Abstract

Robust universal quantum gates with an extremely high fidelity hold an important position in large-scale quantum computing. Here, we propose a scheme for several robust universal photonic quantum gates on a two-or three-photon system, including the controlled-NOT gate, the Toffoli gate, and the Fredkin gate, assisted by low-Q single-sided cavities. In our scheme, the quantum gates are robust against imperfect process occurring with the photons and the electron spins in diamond nitrogen-vacancy (NV) centers inside low-Q cavities. Errors due to the imperfect process are transferred to some heralding responses, which may lead to a direct recycling procedure to remedy the success probability of the quantum gates. As a result, the adverse impact of the imperfect process on fidelity is eliminated, greatly relaxing the restrictions on implementation of various quantum gates in experiments. Furthermore, the scheme is designed in a compact and heralded style, which can increase the robustness against environmental noise and local fluctuation, thus decreasing the operation time, the error probability, and the quantum resource consumption in a large-scale integrated quantum circuit. The near-unity fidelity and not-too-low efficiency with current achievable experimental techniques guarantees the feasibility of the scheme.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (5)

M. Hua, M. J. Tao, A. Alsaedi, T. Hayat, and F. G. Deng, “Universal distributed quantum computing on superconducting qutrits with dark photons,” Ann. Phys. (Berlin) 530, 1700402 (2018).
[Crossref]

J. Zhou, B. J. Liu, Z. P. Hong, and Z. Y. Xue, “Fast holonomic quantum computation based on solid-state spins with all-optical control,” Sci. China: Phys., Mech. Astron. 61, 010312 (2018).

T. Li, J. C. Gao, F. G. Deng, and G. L. Long, “High-fidelity quantum gates on quantum-dot-confined electron spins in low-Q optical microcavities,” Ann. Phys. 391, 150–160 (2018).
[Crossref]

T. Liu, B. Q. Guo, C. S. Yu, and W. N. Zhang, “One-step implementation of a hybrid Fredkin gate with quantum memories and single superconducting qubit in circuit QED and its applications,” Opt. Express 26, 4498–4511 (2018).
[Crossref] [PubMed]

G. Y. Wang, T. Li, Q. Ai, and F. G. Deng, “self-error-corrected hyperparallel photonic quantum computation working with both the polarization and the spatial-mode degrees of freedom,” Opt. Express 26, 23333–23346 (2018).
[Crossref] [PubMed]

2017 (6)

B. C. Ren and F. G. Deng, “Robust hyperparallel photonic quantum entangling gate with cavity QED,” Opt. Express 25, 10863–10873 (2017).
[Crossref] [PubMed]

X. Han, Q. Guo, A. D. Zhu, S. Zhang, and H. F. Wang, “Effective W-state fusion strategies in nitrogen-vacancy centers via coupling to microtoroidal resonators,” Opt. Express 25, 17701–17712 (2017).
[Crossref] [PubMed]

W. Qin, X. Wang, A. Miranowicz, Z. Zhong, and F. Nori, “Heralded quantum controlled-phase gates with dissipative dynamics in macroscopically distant resonators,” Phys. Rev. A 96, 012315 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Distributed secure quantum machine learning,” Sci. Bull. 62, 1025–1029 (2017).
[Crossref]

F. G. Deng, B. C. Ren, and X. H. Li, “Quantum hyperentanglement and its applications in quantum information processing,” Sci. Bull. 62, 46–68 (2017).
[Crossref]

L. Zhou and Y. B. Sheng, “Polarization entanglement purification for concatenated Greenberger-Horne-Zeilinger state,” Ann. Phys. 385, 10–35 (2017).
[Crossref]

2016 (8)

G. Y. Wang, Q. Liu, and F. G. Deng, “Hyperentanglement purification for two-photon six-qubit quantum systems,” Phys. Rev. A 94, 032319 (2016).
[Crossref]

X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
[Crossref]

X. K. Song, Hao Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
[Crossref]

T. Li and F. G. Deng, “Error-rejecting quantum computing with solid-state spins assisted by low-Q optical microcavities,” Phys. Rev. A 94, 062310 (2016).
[Crossref]

