuet, N. Gisin. Experimental Demonstration of Quantum Cryptog-raphy Using Polarized Photons in Optical Fibre Over more Than 1 Km[J]. Euro-physics Letters, 1993, 23(6):383–388.
30 J. D. Franson, H. Ilves. Quantum Cryptography Using Polarization Feedback[J]. J.Mod. Opt., 1994, 41(12):2391–2396.
31 C. Liang, D. H. F. Liang, et al. Quantum Key Distribution Over 1.1 Km in an 850Nm Experimental All-fiber System[J]. Acta Phys. Sin., 2001, 50(8):1429–1433.
32 A. Muller, T. Herzog, B. Huttner, et al. Plug and Play Systems for Quantum Cryp-tography[J]. Appl. Phys. Lett., 1997, 70(7):793–795.
33 M. A. Rowe, D. Kielpinski, V. Meyer, et al. Experimental Violation of a Bell’sInequality with Efficient Detection[J]. Nature, 2001, 409(2):791–794.
34 G. Ribordy, J. D. Gautier. Automated Plug & Play Quantum Key Distribution[J].Electronics letters, 1998, 34(22):2116–2117.
35 S. J. D. Phoenix, S. M. Barnett, P. D. Townsend, et al. Multi-user Quantum Cryp-tography on Optical Networks[J]. J. Mod. Opt., 1995, 42(6):1155–1163.
36 W. T. Buttler, R. J. Hughes, P. G. Kwiat, et al. Practical Free-space Quantum KeyDistribution Over 1 Km[J]. Phys. Rev. Lett., 1998, 81(15):3283–3286.
37 G. Ribordy, J. Brendel, J. D. Fautier, et al. Long-distance Entanglement-basedQuantum Key Distribution[J]. Phys. Rev. A, 2001, 63(1):012309.
38 W. Tittel, J. Brendel, H. Zbinden, et al. Quantum Cryptography Using EntangledPhotons in Energy-time Bell States[J]. Phys. Rev. Lett., 2000, 84(20):4737–4740.
39 W. Tittel, H. Zbinden, N. Gisin. Experimental Demonstration of Quantum SecretSharing[J]. Phys. Rev. A, 2001, 63(4):042301.
40 Y. H. Kim, S. P. Kulik, Y. Shih. Quantum Teleportation of a Polarization State witha Com- Plete Bell State Measurement[J]. Phys. Rev. Lett., 2001, 86(7):1370.
41 A. Furusawa, J. L. Sorensen, S. L. Braunstein, et al. Unconditional Quantum Tele-portation[J]. Science, 1998, 282(5389):706–709.
42 D. Bouwmeester, J. W. Pan, K. Mattle, et al. Experimental Quantum Teleporta-tion[J]. Nature, 1997, 390(6660):575–579.
43 J. W. Pan, M. Daniell, S. Gasparoni, et al. Experimental Demonstration of Four-photon Entanglement and High-fidelity Teleportation[J]. Phys. Rev. Lett., 2001,86(20):4435–4438.
44 M. A. Nielsen, E. Knill, R. La?amme. Complete Quantum Teleportation UsingNuclear Magnetic Resonance[J]. Nature, 1998, 396(6706):52–55.
45 Y. A. Chen, S. Chen, Z. S. Yuan, et al. Memory-built-in Quantum Teleportationwith Photonic and Atomic Qubits[J]. Nature Physics, 2008, 4(2):103–107.
46 P. W. Shor. Algorithms for Quantum Computer Computation: Discrete Logarithmsand Factoring[C]//Proc. 35th Annu. Sym. Found. Comput. Science. l994:124.
47 L. K. Grover. Quantum Mechanics Helps in Searching for a Needle in aHaystack[J]. Phys. Rev. Lett, 1997, 79(23):325–328.
48 R. L. Rivest, A. Shamir, L. M. Adleman. A Method of Obtaining Digital Signaturesand Public-key Cryptosystems[J]. Comm. ACM, 1978, 21(2):120–126.
49 D. E. Knuth. The Art of Computer Programming[C]//Vol. 2: Seminumerical Algo-rithms, Seconded., Addison-Wesley. l981:25–60.
50 C. H. Hardy, E. M. Wright. An Introduction to the Theory of Numbers[C]//NewYork: Oxford University, 1979:1–52.
51 A. Ekert, R. Jozsa. Quantum Computation and Shor’s Factoring Algorithm[J]. Rev.Mod. Phys., 1996, 68(3):733–753.
52 C. Zalka. Grover’s Quantum Searching Algorithm Is Optimal[J]. Phys. Rev. A,1999, 60(4):2746–2751.
53 C. H. Bennett, E. Bernstein, G. Brassard, et al. Strengths and Weaknesses of Quan-tum Computing[J]. SIAM J. Comput., 1997, 60(5):1510–1523.
