多光子量子计算实验研究
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摘要
本论文主要内容是通过操纵多光子纠缠态进行光学量子计算和算法实现的实验研究。
我们在实验上发展了一套高亮度的参量下转换纠缠光源和稳定的六光子干涉技术,首次实现了六光子薛定谔猫态和簇态。这些是现今为止实现的光子数最多的纠缠态。通过进一步相干控制极化和空间两个自由度,我们实验实现了由五个光子的极化状态和空间状态相干叠加形成的超纠缠十量子比特薛定谔猫态。这是现今为止量子比特数目最大的纠缠态。为了克服光子损失这一重要的消相干效应,我们在量子线路和簇态量子计算两种模型中分别实现了四比特和五比特的量子容失编码,首次验证了纠正量子比特丢失错误的可行性。最后,我们在国际上首次用光量子比特演示了Shor大数分解算法,并且实验确认了量子计算中多光子干涉和多体纯纠缠的存在。
Coherent control of multiple photons with linear optics has been a promising approach to quantum computing. Recent years have witness tremedous progresses, both on theory and experiment on this approach. Major challenges ahead lies in manipulation of an increasing number of entangled photonic qubits, fighting against possible decoherence, and constructing quantum circuits for implementing quantum algorithms.
Along this line, this dissertation describes four experiments on multiphoton entanglement and optical quantum computation. We have developed a high-brightness source of entangled photons based on spontaneous parametric down-conversion. Using linear optical networks we are able to entangle them into a six-photon GHZ state, which is the larget photonic Schrodinger cat so far, and a six-photon cluster state, a state-of-the-art one-way quantum computer. Next, we use the hyper-entanglement trick to entangle not only the polarization of the photons, but also their spatial modes. By doing so, we manage to create a six-, eight-and ten-qubit hyper-entangled Schrodinger cat state, expanding the effective Hilbert space up to 1024 dimensions. The third experiment aims to deal with the photon loss error which is a big headache in optical quantum computing. We show both in the quantum circuit model and the one-way model the smallest meaningful quantum codes that are able to tackle the photon-loss problem. Finally, we describe the first optical implementation of Shor's quantum factoring algorithm. We choose the simplest instance of this algorithm-factorization of 15-and exploit a simplified linear optical network to implement the quantum circuits of the modular exponential execution and semiclassical quantum Fourier transformation.
These experiments are necessary steps towards scalable quantum computing with single photons and linear optics.
我们在实验上发展了一套高亮度的参量下转换纠缠光源和稳定的六光子干涉技术,首次实现了六光子薛定谔猫态和簇态。这些是现今为止实现的光子数最多的纠缠态。通过进一步相干控制极化和空间两个自由度,我们实验实现了由五个光子的极化状态和空间状态相干叠加形成的超纠缠十量子比特薛定谔猫态。这是现今为止量子比特数目最大的纠缠态。为了克服光子损失这一重要的消相干效应,我们在量子线路和簇态量子计算两种模型中分别实现了四比特和五比特的量子容失编码,首次验证了纠正量子比特丢失错误的可行性。最后,我们在国际上首次用光量子比特演示了Shor大数分解算法,并且实验确认了量子计算中多光子干涉和多体纯纠缠的存在。
Coherent control of multiple photons with linear optics has been a promising approach to quantum computing. Recent years have witness tremedous progresses, both on theory and experiment on this approach. Major challenges ahead lies in manipulation of an increasing number of entangled photonic qubits, fighting against possible decoherence, and constructing quantum circuits for implementing quantum algorithms.
Along this line, this dissertation describes four experiments on multiphoton entanglement and optical quantum computation. We have developed a high-brightness source of entangled photons based on spontaneous parametric down-conversion. Using linear optical networks we are able to entangle them into a six-photon GHZ state, which is the larget photonic Schrodinger cat so far, and a six-photon cluster state, a state-of-the-art one-way quantum computer. Next, we use the hyper-entanglement trick to entangle not only the polarization of the photons, but also their spatial modes. By doing so, we manage to create a six-, eight-and ten-qubit hyper-entangled Schrodinger cat state, expanding the effective Hilbert space up to 1024 dimensions. The third experiment aims to deal with the photon loss error which is a big headache in optical quantum computing. We show both in the quantum circuit model and the one-way model the smallest meaningful quantum codes that are able to tackle the photon-loss problem. Finally, we describe the first optical implementation of Shor's quantum factoring algorithm. We choose the simplest instance of this algorithm-factorization of 15-and exploit a simplified linear optical network to implement the quantum circuits of the modular exponential execution and semiclassical quantum Fourier transformation.