T. Li and G. L. Long, “Hyperparallel optical quantum computation assisted by atomic ensembles embedded in double-sided optical cavities,” Phys. Rev. A 94, 022343 (2016).
[Crossref]

L. Y. He, T. J. Wang, and C. Wang, “Construction of high-dimensional universal quantum logic gates using a Λ system coupled with a whispering-gallery-mode microresonator,” Opt. Express 24, 15429–15445 (2016).
[Crossref] [PubMed]

H. R. Wei, F. G. Deng, and G. L. Long, “Hyper-parallel Toffoli gate on three-photon system with two degrees of freedom assisted by single-sided optical microcavities,” Opt. Express 24, 18619–18630 (2016).
[Crossref] [PubMed]

G. Y. Wang, Q. Ai, B. C. Ren, T. Li, and F. G. Deng, “Error-detected generation and complete analysis of hyperentangled Bell states for photons assisted by quantum-dot spins in double-sided optical microcavities,” Opt. Express 24, 28444–28458 (2016).
[Crossref] [PubMed]

2015 (7)

J. Borregaard, P. Komar, E. M. Kessler, A. S. Sørensen, and M. D. Lukin, “Heralded quantum gates with integrated error detection in optical cavities,” Phys. Rev. Lett. 114, 110502 (2015).
[Crossref] [PubMed]

H. R. Wei and G. L. Long, “Universal photonic quantum gates assisted by ancilla diamond nitrogen-vacancy centers coupled to resonators,” Phys. Rev. A 91, 032324 (2015).
[Crossref]

B. C. Ren, G. Y. Wang, and F. G. Deng, “Universal hyperparallel hybrid photonic quantum gates with the dipole induced transparency in weak-coupling regime,” Phys. Rev. A 91, 032328 (2015).
[Crossref]

Q. Liu and M. Zhang, “Generation and complete nondestructive analysis of hyperentanglement assisted by nitrogen-vacancy centers in resonators,” Phys. Rev. A 91, 062321 (2015).
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M. J. Tao, M. Hua, Q. Ai, and F. G. Deng, “Quantum-information processing on nitrogen-vacancy ensembles with the local resonance assisted by circuit QED,” Phys. Rev. A 91, 062325 (2015).
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Y. B. Sheng, J. Pan, R. Guo, L. Zhou, and L. Wang, “Efficient N-particle W state concentration with different parity check gates,” Sci. China: Phys., Mech. Astron. 58, 060301 (2015).

L. Zhou and Y. B. Sheng, “Complete logic Bell-state analysis assisted with photonic Faraday rotation,” Phys. Rev. A,  92, 042314 (2015).
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2014 (5)

L. Zhou and Y. B. Sheng, “Detection of nonlocal atomic entanglement assisted by single photons,” Phys. Rev. A 90, 024301 (2014).
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J. D. Pritchard, J. A. Isaacs, M. A. Beck, R. McDermott, and M. Saffman, “Hybrid atom-photon quantum gate in a superconducting microwave resonator,” Phys. Rev. A 89, 010301 (2014).
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T. J. Wang and C. Wang, “Universal hybrid three-qubit quantum gates assisted by a nitrogen-vacancy center coupled with a whispering-gallery-mode microresonator,” Phys. Rev. A 90, 052310 (2014).
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B. C. Ren and F. G. Deng, “Hyper-parallel photonic quantum computing with coupled quantum dots,” Sci. Rep. 4, 4623 (2014).
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H. R. Wei and F. G. Deng, “Universal quantum gates on electron-spin qubits with quantum dots inside single-side optical microcavities,” Opt. Express 22, 593–607 (2014).
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2013 (10)