54 M. Boyer, G. Brassard, P. Hoyer, et al. Tight Bounds on Quantum Searching[J].Fortsch. Phys. Prog. Phys., 1998, 46(4 & 5):493–505.
55 G. Brassard, P. Hoyer, A. Tapp. Quantum Counting[J]. quantum-ph/9805082.
56 M. Mosca. Quantum Searching, Counting and Amplitude Amplification by Eigen-vector Analysis[C]//Proceedings of International Workshop on Randomized Algo-rithms. l998:90–100.
57 A. Einstein, B. Podolsky, N. Rosen. Can Quantum Mechanical Description of Phys-ical Reality Be Considered Complete[J]. Phys. Rev., 1935, 47(10):777–780.
58 E. Schro¨dinger. Die Genenwartige Situation in Der Quantenme Chanik[J]. Natur-wissenschafter, 1935, 23(8):807–812.
59 D. Gottesman, I. L. Chuang. Demonstrating the Viability of Universal QuantumComputation Using Teleportation and Single-qubit Operations[J]. Nature, 1999,402(11):390–393.
60 Y. Yeo, W. K. Chua. Teleportation and Dense Coding with Genuine MultipartiteEntanglement[J]. Phys. Rev. Lett., 2006, 96(6):060502.
61 X. W. Wang, G. J. Yang. Generation and Discrimination of a Type of Four-partiteEntangled State[J]. Phys. Rev. A, 2008, 78(2):024301.
62 J. Kempe, D. Bacon, D. A. Lidar, et al. Theory of Decoherence-free Fault-tolerantUniversal Quantum Computation[J]. Phys. Rev. A, 2001, 63(4):042307.
63 H. W. P. G. Kwiat, K. Mattle, et al. New High-intensity Source of Polarization-entangled Photon Pairs[J]. Phys. Rev. Lett., 1995, 75(24):4337–4341.
64 X. B. Zou, K. Pahlke, W. Mathis. Generation of an Entangled Four-photon WState[J]. Phys. Rev. A, 2002, 66(4):044302.
65 H. F. Wang, S. Zhang. Linear Optical Generation of Multipartite Entanglement withConventional Photon Detectors[J]. Phys. Rev. A, 2009, 79(4):042336.
66 Y. Tokunaga, T. Yamamoto, M. Koashi, et al. Simple Experimental Scheme ofPreparing a Four-photon Entangled State for the Teleportation-based Realization ofa Linear Optical Controlled-not Gate[J]. Phys. Rev. A, 2005, 71(3):030301.
67 D. Bouwmeester, J. W. Pan, M. Daniell, et al. Observation of Three-photonGreenberger-horne-zeilinger Entanglement[J]. Phys. Rev. Lett., 1999, 82(7):1345–1349.
68 E. Knill, R. La?amme, G. Milburn. A Scheme for Efficient Quantum Computationwith Linear Optics[J]. Nature, 2001, 409(1):46–52.
69 T. C. Ralph, N. K. Langford, T. B. Bell. Linear Optical Controlled-not Gate in theCoincidence Basis[J]. Phys. Rev. A, 2002, 65(6):062324.
70 H. F. Hofmann, S. Takeuchi. Quantum Phase Gate for Photonic Qubits Using onlyBeam Splitters and Postselection[J]. Phys. Rev. A, 2002, 66(2):024308.
71 J. L. O’Brien, G. J. Pryde, A. G. White, et al. Demonstration of an All-opticalQuantum Controlled-not Gate[J]. Nature, 2003, 426(11):264–267.
72 T. B. Pittman, M. J. Fitch, B. C. Jacobs, et al. Experimental Controlled-notLogic Gate for Single Photons in the Coincidence Basis[J]. Phys. Rev. A, 2003,68(3):032316.
73 T. B. Pittman, B. C. Jacobs, J. D. Franson. Demonstration of Feed-forward Controlfor Linear Optics Quantum Computation[J]. Phys. Rev. A, 2002, 66(5):052305.
74 T. B. Pittman, B. C. Jacobs, J. D. Franson. Probabilistic Quantum Logic OperationsUsing Polarizing Beam Splitters[J]. Phys. Rev. A, 2002, 64(6):062311.