These experiments are necessary steps towards scalable quantum computing with single photons and linear optics.
引文
[Aliferis and Terhal,2007] P. Aliferis and B. Terhal, Fault-tolerant quantum computation for local leak-age faults, Quant. Inf. Comp., vol.7, pp.139-156, (2007).
[Beckham et al.1996] D. Beckman et al., Efficient networks for quantum factoring, Phys. Rev. A 54,1034 (1996)
[Bennett et al.1993] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Physical Review Letters, vol.70, no.13, pp. 1895-1899,(1993).
[Bourennane et al.,2004] M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H.Weinfurter, O.GAuhne, P. Hyllus, D. Brub, M. Lewenstein, and A. Sanpera, Experimental Detection of Multipartite Entanglement using Witness Operators. Physical Review Letters, vol.92, no.8, p.087902, (2004).
[Braunstein et al.,1999] S. L. Braunstein et al., Separability of very noisy mixed states and implications for NMR quantum computing, Phys. Rev. Lett.83,1054, (1999)
[Briegel and Raussendorf,2001] H. Briegel and R. Raussendorf. Persistent entanglement in arrays of interacting particles. Physical Review Letters, vol.86, pp.910-913, (2001).
[Browne and Rudolph,2005] D. E. Browne and T. Rudolph, Resource-Efficient Linear Optical Quantum Computation. Phys. Rev. Lett.95,010501, (2005).
[Bouwmeester et al.,1997] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A.
Zeilinger, Experimental quantum teleportation, Nature, vol.390, pp.575-579, (1997).
[Bouwmeester et al.,1999] D. Bouwmeester, J. W. Pan, M. Doniell, H. Weinfurter, and A.Zeilinger, Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement Phys. Rev. Lett.82,1345,(1999).
[Calderbank and Shor,1996] A. Calderbank and P. Shor, Good quantum error-correcting codes exist. Phys. Rev. A, vol.54, pp.1098-1105, (1996).
[Cleve et al.,1999] R. Cleve, D. Gottesman, and H. Lo, How to Share a Quantum Secret, Physical Review Letters, vol.83, p.648, (1999).
[DiCarlo et al.,2010] L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, R. J. Schoelkopf, Preparation and Measurement of Three-Qubit Entanglement in a Superconducting Circuit, arXiv:1004.4324
[Deutsch,1985] D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. R. Soc. Lond., Ser. A.400,97, (1985).
[Devitt et al.2007] S. Devitt, S. Devitt, S. Schirmer, D. Oi, J. Cole, and L. Hollenberg, Subspace confinement:how good is your qubit?, New J. Phys., vol.9, p.384, (2007).
[Feyman 1982] R. P. Feynman, Simulating physics with computers," Int. J. Theor. Phys., vol.21, no.67, pp.467-488, (1982).
[Fuchs,1996] C. A. Fuchs, Ph.D. thesis, Univ. of New Mexico, (1996).
[Giovannetti et al.,2004] Giovannetti, V., Lloyd, S.& Maccone, L, Quantum-enhanced measurements:Beating the standard quantum limit, Science 306,1330-1336, (2004).
[Gottesman,1997] D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, (1997).
[Gottesman and Chuang,1999] Gottesman, and I. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit," Nature, vol.402, p. 390, (1999).
[Grassl et al.,1997] M. Grassl, T. Beth, and T. Pellizari. Codes for the quantum erasure channel. Phys. Rev. A, vol.56, pp.33-38, (1997).
[Greenberger et al.1990] D. Greenberger, M. Home, A. Shimony, and A. Zeilinger, Bell's theorem without inequalities, Am. J. Phys., vol.58, pp.1131-1143, (1990).