H. R. Wei and F. G. Deng, “Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity,” Opt. Express 21, 17671–17685 (2013).
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C. Wang, Y. Zhang, R. Z. Jiao, and G. S. Jin, “Universal quantum controlled phase gate on photonic qubits based on nitrogen vacancy centers and microcavity resonators,” Opt. Express 21, 19252–19260 (2013).
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H. R. Wei and F. G. Deng, “Compact quantum gates on electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities,” Phys. Rev. A 88, 042323 (2013).
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H. R. Wei and F. G. Deng, “Universal quantum gates for hybrid systems assisted by quantum dots inside double-sided optical microcavities,” Phys. Rev. A 87, 022305 (2013).
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B. C. Ren, H. R. Wei, and F.G. Deng, “Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by quantum dot inside one-side optical microcavity,” Laser Phys. Lett. 10, 095202 (2013).
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G. R. Feng, G. F. Xu, and G. L. Long, “Experimental realization of nonadiabatic Holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
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H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, “Heralded entanglement between solid-state qubits separated by 3 meters,” Nature 497, 86–90 (2013).
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J. H. Shim, I. Niemeyer, J. Zhang, and D. Suter, “Room-temperature high-speed nuclear-spin quantum memory in diamond,” Phys. Rev. A 87, 012301 (2013).
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P. F. Wang, C. Y. Ju, F. Z. Shi, and J. F. Du, “Optimizing ultrasensitive single electron magnetometer based on nitrogen-vacancy center in diamond,” Chin. Sci. Bull. 58, 2920–2923 (2013).

B. C. Ren and F. G. Deng, “Hyperentanglement purification and concentration assisted by diamond NV centers inside photonic crystal cavities,” Laser Phys. Lett. 10, 115201 (2013).
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2012 (4)

T. Peyronel, O. Firstenberg, Q. Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
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M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Realization of three-qubit quantum error correction with superconducting circuits,” Nature 482, 382–385 (2012).
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A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, “Implementation of a Toffoli gate with superconducting circuits,” Nature 481, 170–172 (2012).
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Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012).
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2011 (5)

Y. Eto, A. Noguchi, P. Zhang, M. Ueda, and M. Kozuma, “Projective measurement of a single nuclear spin qubit by using two-mode cavity QED,” Phys. Rev. Lett. 106, 160501 (2011).
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S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
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G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789–793 (2011).
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L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578 (2011).
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Q. Chen, W. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
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2010 (3)

B. B. Buckley, G. D. Fuchs, L. C. Bassett, and D. D. Awschalom, “Spin-light coherence for single-spin measurement and control in diamond,” Science 330, 1212–1215 (2010).
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E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Srensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730 (2010).
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C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. lett. 104, 160503 (2010).
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2009 (6)

C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009).
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L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, and M. D. Lukin, “Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae,” Science 326, 267–272 (2009).
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G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
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G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520–1522 (2009).
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P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009).
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2008 (2)

C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: Applications to entangling remote spins via a single photon,” Phys. Rev. B 78, 085307 (2008).
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Y. Liu, G. L. Long, and Y. Sun, “Analytic one-bit and CNOT gate constructions of general n-qubit controlled gates,” Int. J. Quant. Inform. 6, 447–462 (2008).
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2007 (1)

M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science 316, 1312–1316 (2007).
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2006 (4)

L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
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T. Gaebel, M. Domhan, I. Popa, C. Wittmann, P. Neumann, F. Jelezko, J. R. Rabeau, N. Stavrias, A. D. Greentree, S. Prawer, J. Meijer, J. Twamley, P. R. Hemmer, and J. Wrachtrup, “Room-temperature coherent coupling of single spins in diamond,” Nat. Phys. 2, 408–413 (2006).
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M. Wallquist, V. S. Shumeiko, and G. Wendin, “Selective coupling of superconducting charge qubits mediated by a tunable stripline cavity,” Phys. Rev. B 74, 224506 (2006).
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N. B. Manson, J. P. Harrison, and M. J. Sellars, “Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics,” Phys. Rev. B 74, 104303 (2006).
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2005 (2)

L. M. Liang and C. Z. Li, “Realization of quantum SWAP gate between flying and stationary qubits,” Phys. Rev. A 72, 024303 (2005).
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I. Friedler, D. Petrosyan, M. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
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2004 (8)