75 X. B. Zou, S. L. Zhang, K. Li, et al. Linear Optical Implementation of the Two-qubit Controlled Phase Gate with Conventional Photon Detectors[J]. Phys. Rev. A,2007, 75(3):034302.
76 X. B. Zou, K. Li, G. C. Guo. Linear Optical Scheme for Direct Implementa-tion of a Nondestructive N-qubit Controlled Phase Gate[J]. Phys. Rev. A, 2006,74(4):044305.
77 H. F. Wang, S. Zhang, K. H. Yeon. Linear Optical Scheme for Entanglement Con-centration of Two Partially Entangled Three-photon W States[J]. Eur. Phys. J. D,2010, 56(2):271–275.
78 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Linear Optical Implementation of anAncilla-free Quantum Swap Gate[J]. Phys. Scr., 2010, 81(1):015011.
79 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Scheme for Implementing Linear OpticalQuantum Iswap Gate with Conventional Photon Detectors[J]. J. Opt. Soc. Am. B,2010, 27(1):27–31.
80 Y. Xia, J. Song, H. S. Song. Linear Optical Protocol for Preparation of N-photonGreenberger-horne-zeilinger State with Conventional Photon Detectors[J]. Appl.Phys. Lett., 2008, 92(12):021127.
81 Y. Xia, J. Song, H. S. Song, et al. Controlled Generation of Four-photonPolarization-entangled Decoherence-free States with Conventional Photon Detec-tors[J]. J. Opt. Soc. Am. B, 2009, 26(1):129–132.
82 S. Glancy, J. M. LoSecco, H. M. Vasconcelos, et al. Imperfect Detectors in LinearOptical Quantum Computers[J]. Phys. Rev. A, 2002, 65(6):062317.
83 Z. Y. Ou, L. Mandel. Observation of Spatial Quantum Beating with SeparatedPhotodetectors[J]. Phys. Rev. Lett., 1998, 61(1):54–57.
84 F. DeMartini, G. DiGiuseppe, M. Marrocco. Single-mode Generation of QuantumPhoton States by Excited Single Molecules in a Microcavity Trap[J]. Phys. Rev.Lett., 1996, 76(6):900–903.
85 H. J. Kimble, M. Dagenais, L. Mandel. Photon Antibunching in Resonance Fluo-rescence[J]. Phys. Rev. Lett., 1977, 39(11):691–695.
86 S. Brattke, B. T. H. Varcoe, H. Walther. Generation of Photon Number Stateson Demand via Cavity Quantum Electrodynamics[J]. Phys. Rev. Lett., 2001,86(16):3534–3537.
87 P. Michler, A. Imamoglu, M. D. Mason, et al. Quantum Correlation Among Photonsfrom a Single Quantum Dot at Room Temperature[J]. Nature, 2000, 406(8):968–970.
88 J. Kim, O. Benson, H. Kan, et al. A Single-photon Turnstile Device[J]. Nature,1999, 397(2):500–503.
89 N. Akopian, N. H. Lindner, E. Poem, et al. Entangled Photon Pairs from Semicon-ductor Quantum Dots[J]. Phys. Rev. Lett., 2006, 96(13):130501.
90 O. Benson, C. Santori, M. Pelton, et al. Regulated and Entangled Photons from aSingle Quantum Dot[J]. Phys. Rev. Lett., 2000, 84(11):2513–2516.
91 A. Barenco, D. Deutsch, A. Ekert. Conditional Quantum Dynamics and LogicGates[J]. Phys. Rev. Lett., 1995, 74(5):4083–4086.
92 T. Sleator, H. Weinfurter. Realizable Universal Quantum Logic Gates[J]. Phys.Rev. Lett., 1995, 74(5):4087–4090.
93 X. B. Zou, K. Pahlke, W. Mathis. Quantum Entanglement of Four Distant AtomsTrapped in Different Optical Cavities[J]. Phys. Rev. A, 2004, 69(5):052314.
94 Y. X. Gong, X. B. Zou, X. L. Niu, et al. Generation of Arbitrary Four-photon Polarization-entangled Decoherence-free States[J]. Phys. Rev. A, 2008,77(4):042317.
95 G. C. Guo, Y. S. Zhang. Scheme for Preparation of the W State via Cavity QuantumElectrodynamics[J]. Phys. Rev. A, 2002, 65(5):054302.
96 X. B. Zou, K. Pahlke, W. Mathis. Conditional Generation of the Greenberger-horne-zeilinger State of Four Distant Atoms via Cavity Decay[J]. Phys. Rev. A,2003, 68(7):024302.