[Grover,1997] L. K. Grover. Quantum Computers Can Search Arbitrarily Large Databases by a Single Query. Phys. Rev. Lett.79,4709, (1997).
[Haffner et al.,2005] Haffner, H. et al. Scalable multiparticle entanglement of trapped ions. Nature 438,643-646, (2005).
[Han et al.,2007] Y.-J. Han, R. Raussendorf, and L.-M. Duan, Scheme for Demonstration of Fractional Statistics of Anyons in an Exactly Solvable Model, Physical Review Letters, vol.98, p.150404,(2007).
[Hein et al.,2004] M. Hein, J. Eisert, and H. Briegel. Multiparty entanglement in graph states. Physical Review A, vol.69, no.6, p.62311, (2004).
[Hong et al.,1987] C. K. Hong, Z. Y. Ou and L. Mandel. Phys. Rev. Lett.59,2044, (1987).
[Knill et al.,2001] E. Knill, R. Laflamme, and G. Milburn. A scheme for efficient quantum computation with linear optics. Nature (London) 409,46 (2001).
[Knill et al.,2005] E. Knill, Quantum computing with realistically noisy devices, Nature. vol.434, pp.39-44,2005.
[Kok et al.,2007] P. Kok et al., Linear optical quantum computing with photonic qubits
Rev. Mod. Phys.79,135 (2007).
[Kwiat et al.,1995] P. G. Kwiat, K. Mattle, H.Weinfurter, A. Zeilinger, A.V. Sergienko, and Y.H. Shih. New high intensity source of polarizationentangled photon pairs. Phys. Rev. Lett.75,4337-4341,1995.
[Kwiat,2000] P. Kwiat, J. Mitchell, P. Schwindt, and A. White, Grover's search algorithm:an optical approach, J. Mod. Opt, vol.47, p.257,2000.
[Lanyon et al.2007] B. Lanyon, T. Weinhold, N. Langford, M. Barbieri, D. James, A. Gilchrist, and A. White, Experimental demonstration of a complied version of Shor's algorithm with quantum entanglement," Physical Review Letters, vol.99, p.250505, (2007).
[Leibfried et al.,2005] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. Blakestad, J. Chiaverini, D. Hume, W. Itano, J. Jost, C. Langer, et al., Creation of a six-atom Schroedinger cat' state, Nature, vol.438, no.7068, pp.639-642,2005.
[Lloyd,1996] S. Lloyd, Universal Quantum Simulators, Science, vol.273, p.1073,1996.
[N. Linden and S. Popescu,2001] N. Linden and S. Popescu, Good Dynamics versus Bad Kinematics:Is Entanglement Needed for Quantum Computation? Phys. Rev. Lett.87, 047901
[Neumann et al.,2008] P. Neumann et al., Multipartite Entanglement Among Single Spins in Diamond, Science 320,1326,2008.
[Nielsen and Chuang,2000] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge Univ. Press,2000.
[Nielsen]M. Nielsen, Optical Quantum Computation Using Cluster States, Phys. Rev. Lett. 93,040503 (2004)
[O'Brien,2003] J. L. O'Brien et al., Demonstration of an all-optical quantum controlled-NOT gate, Nature (London) 426,264 (2003).
[Pan et al.,2000] J.-W. Pan et al. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature (London) 403,515,2000.
[Pan et al.,2001] J-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and HA. Zeilinger. Experimental demonstration of four-photon entanglement and high-fidelity teleportation. Phys. Rev. Lett.86,4435,2001.
[Preskill,1998] J. Preskill, Reliable quantum computers, Proc. R. Soc. Lond.A, vol.454, pp. 385-410,1998.
[Ralph et al.,2005] T. Ralph, A. Hayes, and A. Gilchrist. Loss-tolerant optical qubits. Physical Review Letters, vol.95, p.100501,2005.
[Rarity and Tapster,1999] J. Rarity and P. Tapster. Three-particle entanglement from entangled photon pairs and a weak coherent state. Phys. Rev. A, vol.59, pp. R35-R38, 1999.