K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004).
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L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
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F. Jelezko, T. Gaebel, I. Popa, A. Gruber, and J. Wrachtrup, “Observation of coherent oscillations in a single electron spin,” Phys. Rev. Lett. 92, 076401 (2004).
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G. Vidal and C. M. Dawson, “Universal quantum circuit for two-qubit transformations with three controlled-NOT gates,” Phys. Rev. A 69, 010301 (2004).
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F. Vatan and C. Williams, “Optimal quantum circuits for general two-qubit gates,” Phys. Rev. A 69, 032315 (2004).
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J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, “Optimal quantum circuit synthesis from controlled-unitary gates,” Phys. Rev. A 69, 042309 (2004).
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V. V. Shende, I. L. Markov, and S. S. Bullock, “Minimal universal two-qubit controlled-NOT-based circuits,” Phys. Rev. A 69, 062321 (2004).
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V. V. Shende, S. S. Bullock, and I. L. Markov, “Recognizing small-circuit structure in two-qubit operators,” Phys. Rev. A 70, 012310 (2004).
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2003 (2)

Y. Y. Shi, “Both Toffoli and controlled-NOT need little help to universal quantum computing,” Quantum Inf. Comput. 3, 84–92 (2003).

A. O. Niskanen, J. J. Vartiainen, and M. M. Salomaa, “Optimal multiqubit operations for Josephson charge qubits,” Phys. Rev. Lett. 90, 197901 (2003).
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2001 (3)

E. Dennis, “Toward fault-tolerant quantum computation without concatenation,” Phys. Rev. A 63, 052314 (2001).
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G. L. Long, “Grover algorithm with zero theoretical failure rate,” Phys. Rev. A 64, 022307 (2001).
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E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001).
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1998 (2)

D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, “Experimental quantum error correction,” Phys. Rev. Lett. 81, 2152 (1998).
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I. L. Chuang, N. Gershenfeld, M. G. Kubinec, and D. W. Leung, “Bulk quantum computation with nuclear magnetic resonance: theory and experiment,” Proc. R. Soc. Lond. A 454, 447–467 (1998).
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1997 (2)

P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Sci. Stat. Comput. 26, 1484–1509 (1997).
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1996 (2)

A. Lenef and S. C. Rand, “Electronic structure of the N-V center in diamond: Theory,” Phys. Rev. B 53, 13441–13455 (1996).
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1995 (3)

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457 (1995).
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T. Sleator and H. Weinfurter, “Realizable universal quantum logic gate,” Phys. Rev. Lett. 74, 4087 (1995).
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Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710 (1995).
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1982 (1)

E. Fredkin and T. Toffoli, “Conservative logic,” Int. J. Theor. Phys. 21, 219–253 (1982).
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Achard, J.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
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Ai, Q.

G. Y. Wang, T. Li, Q. Ai, and F. G. Deng, “self-error-corrected hyperparallel photonic quantum computation working with both the polarization and the spatial-mode degrees of freedom,” Opt. Express 26, 23333–23346 (2018).
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G. Y. Wang, Q. Ai, B. C. Ren, T. Li, and F. G. Deng, “Error-detected generation and complete analysis of hyperentangled Bell states for photons assisted by quantum-dot spins in double-sided optical microcavities,” Opt. Express 24, 28444–28458 (2016).
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X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
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X. K. Song, Hao Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
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M. J. Tao, M. Hua, Q. Ai, and F. G. Deng, “Quantum-information processing on nitrogen-vacancy ensembles with the local resonance assisted by circuit QED,” Phys. Rev. A 91, 062325 (2015).
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Alkemade, P. F. A.

L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578 (2011).
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Alsaedi, A.

M. Hua, M. J. Tao, A. Alsaedi, T. Hayat, and F. G. Deng, “Universal distributed quantum computing on superconducting qutrits with dark photons,” Ann. Phys. (Berlin) 530, 1700402 (2018).
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An, J. H.

J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009).
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Aolita, L.

Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012).
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Ates, C.

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
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Awschalom, D. D.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789–793 (2011).
[Crossref]

B. B. Buckley, G. D. Fuchs, L. C. Bassett, and D. D. Awschalom, “Spin-light coherence for single-spin measurement and control in diamond,” Science 330, 1212–1215 (2010).
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G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520–1522 (2009).
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Balasubramanian, G.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
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Barclay, P. E.