97 Z. J. Deng, M. Feng, K. L. Gao. Preparation of Entangled States of Four Re-mote Atomic Qubits in Decoherence-free Subspace[J]. Phys. Rev. A, 2007,75(2):024302.
98 S. Bose, P. L. Knight, M. B. Plenio, et al. Proposal for Teleportation of an AtomicState via Cavity Decay[J]. Phys. Rev. Lett., 1999, 83(24):5158–5161.
99 H. F. Wang, S. Zhang. Scheme for Linear Optical Preparation of a Type of Four-photon Entangled State with Conventional Photon Detectors[J]. Eur. Phys. J. D,2009, 53(3):359–363.
100 L. M. Duan, M. D. Lukin, J. Cirac, et al. Long-distance Quantum Communicationwith Atomic Ensembles and Linear Optics[J]. Nature, 2001, 414(9):413–418.
101 C. Cabrillo, J. I. Cirac, P. Garcia-Fernandez, et al. Creation of Entangled States ofDistant Atoms by Interference[J]. Phys. Rev. A, 1999, 59(2):1025–1033.
102 X. L. Feng, Z. M. Zhang, X. D. Li, et al. Entangling Distant Atoms by Interferenceof Polarized Photons[J]. Phys. Rev. Lett., 2003, 90(21):217902.
103 C. Simon, W. T. M. Irvine. Robust Long-distance Entanglement and a Loophole-free Bell Test with Ions and Photons[J]. Phys. Rev. Lett., 2003, 91(11):110405.
104 J. Hong, H. W. Lee. Quasideterministic Generation of Entangled Atoms in a Cav-ity[J]. Phys. Rev. Lett., 2002, 89(23):237901.
105 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Schemes for Generation of MultipartiteEntanglement of Atoms Trapped in Separate Optical Cavities[J]. J. Phys. B: At.Mol. Phys., 2009, 42(17):175506.
106 J. Song, Y. Xia, H. S. Song, et al. Quantum Computation and Entangled-stateGeneration Through Adiabatic Evolution in Two Distant Cavities[J]. Eur. Phys.Lett., 2007, 80(12):60001.
107 J. Song, Y. Xia, H. S. Song, et al. Four-dimensional Entangled State Generation inRemote Cavities[J]. Eur. Phys. J. D, 2008, 50(1):91–96.
108 J. Song, Y. Xia, H. S. Song. Quantum Gate Operations Using Atomic QubitsThrough Cavity Input-output Process[J]. Eur. Phys. Lett., 2009, 87(10):50005.
109 J. Shu, X. B. Zou, Y. F. Xiao, et al. Generating Four-mode Multiphoton EntangledStates in Cavity Qed[J]. Phys. Rev. A, 2006, 74(4):044301.
110 G. S. Jin, S. S. Li, S. L. Feng, et al. Generation of a Supersinglet of Three Three-level Atoms in Cavity Qed[J]. Phys. Rev. A, 2005, 71(3):034307.
111 X. B. Zou, K. Pahlke, W. Mathis. Generation of an Entangled State of Two Three-level Atoms in Cavity Qed[J]. Phys. Rev. A, 2003, 67(4):044301.
112 Y. F. Xiao, X. B. Zou, G. C. Guo. Generation of Atomic Entangled States withSelective Resonant Interaction in Cavity Quantum Electrodynamics[J]. Phys. Rev.A, 2007, 75(1):012310.
113 P. Dong, Z. Y. Xue, M. Yang, et al. Generation of Cluster States[J]. Phys. Rev. A,2006, 73(3):033818.
114 L. Ye, L. B. Yu, G. C. Guo. Generation of Entangled States in Cavity Qed[J]. Phys.Rev. A, 2005, 72(3):034304.
115 G. W. Lin, M. Y. Ye, L. B. Chen, et al. Generation of the Singlet State for ThreeAtoms in Cavity Qed[J]. Phys. Rev. A, 2007, 76(1):014308.
116 X. B. Zou, J. Kim, H. W. Lee. Generation of Two-mode Nonclassical MotionalStates and a Fredkin Gate Operation in a Two-dimensional Ion Trap[J]. Phys. Rev.A, 2001, 63(6):065801.
117 X. B. Zou, K. Pahlke, W. Mathis. Generation of Two-mode Nonclassical Statesand a Quantum-phase-gate Operation in Trapped-ion Cavity Qed[J]. Phys. Rev. A,2002, 65(6):064303.