[Raussendorf et al.,2007] R. Raussendorf, J. Harrington, and K. Goyal, Topological fault-tolerance in cluster state quantum computation, New J. Phys., vol.9, p.199,2007.
[Raussendorf and Briegel,2001] R. Raussendorf et al., A One-Way Quantum Computer, Phys. Rev. Lett.86,5188(2001).
[Sackett et al.,2000] C. A. Sackett et al., experimental entanglement of four particles, Nature (London)404,256,2000.
[Schlingemann and Werner,2002] D. Schlingemann and R.Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A, vol.65, p.012308,2001.
[Shor,1994] P. W. Shor, Algorithms for quantum computation:discrete logarithms and factoring, Proceedings,35th Annual Symposium on Fundamentals of Computer Science, 1994.
[Spedalieri et al.2006] F. Spedalieri, H. Lee, and J. Dowling. High-fidelity linear optical quantum
computing with polarization encoding. Phys. Rev. A, vol.73, p.012334,2006.
[Steane,1996] A. M. Steane. Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A, vol.425, pp.2551-2576,1996. quant-ph/9601029.
[Toth and Guhne,2005] G. Toth and O. Guhne. Entanglement detection in the stabilizer formalism. Phys. Rev. A, vol.72, p.022340,2005.
[Vandersypen et al.2001] L. M. K. Vandersypen et al., Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance, Nature (London) 414, 883
[Vala et al.2005] J. Vala, K. Whaley, and D.Weiss, Quantum error correction of a qubit loss in an addressable atomic system, Phys. Rev. A, vol.72, p.052318,2005.
[Walther et al.,2005] P. Walther et al. Experimental One-Way Quantum Computing. Nature 434 169-176,2005.
[Varnava et al.,2006] M. Varnava, D. Browne, and T. Rudolph. Loss Tolerance in One-Way Quantum Computation via Counterfactual Error Correction. Physical Review Letters, vol. 97, no.12, p.120501,2006.
[Wu et al.2002] L.-A. Wu, M. Byrd, and D. Lidar, Efficient Universal Leakage Elimination for Physical and Encoded Qubits, Physical Review Letters, vol.89, p.127901,2002.
[Zhao et al.,2004] Z. Zhao et al. Experimental demonstration of five-photon entanglement and open-destination teleportation. Nature 430,54-58,2004.
[Beckham et al.1996] D. Beckman et al., Efficient networks for quantum factoring, Phys. Rev. A 54,1034 (1996)
[Bennett et al.1993] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Physical Review Letters, vol.70, no.13, pp. 1895-1899,(1993).
[Bourennane et al.,2004] M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H.Weinfurter, O.GAuhne, P. Hyllus, D. Brub, M. Lewenstein, and A. Sanpera, Experimental Detection of Multipartite Entanglement using Witness Operators. Physical Review Letters, vol.92, no.8, p.087902, (2004).
[Braunstein et al.,1999] S. L. Braunstein et al., Separability of very noisy mixed states and implications for NMR quantum computing, Phys. Rev. Lett.83,1054, (1999)
[Briegel and Raussendorf,2001] H. Briegel and R. Raussendorf. Persistent entanglement in arrays of interacting particles. Physical Review Letters, vol.86, pp.910-913, (2001).
[Browne and Rudolph,2005] D. E. Browne and T. Rudolph, Resource-Efficient Linear Optical Quantum Computation. Phys. Rev. Lett.95,010501, (2005).
[Bouwmeester et al.,1997] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A.
Zeilinger, Experimental quantum teleportation, Nature, vol.390, pp.575-579, (1997).
[Bouwmeester et al.,1999] D. Bouwmeester, J. W. Pan, M. Doniell, H. Weinfurter, and A.Zeilinger, Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement Phys. Rev. Lett.82,1345,(1999).
[Calderbank and Shor,1996] A. Calderbank and P. Shor, Good quantum error-correcting codes exist. Phys. Rev. A, vol.54, pp.1098-1105, (1996).