P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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Barenco, A.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457 (1995).
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Bassett, L. C.

B. B. Buckley, G. D. Fuchs, L. C. Bassett, and D. D. Awschalom, “Spin-light coherence for single-spin measurement and control in diamond,” Science 330, 1212–1215 (2010).
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Baur, M.

A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, “Implementation of a Toffoli gate with superconducting circuits,” Nature 481, 170–172 (2012).
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Beausoleil, R. G.

P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009).
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Beck, J.

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009).
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Beck, M. A.

J. D. Pritchard, J. A. Isaacs, M. A. Beck, R. McDermott, and M. Saffman, “Hybrid atom-photon quantum gate in a superconducting microwave resonator,” Phys. Rev. A 89, 010301 (2014).
[Crossref]

Bennett, C. H.

A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457 (1995).
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Bernien, H.

H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, “Heralded entanglement between solid-state qubits separated by 3 meters,” Nature 497, 86–90 (2013).
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L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578 (2011).
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Blok, M. S.

H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, “Heralded entanglement between solid-state qubits separated by 3 meters,” Nature 497, 86–90 (2013).
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Bonato, C.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. lett. 104, 160503 (2010).
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Borregaard, J.

J. Borregaard, P. Komar, E. M. Kessler, A. S. Sørensen, and M. D. Lukin, “Heralded quantum gates with integrated error detection in optical cavities,” Phys. Rev. Lett. 114, 110502 (2015).
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Bouwmeester, D.

C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Bell-state analysis in the weak-coupling cavity QED regime,” Phys. Rev. lett. 104, 160503 (2010).
[Crossref]

Buckley, B. B.

B. B. Buckley, G. D. Fuchs, L. C. Bassett, and D. D. Awschalom, “Spin-light coherence for single-spin measurement and control in diamond,” Science 330, 1212–1215 (2010).
[Crossref] [PubMed]

Bullock, S. S.

V. V. Shende, I. L. Markov, and S. S. Bullock, “Minimal universal two-qubit controlled-NOT-based circuits,” Phys. Rev. A 69, 062321 (2004).
[Crossref]

V. V. Shende, S. S. Bullock, and I. L. Markov, “Recognizing small-circuit structure in two-qubit operators,” Phys. Rev. A 70, 012310 (2004).
[Crossref]

Burkard, G.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centres in diamond,” Nat. Phys. 7, 789–793 (2011).
[Crossref]

Chang, D. E.

Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012).
[Crossref] [PubMed]

Chen, Q.

Q. Chen, W. Yang, M. Feng, and J. F. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011).
[Crossref]

Childress, L.

H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, “Heralded entanglement between solid-state qubits separated by 3 meters,” Nature 497, 86–90 (2013).
[Crossref] [PubMed]

L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578 (2011).
[Crossref] [PubMed]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Srensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730 (2010).
[Crossref] [PubMed]

M. V. G. Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science 316, 1312–1316 (2007).
[Crossref] [PubMed]

L. Childress, M. V. G. Dutt, J. M. Taylor, A. S. Zibrov, F. Jelezko, J. Wrachtrup, P. R. Hemmer, and M. D. Lukin, “Coherent dynamics of coupled electron and nuclear spin qubits in diamond,” Science 314, 281–285 (2006).
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Figures (5)