118 N. Gisin, G. Ribordy, W. Tittel, et al. Quantum Cryptography[J]. Rev. Mod. Phys.,2002, 74(1):145–195.
119 S. Tanzilli, W. Tittel, M. Halder, et al. A Photonic Quantum Information Inter-face[J]. Nature, 2005, 437(9):116–120.
120 T. Yamamoto, K. Tamaki, M. Koashi, et al. Polarization-entangled W State UsingParametric Down-conversion[J]. Phys. Rev. A, 2002, 66(6):064301.
121 G. L. Long, Y. S. Li, W. L. Zhang, et al. Phase Matching in Quantum Searching[J].Phys. Lett. A, 1999, 262(1):27–34.
122 G. L. Long, W. L. Zhang, Y. S. Li, et al. Arbitrary Phase Rotation of the MarkedState Can Not Be Used for Grover’s Quantum Search Algorithm[J]. quantum-ph/9904077v1.
123 P. H?yer. Arbitrary Phases in Quantum Amplitude Amplification[J]. Phys. Rev. A,2000, 62(5):052304.
124 G. L. Long. Grover Algorithm with Zero Theoretical Failure Rate[J]. Phys. Rev.A, 2001, 64(2):022307.
125 G. L. Long, S. Li, Y. Sun. Phase Matching Condition for Quantum Search with aGeneralized Initial State[J]. Phys. Lett. A, 2002, 294(3-4):143–152.
126 L. K. Grover. Fixed-point Quantum Search[J]. Phys. Rev. Lett., 2005,95(15):150501.
127 L. K. Grover. Quantum Computers Can Search Rapidly by Using Almost anyTransformation[J]. Phys. Rev. Lett., 1998, 80(19):4329–4332.
128 P. C. Li, S. Y. Li. Phase Matching in Grover’s Algorithm[J]. Phys. Lett. A, 2007,366(1-2):42–46.
129 F. Yamaguchi, P. Milman, M. Brune, et al. Quantum Search with Two-atom Colli-sions in Cavity Qed[J]. Phys. Rev. A, 2002, 66(1):010302.
130 Z. J. Deng, M. Feng, K. L. Gao. Simple Scheme for the Two-qubit Grover Searchin Cavity Qed[J]. Phys. Rev. A, 2005, 72(3):034306.
131 W. L. Yang, C. Y. Chen, M. Feng. Implementation of Three-qubit Grover Search inCavity Quantum Electrodynamics[J]. Phys. Rev. A, 2007, 76(5):054301.
132 H. F. Wang, S. Zhang, K. H. Yeon. Implementation of Grover Quantum Searchvia Cavity Quantum Electrodynamics[J]. J. Korean Phys. Soc., 2008, 53(6):3144–3150.
133 N. Bhattacharya, H. B. van Linden van den Heuvell, R. J. C. Spreeuw. Implemen-tation of Quantum Search Algorithm Using Classical Fourier Optics[J]. Phys. Rev.Lett., 2002, 88(13):137901.
134 I. L. Chuang, N. Gershenfeld, M. Kubinec. Experimental Implementation of FastQuantum Searching[J]. Phys. Rev. Lett., 1998, 80(15):3408–3411.
135 M. Feng. Grover Search with Pairs of Trapped Ions[J]. Phys. Rev. A, 2001,63(5):052308.
136 S. Fujiwara, S. Hasegawa. General Method for Realizing the Conditional Phase-shift Gate and a Simulation of Grover’s Algorithm in an Ion-trap System[J]. Phys.Rev. A, 2005, 71(1):012337.
137 B. Deb, G. Kurizk. Formation of Giant Quasibound Cold Diatoms by Strong Atom-cavity Coupling[J]. Phys. Rev. Lett., 1999, 83(4):714–717.
138 A. S?rensen, K. M?Imer. Entanglement and Quantum Computation with Ions inThermal Motion[J]. Phys. Rev. A, 2000, 62(2):022311.
139 A. Rauschenbeutel, G. Nogues, S. Osnaghi, et al. Coherent Operation of a TunableQuantum Phase Gate in Cavity Qed[J]. Phys. Rev. Lett., 1999, 83(24):5166–5169.
140 A. Rauschenbeutel, G. Nogues, S. Osnaghi, et al. Step-by-step Engineered Multi-particle Entanglement[J]. Science, 2000, 288(6):2024–2028.
141 A. Rauschenbeutel, P. Bertet, S. Osnaghi, et al. Controlled Entanglement of TwoField Modes in a Cavity Quantum Electrodynamics Experiment[J]. Phys. Rev. A,2001, 64(5):050301.