[Cleve et al.,1999] R. Cleve, D. Gottesman, and H. Lo, How to Share a Quantum Secret, Physical Review Letters, vol.83, p.648, (1999).
[DiCarlo et al.,2010] L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, R. J. Schoelkopf, Preparation and Measurement of Three-Qubit Entanglement in a Superconducting Circuit, arXiv:1004.4324
[Deutsch,1985] D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer, Proc. R. Soc. Lond., Ser. A.400,97, (1985).
[Devitt et al.2007] S. Devitt, S. Devitt, S. Schirmer, D. Oi, J. Cole, and L. Hollenberg, Subspace confinement:how good is your qubit?, New J. Phys., vol.9, p.384, (2007).
[Feyman 1982] R. P. Feynman, Simulating physics with computers," Int. J. Theor. Phys., vol.21, no.67, pp.467-488, (1982).
[Fuchs,1996] C. A. Fuchs, Ph.D. thesis, Univ. of New Mexico, (1996).
[Giovannetti et al.,2004] Giovannetti, V., Lloyd, S.& Maccone, L, Quantum-enhanced measurements:Beating the standard quantum limit, Science 306,1330-1336, (2004).
[Gottesman,1997] D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, (1997).
[Gottesman and Chuang,1999] Gottesman, and I. Chuang, Demonstrating the viability of universal quantum computation using teleportation and single-qubit," Nature, vol.402, p. 390, (1999).
[Grassl et al.,1997] M. Grassl, T. Beth, and T. Pellizari. Codes for the quantum erasure channel. Phys. Rev. A, vol.56, pp.33-38, (1997).
[Greenberger et al.1990] D. Greenberger, M. Home, A. Shimony, and A. Zeilinger, Bell's theorem without inequalities, Am. J. Phys., vol.58, pp.1131-1143, (1990).
[Grover,1997] L. K. Grover. Quantum Computers Can Search Arbitrarily Large Databases by a Single Query. Phys. Rev. Lett.79,4709, (1997).
[Haffner et al.,2005] Haffner, H. et al. Scalable multiparticle entanglement of trapped ions. Nature 438,643-646, (2005).
[Han et al.,2007] Y.-J. Han, R. Raussendorf, and L.-M. Duan, Scheme for Demonstration of Fractional Statistics of Anyons in an Exactly Solvable Model, Physical Review Letters, vol.98, p.150404,(2007).
[Hein et al.,2004] M. Hein, J. Eisert, and H. Briegel. Multiparty entanglement in graph states. Physical Review A, vol.69, no.6, p.62311, (2004).
[Hong et al.,1987] C. K. Hong, Z. Y. Ou and L. Mandel. Phys. Rev. Lett.59,2044, (1987).
[Knill et al.,2001] E. Knill, R. Laflamme, and G. Milburn. A scheme for efficient quantum computation with linear optics. Nature (London) 409,46 (2001).
[Knill et al.,2005] E. Knill, Quantum computing with realistically noisy devices, Nature. vol.434, pp.39-44,2005.
[Kok et al.,2007] P. Kok et al., Linear optical quantum computing with photonic qubits
Rev. Mod. Phys.79,135 (2007).
[Kwiat et al.,1995] P. G. Kwiat, K. Mattle, H.Weinfurter, A. Zeilinger, A.V. Sergienko, and Y.H. Shih. New high intensity source of polarizationentangled photon pairs. Phys. Rev. Lett.75,4337-4341,1995.
[Kwiat,2000] P. Kwiat, J. Mitchell, P. Schwindt, and A. White, Grover's search algorithm:an optical approach, J. Mod. Opt, vol.47, p.257,2000.
[Lanyon et al.2007] B. Lanyon, T. Weinhold, N. Langford, M. Barbieri, D. James, A. Gilchrist, and A. White, Experimental demonstration of a complied version of Shor's algorithm with quantum entanglement," Physical Review Letters, vol.99, p.250505, (2007).
[Leibfried et al.,2005] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. Blakestad, J. Chiaverini, D. Hume, W. Itano, J. Jost, C. Langer, et al., Creation of a six-atom Schroedinger cat' state, Nature, vol.438, no.7068, pp.639-642,2005.