Fig. 1
Fig. 1 The schematic of a diamond NV center in a cavity and the possible Λ-type energy level. Quantum information is encoded in the spin states |+〉 and |−〉. The optical transitions |+〉→|A2〉 and |−〉 → |A2〉 are driven by a right-circular polarized photon (denoted by |R〉) and a left-circular polarized photon (denoted by |L〉), respectively.
Fig. 2
Fig. 2 Schematic of the robust photonic CNOT gate completing a bit-flip operation on photon 2 (the target qubit) when photon 1 (the control qubit) is in the right-circular polarization |R〉. PBS is the circular polarization beam splitter which transmits the right-circular polarized photon and reflects the left-circular polarized photon. H and H′ are half-wave plates which achieve the rotations, | R 1 2 ( | R + | L ), | L 1 2 ( | R | L ) and the rotations, | R 1 2 ( | R + | L ), | L 1 2 ( | L | R ), respectively. Rθ modifies the shape and the intensity of the photon passing through. D is a single-photon detector. SW is an optical switch which controls photons 1 and 2 passing through the system successively.
Fig. 3
Fig. 3 (a) Schematic of the robust photonic quantum Toffoli gate completing a bit-flip operation on photon 3 (the target qubit) when both photon 1 and photon 2 (the control qubits) are in the right-circular polarization |R〉. X achieves a bit-flip operation on the polarization of a photon passing through. (b) The order of the operations that photons interact with NV centers for Toffoli gate.
Fig. 4
Fig. 4 (a) Schematic of the robust photonic quantum Fredkin gate completing a polarization swap operation between photon 2 and photon 3 (the target qubits) if photon 1 (the control qubit) is in right-circular polarization |R〉. X achieves a bit-flip operation on the polarization of a photon passing through. T is the robust photonic Toffoli gate depicted in Section 4. (b) The order of operation that photons interact with NV centers for Fredkin gate.
Fig. 5
Fig. 5 The efficiencies vs. the cooperativity C = g ( κ s κ ) γ with leak rates κs = 0 (the red solid line), κs = 0.01κ (the blue dashed line), κs = 0.05κ (the brown dash-dot line), and κs = 0.10κ (the orange dotted line):(a) the efficiency ηC of the CNOT gate, (b) the efficiency ηT of the Toffoli gate, and (c) the efficiency ηF of the Fredkin gate. The pink solid vertical line indicates the achievable cooperativity C = 3 with current technology.

Equations (23)