142 P. Bertet, A. Auffeves, P. Maioli, et al. Direct Measurement of the Wigner Functionof a One-photon Fock State in a Cavity[J]. Phys. Rev. Lett., 2002, 89(20):200402.
143 G. Lenz, P. Meystre. Resonance Fluorescence from Two Identical Atoms in aStanding-wave Field[J]. Phys. Rev. A, 1993, 48(4):3365–3374.
144 R. G. Brewer. Two-ion Superradiance Theory[J]. Phys. Rev. A, 1995, 52(4):2965–2970.
145 E. V. Goldstein, P. Pax, P. Meystre. Dipole-dipole Interaction in Three-dimensionalOptical Lattices[J]. Phys. Rev. A, 1996, 53(4):2604–2615.
146 A. Beige, G. C. Hegerfeldt. Transition from Antibunching to Bunching for TwoDipole-interacting Atoms[J]. Phys. Rev. A, 1998, 58(5):4133–4139.
147 G. K. Brennen, I. H. Deutsch, P. S. Jessen. Entangling Dipole-dipole Interactionsfor Quantum Logic with Neutral Atoms[J]. Phys. Rev. A, 2000, 61(6):062309.
148 J. Guo. Effects of Resonant Dipole Interaction between Atoms on Laser Cooling:The Case of Two Identical Atoms[J]. Phys. Rev. A, 1994, 50(4):R2830–R2833.
149 A. Beige, S. F. Huelga, P. L. Knight, et al. Coherent Manipulation of Two Dipole―dipole Interacting Ions[J]. J. Mod. Opt., 2000, 47(1-2):401–414.
150 S. B. Zheng. Quantum-information Processing and Multiatom-entanglement Engi-neering with a Thermal Cavity[J]. Phys. Rev. A, 2002, 66(6):060303.
151 E. Farhi, J. Goldstone, S. Gutmann, et al. Limit on the Speed of Quantum Compu-tation in Determining Parity[J]. Phys. Rev. Lett., 1998, 81(24):5442–5444.
152 R. Stadelhofer, D. Suter, W. Banzhaf. Quantum and Classical Parallelism inParity Algorithms for Ensemble Quantum Computers[J]. Phys. Rev. A, 2005,71(3):032345.
153 M. O. Rabin. Digitalized Signatures and Public-key Functions as Intractable as Fac-torizations[C]//Technical Report, MIT/LCS/TR212, MIT Lab., Computer Science,Cambridge, Mass. l979.
154 R. L. Rivest, A. Shamir, L. Adleman. Public Key Cryptography[J]. Comm. Assoc.Comput. Mach., 1978, 21(6):120–125.
155 L. Harn, T. Kiesler. Improved Rabin’s Scheme with High Efficiency[J]. ElectronicsLetter, 1989, 25(11):726–728.
156 D. H. Nyang, J. S. Song. Fast Digital Signature Scheme Based on the QuadraticResidue Problem[J]. Electronics Letter, 1997, 33(3):205–206.
157 Y. Lecerf. Reversible Turing Machines: Recursive Insolubility[J]. C. R Acad.Francaise Sci., 1963, 257(7):2597–2600.
158 C. H. Bennett. Logical Reversibility of Computation[J]. IBM J. Res. Develop,1973, 17(6):525–532.
159 A. Scho¨nhage, V. Strassen. Schnelle Multiplikation GroSSer Zahlen[J]. Comput-ing, 1971, 7(3-4):281–292.
160 G. Chen, S. A. Fulling, J. Chen. Generalization of Grover’s Algorithm to Multiob-ject Search in Quantum Computing, Part I: Continuous Time and Discrete Time[J].quantum-ph/0007123.
161 H. F. Wang, S. Zhang. Implementation of N-qubit Deutsch–jozsa Algorithm UsingResonant Interaction in Cavity Qed[J]. Chin. Phys. B, 2008, 17(4):1165–1173.
162 C. P. Yang, S. Han. N-qubit-controlled Phase Gate with Superconducting Quantum-interference Devices Coupled to a Resonator[J]. Phys. Rev. A, 2005, 72(3):032311.
163 H. F. Wang, S. Zhang, K. H. Yeon. Implementing Quantum Discrete Fourier Trans-form by Using Cavity Quantum Electrodynamics[J]. J Korean Phys. Soc., 2008,53(4):1787–1790.
164 A. Gabris, G. S. Agarwal. Vacuum-induced Stark Shifts for Quantum Logic Usinga Collective System in a High-quality Dispersive Cavity[J]. Phys. Rev. A, 2005,71(5):052316.