[Lloyd,1996] S. Lloyd, Universal Quantum Simulators, Science, vol.273, p.1073,1996.
[N. Linden and S. Popescu,2001] N. Linden and S. Popescu, Good Dynamics versus Bad Kinematics:Is Entanglement Needed for Quantum Computation? Phys. Rev. Lett.87, 047901
[Neumann et al.,2008] P. Neumann et al., Multipartite Entanglement Among Single Spins in Diamond, Science 320,1326,2008.
[Nielsen and Chuang,2000] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge Univ. Press,2000.
[Nielsen]M. Nielsen, Optical Quantum Computation Using Cluster States, Phys. Rev. Lett. 93,040503 (2004)
[O'Brien,2003] J. L. O'Brien et al., Demonstration of an all-optical quantum controlled-NOT gate, Nature (London) 426,264 (2003).
[Pan et al.,2000] J.-W. Pan et al. Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature (London) 403,515,2000.
[Pan et al.,2001] J-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and HA. Zeilinger. Experimental demonstration of four-photon entanglement and high-fidelity teleportation. Phys. Rev. Lett.86,4435,2001.
[Preskill,1998] J. Preskill, Reliable quantum computers, Proc. R. Soc. Lond.A, vol.454, pp. 385-410,1998.
[Ralph et al.,2005] T. Ralph, A. Hayes, and A. Gilchrist. Loss-tolerant optical qubits. Physical Review Letters, vol.95, p.100501,2005.
[Rarity and Tapster,1999] J. Rarity and P. Tapster. Three-particle entanglement from entangled photon pairs and a weak coherent state. Phys. Rev. A, vol.59, pp. R35-R38, 1999.
[Raussendorf et al.,2007] R. Raussendorf, J. Harrington, and K. Goyal, Topological fault-tolerance in cluster state quantum computation, New J. Phys., vol.9, p.199,2007.
[Raussendorf and Briegel,2001] R. Raussendorf et al., A One-Way Quantum Computer, Phys. Rev. Lett.86,5188(2001).
[Sackett et al.,2000] C. A. Sackett et al., experimental entanglement of four particles, Nature (London)404,256,2000.
[Schlingemann and Werner,2002] D. Schlingemann and R.Werner, Quantum error-correcting codes associated with graphs, Phys. Rev. A, vol.65, p.012308,2001.
[Shor,1994] P. W. Shor, Algorithms for quantum computation:discrete logarithms and factoring, Proceedings,35th Annual Symposium on Fundamentals of Computer Science, 1994.
[Spedalieri et al.2006] F. Spedalieri, H. Lee, and J. Dowling. High-fidelity linear optical quantum
computing with polarization encoding. Phys. Rev. A, vol.73, p.012334,2006.
[Steane,1996] A. M. Steane. Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A, vol.425, pp.2551-2576,1996. quant-ph/9601029.
[Toth and Guhne,2005] G. Toth and O. Guhne. Entanglement detection in the stabilizer formalism. Phys. Rev. A, vol.72, p.022340,2005.
[Vandersypen et al.2001] L. M. K. Vandersypen et al., Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance, Nature (London) 414, 883
[Vala et al.2005] J. Vala, K. Whaley, and D.Weiss, Quantum error correction of a qubit loss in an addressable atomic system, Phys. Rev. A, vol.72, p.052318,2005.
[Walther et al.,2005] P. Walther et al. Experimental One-Way Quantum Computing. Nature 434 169-176,2005.
[Varnava et al.,2006] M. Varnava, D. Browne, and T. Rudolph. Loss Tolerance in One-Way Quantum Computation via Counterfactual Error Correction. Physical Review Letters, vol. 97, no.12, p.120501,2006.
[Wu et al.2002] L.-A. Wu, M. Byrd, and D. Lidar, Efficient Universal Leakage Elimination for Physical and Encoded Qubits, Physical Review Letters, vol.89, p.127901,2002.
[Zhao et al.,2004] Z. Zhao et al. Experimental demonstration of five-photon entanglement and open-destination teleportation. Nature 430,54-58,2004.