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H ^ = ω 0 | A A | + ω c , L a ^ L a ^ L + ω c , R a ^ R a ^ R + g ( a ^ L σ ^ + a ^ L σ ^ + a ^ R σ ^ + + a ^ R σ ^ + ) ,
d a ^ d t = [ i ( ω c ω p ) + κ 2 + κ s 2 ] a ^ g σ ^ κ a ^ i n κ s s ^ i n , d σ ^ d t = [ i ( ω 0 ω p ) + γ 2 ] σ ^ ( t ) g σ ^ z a ^ + γ σ ^ z N ^ ,
r ( ω p ) = [ i ( ω c ω p ) κ 2 + κ s 2 ] [ i ( ω 0 ω p ) + γ 2 ] + g 2 [ i ( ω c ω p ) + κ 2 + κ s 2 ] [ i ( ω 0 ω p ) + γ 2 ] + g 2 .
r 1 = ( κ s κ ) γ + 4 g 2 ( κ s + κ ) γ + 4 g 2 , r 0 = ( κ s κ ) γ ( κ s + κ ) γ .
| ψ 1 = α 1 2 | R 1 ( | + | ) + β 1 2 | L 1 ( | + + | ) .
| ψ 2 = ( α 1 | R 1 | + β 1 | L 1 | + ) | ϕ p 2 .
| ψ 3 = α 1 | R 1 ( α 2 | L 2 + β 21 | R 2 ) | + β 1 | L 1 ( α 2 | R 2 + β 2 | L 2 ) | + .
| ψ 4 = | [ α 1 | R 1 ( α 2 | L 2 + β 2 | R 2 ) + β 1 | L 1 ( α 2 | R 2 + β 2 | L 2 ) ] + | + [ α 1 | R 1 ( α 2 | L 2 + β 2 | R 2 ) + β 1 | L 1 ( α 2 | R 2 + β 2 | L 2 ) ] .
| φ 1 = [ α 1 | R 1 ( | + 1 | 1 ) + β 1 | L 1 ( | + 1 + | 1 ) ] [ α 2 | R 2 ( | + 2 | 2 ) + β 2 | L 2 ( | + 2 + | 2 ) ] .
| φ 2 = ( α 1 | R 1 | 1 + β 1 | L 1 | + 1 ) ( α 2 | R 2 | 2 + β 2 | L 2 | + 2 ) .
| φ 3 = 1 2 ( α 3 β 3 ) | R 3 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 1 + β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) + 1 2 ( α 3 + β 3 ) | L 3 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 1 + β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) .
| τ 0 = | R 3 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 2 + β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) .
| τ 1 = 1 2 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 2 ) ( | R 3 + | L 3 ) + 1 2 ( β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) ( | R 3 + | L 3 ) .
| τ 2 = ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 2 ) | L 3 + ( β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) | R 3 .
| τ 3 = ( α 1 α 2 | R 1 | R 2 | 1 | 2 α 1 β 2 | R 1 | L 2 | 1 | + 2 ) | L 3 + ( β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) | R 3 .
| τ 4 = | R 3 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 2 + β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) .
| φ 5 = 1 2 ( α 3 + β 3 ) | L 3 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 1 + β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) + 1 2 ( α 3 β 3 ) | R 3 ( α 1 α 2 | R 1 | R 2 | 1 | 2 + α 1 β 2 | R 1 | L 2 | 1 | + 1 + β 1 α 2 | L 1 | R 2 | + 1 | 2 + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ) .
| φ 4 = α 1 α 2 | R 1 | R 2 | 1 | 2 ( α 3 | L 3 + β 3 | R 3 ) + α 1 β 2 | R 1 | L 2 | 1 | + 2 ( α 3 | R 3 + β 3 | L 3 ) + β 1 α 2 | L 1 | R 2 | + 1 | 2 ( α 3 | R 3 + β 3 | L 3 ) + β 1 β 2 | L 1 | L 2 | + 1 | + 2 ( α 3 | R 3 + β 3 | L 3 ) .
| φ 5 = α 1 α 2 | R 1 | R 2 ( α 3 | L 3 + β 3 | R 3 ) + α 1 β 2 | R 1 | L 2 ( α 3 | R 3 + β 3 | L 3 ) + β 1 α 2 | L 1 | R 2 ( α 3 | R 3 + β 3 | L 3 ) + β 1 β 2 | L 1 | L 2 ( α 3 | R 3 + β 3 | L 3 ) .
| ϑ 1 = 1 2 | x + 1 [ α 3 | R 3 ( α 2 | L 2 + β 2 | R 2 ) + β 3 | L 3 ( α 2 | R 2 + β 2 | L 2 ) ] | ϕ p 1 + 1 2 | x 1 [ α 3 | R 3 ( α 2 | L 2 + β 2 | R 2 ) + β 3 | L 3 ( α 2 | R 2 + β 2 | L 2 ) ] | ϕ p 1 .
| ϑ = α 1 | R 1 ( α 2 α 3 | L 2 | R 3 + β 2 α 3 | R 2 | L 3 + α 2 β 3 | R 2 | R 3 + β 2 β 3 | L 2 | L 3 ) | x + + β 1 | L 1 ( α 2 α 3 | L 2 | R 3 + β 2 α 3 | R 2 | R 3 + α 2 β 3 | R 2 | L 3 + β 2 β 3 | L 2 | L 3 ) | x + .
| ϑ = | 1 { α 1 | R 1 ( α 2 α 3 | R 2 | R 3 + β 2 α 3 | R 2 | L 3 + α 2 β 3 | L 2 | R 3 + β 2 β 3 | L 2 | L 3 ) + β 1 | L 1 ( α 2 α 3 | R 2 | R 3 + β 2 α 3 | L 2 | R 3 + α 2 β 3 | R 2 | L 3 + β 2 β 3 | L 2 | L 3 ) } + | + 1 { α 1 | R 1 ( α 2 α 3 | R 2 | R 3 + β 2 α 3 | R 2 | L 3 α 2 β 3 | L 2 | R 3 + β 2 β 3 | L 2 | L 3 ) + β 1 | L 1 ( α 2 α 3 | R 2 | R 3 β 2 α 3 | L 2 | R 3 + α 2 β 3 | R 2 | L 3 + β 2 β 3 | L 2 | L 3 ) } .
η C = T 1 , η T = T 1 + T 2 + T 1 2 T 2 3 , η F = T 2 + T 1 2 T 3 + T 1 2 T 2 2 T 3 3 ,