165 H. F. Wang, S. Zhang, K. H. Yeon. Implementation of Grover Quantum Search viaCavity Quantum Electrodynamics[J]. J Korean Phys. Soc., 2008, 53(6):3144–3150.
166 Y. F. Xiao, X. B. Zou, G. C. Guo. Implementing a Conditional N-qubit Phase Gatein a Largely Detuned Optical Cavity[J]. Phys. Rev. A, 2007, 75(1):014302.
167 C. Y. Chen, M. Feng, K. L. Gao. Toffoli Gate Originating from a Single ResonantInteraction with Cavity Qed[J]. Phys. Rev. A, 2006, 73(6):064304.
168 A. O. Niskanen, J. J. Vartiainen, M. M. Salomaa. Optimal Multiqubit Operationsfor Josephson Charge Qubits[J]. Phys. Rev. Lett., 2003, 90(19):197901.
169 J. I. Cirac, P. Zoller. Quantum Computations with Cold Trapped Ions[J]. Phys. Rev.Lett., 1995, 74(20):4091–4094.
170 M. Sˇasˇura, V. Buzˇck. Multiparticle Entanglement with Quantum Logic Networks:Application to Cold Trapped Ions[J]. Phys. Rev. A, 2001, 64(1):012305.
171 V. Vedral, M. A. Rippin, P. L. Knight. Quantifying Entanglement[J]. Phys. Rev.Lett., 1997, 78(12):2275–2279.
172 O. Gu¨hne, P. Hyllus, A. Ekert, et al. Detection of Entanglement with Few LocalMeasurements[J]. Phys. Rev. A, 2002, 66(6):062305.
173 A. Salles, F. de MELO, J. C. Retamal, et al. Single Observable Voncurrence Mea-surement without Simultaneous Copies[J]. Phys. Rev. A, 2006, 74(6):060303.
174 P. Horodecki. Measuring Quantum Entanglement without Prior State Reconstruc-tion[J]. Phys. Rev. Lett., 2003, 90(16):167901.
175 R. T. Thew, K. Nemoto, A. G. White, et al. Qudit Quantum-state Tomography[J].Phys. Rev. A, 2002, 66(1):012303.
176 D. F. V. James, P. G. Kwiat, W. J. Munro, et al. Measurement of Qubits[J]. Phys.Rev. A, 2001, 64(5):052312.
177 N. Kiesel, C. Schmid, G. To′th, et al. Experimental Observation of Four-photonEntangled Dicke State with High Fidelity[J]. Phys. Rev. Lett., 2007, 98(6):063604.
178 A. G. White, D. F. V. James, P. H. Eberhard, et al. Nonmaximally EntangledStates: Production, Characterization, and Utilization[J]. Phys. Rev. Lett., 1999,83(16):3103–3107.
179 J. Rehacek, B. G. Englert, D. Kaszlikowski. Minimal Qubit Tomography[J]. Phys.Rev. A, 2004, 70(5):052321.
180 A. Ling, K. P. Soh, A. Lamas-Linares, et al. Experimental Polarization State To-mography Using Optimal Polarimeters[J]. Phys. Rev. A, 2006, 74(2):022309.
181 G. To′thl, O. Gu¨hne. Detecting Genuine Multipartite Entanglement with Two LocalMeasurements[J]. Phys. Rev. Lett., 2005, 94(6):060501.
182 F. A. Bovino, G. Castagnoli, A. Ekert, et al. Direct Measurement of NonlinearProperties of Bipartite Quantum States[J]. Phys. Rev. Lett., 2005, 95(24):240407.
183 M. Lewenstein, B. Kraus, J. I. Cirac, et al. Optimization of Entanglement Wit-nesses[J]. Phys. Rev. A, 2000, 62(5):052310.
184 S. P. Walborn, P. H. S. Ribeiro, L. Davidovich, et al. Experimental Determinationof Entanglement with a Single Measurement[J]. Nature, 2006, 440(4):1022–1024.
185 G. Romero, C. E. Lo′pez, F. Lastra, et al. Direct Measurement of Concurrence forAtomic Two-qubit Pure States[J]. Phys. Rev. A, 2007, 75(3):032303.
186 J. M. Cai, Z. W. Zhou, G. C. Guo. Entanglement Measurement Based on Two-particle Interference[J]. Phys. Rev. A, 2006, 73(2):024301.
187 S. M. Lee, S. W. Ji, H. W. Lee, et al. Proposal for Direct Measurement of Concur-rence via Visibility in a Cavity Qed System[J]. Phys. Rev. A, 2008, 77(4):040301.
188 H. F. Wang, S. Zhang. Application of Quantum Algorithms to Direct Measurementof Concurrence of a Two-qubit Pure State[J]. Chin. Phys. B, 2009, 18(7):2642–2648.
189 O. Biham, M. A. Nielsen, T. J. Osborne. Entanglement Monotone Derived fromGrover’s Algorithm[J]. Phys. Rev. A, 2002, 65(6):062312.
190 D. Shapira, Y. Shimoni, O. Biham. Groverian Measure of Entanglement for MixedStates[J]. Phys. Rev. A, 2006, 73(4):044301.
191 P. Benioff. Quantum Mechanical Hamiltonian Models of Turing Machines[J]. J.Stat. Phys., 1982, 29(6):515–518.
192 P. Benioff. Quantum Mechanical Hamiltonian Models of Turing Machines ThatDissipates No Energy[J]. Phys. Rev. Lett., 1982, 48(23):1581–1585.
193 D. Deutsch. Quantum Computational Networks[J]. Proc. Roy. Soc. lond. A, 1989,425(13):73–78.
194 S. Lloyd. A Potentially Realizable Quantum Computer[J]. Science, 1993,261(9):1569–1571.
195 J. Watrous. On One Dimensional Quantum Cellular Automata[C]//Proceeding ofthe 36th IEEE Symposium on Foundations of Computer Science. l995:528.
196 S. L. Braunstein, C. M. Caves, R. Jozsa, et al. Separability of Very Noisy MixedStates and Implications for Nmr Quantum Computing[J]. Phys. Rev. Lett., 1999,83(5):1054–1057.
197 R. Schack, C. M. Caves. Explicit Product Ensembles for Separable QuantumStates[J]. quantum-ph/9904109.
198 R. Schack, C. M. Caves. Classical Model for Bulk-ensemble Nmr Quantum Com-putation[J]. quantum-ph/9903101.
199 M. Brune, P. Nussenzveig, F. Schmidt-Kaler, et al. From Lamb Shift to Light Shifts:Vacuum and Subphoton Cavity Fields Measured by Atomic Phase Sensitive Detec-tion[J]. Phys. Rev. Lett., 1994, 72(21):3339–3342.
200 Q. A. Turchette, C. J. Hood, W. Lange, et al. Measurement of Conditional PhaseShifts for Quantum Logic[J]. Phys. Rev. Lett., 1995, 75(25):4710–4713.
201 M. O. Scully, M. S. Zubairy. Cavity Qed Implementation of the Discrete QuantumFourier Transform[J]. Phys. Rev. A, 2002, 65(5):052324.
202 Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, et al. Implementation of the Quan-tum Fourier Transform[J]. Phys. Rev. Lett., 2001, 86(9):1889–1891.
203 H. F. Wang, X. Q. Shao, Y. F. Zhao, et al. Protocol and Quantum Circuit for Imple-menting N-bit Discrete Quantum Fourier Transform in Cavity Qed[J]. J. Phys. B:At. Mol. Phys., 2010, 53(4):in press.
204 M. Brune, E. Hagley, J. Dreyer, et al. Observing the Progressive Decoherence ofthe“meter”in a Quantum Measurement[J]. Phys. Rev. Lett., 1996, 77(24):4887–4890.
205 S. B. Zheng, G. C. Guo. Generation of Schro¨dinger Cat States via the Jaynes-cummings Model with Large Detuning[J]. Phys. Lett. A, 1996, 223(5):332–4890.
206 M. J. Holland, D. F. Walls, P. Zoller. Quantum Nondemolition Measurementsof Photon Number by Atomic Beam De?ection[J]. Phys. Rev. Lett., 1991,67(13):1716–1719.
207 S. B. Zheng, G. C. Guo. Efficient Scheme for Two-atom Entanglement and Quan-tum Information Processing in Cavity Qed[J]. Phys. Rev. Lett., 2000, 85(11):2392–2395.
208 J. M. Raimond, M. Brune, S. Haroche. Manipulating Quantum Entanglement withAtoms and Photons in a Cavity[J]. Rev. Mod. Phys., 2001, 73(3):565–582.
209 A. Barenco, C. H. Bennett, R. Cleve, et al. Elementary Gates for Quantum Compu-tation[J]. Phys. Rev. A, 1995, 52(5):3457–3467.
210 J. F. Poyatos, J. I. Cirac, P. Zoller. Complete Characterization of a Quantum Process:The Two-bit Quantum Gate[J]. Phys. Rev. Lett., 1997, 78(2):390–393.