Abstract

In a recent remarkable experiment [Sci. Adv. 2, e1501531 (2016)], a 3-qubit quantum Fredkin (i.e., controlled-SWAP) gate was demonstrated by using linear optics. Here we propose a simple experimental scheme by utilizing the dispersive interaction in superconducting quantum circuit to implement a hybrid Fredkin gate with a superconducting flux qubit as the control qubit and two separated quantum memories as the target qudits. The quantum memories considered here are prepared by the superconducting coplanar waveguide resonators or nitrogen-vacancy center ensembles. In particular, it is shown that this Fredkin gate can be realized using a single-step operation and more importantly, each target qudit can be in an arbitrary state with arbitrary degrees of freedom. Furthermore, we show that this experimental scheme has many potential applications in quantum computation and quantum information processing such as generating arbitrary entangled states (discrete-variable states or continuous-variable states) of the two memories, measuring the fidelity and the entanglement between the two memories. With state-of-the-art circuit QED technology, the numerical simulation is performed to demonstrate that two-memory NOON states, entangled coherent states, and entangled cat states can be efficiently synthesized.

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

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2017 (6)

G. Wendin, “Quantum information processing with superconducting circuits: a review,” Rep. Prog. Phys. 80, 106001 (2017).
[Crossref] [PubMed]

F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, “Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime,” Nat. Phys. 13, 44–47 (2017).
[Crossref]

C. P. Yang, Q. P. Su, S. B. Zheng, F. Nori, and S. Han, “Entangling two oscillators with arbitrary asymmetric initial states,” Phys. Rev. A 95, 052341 (2017).
[Crossref]

T. Liu, Y. Zhang, B. Q. Guo, C. S. Yu, and W. N. Zhang, “Circuit QED: cross-Kerr effect induced by a superconducting qutrit without classical pulses,” Quantum Inf. Process. 16, 209 (2017).
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C. Song, K. Xu, W. Liu, C. P. Yang, S. B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y. A. Chen, C. Y. Lu, S. Han, and J. W. Pan, “10-qubit entanglement and parallel logic operations with a superconducting circuit,” Phys. Rev. Lett. 119, 180511 (2017).
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R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, “Implementing a universal gate set on a logical qubit encoded in an oscillator,” Nat. Commun. 8, 94 (2017).
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2016 (7)

F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, “The flux qubit revisited to enhance coherence and reproducibility,” Nat. Commun. 7, 12964 (2016).
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M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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C. Axline, M. Reagor, R. Heeres, P. Reinhold, C. Wang, K. Shain, W. Pfaff, Y. Chu, L. Frunzio, and R. J. Schoelkopf, “An architecture for integrating planar and 3D cQED devices,” Appl. Phys. Lett. 109, 042601 (2016).
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R. B. Patel, J. Ho, F. Ferreyrol, T. C. Ralph, and G. J. Pryde, “A quantum Fredkin gate,” Sci. Adv. 2, e1501531 (2016).
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R. Sharma and F. W. Strauch, “Quantum state synthesis of superconducting resonators,” Phys. Rev. A 93, 012342 (2016).
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Y. J. Zhao, C. Q. Wang, X. B. Zhu, and Y. X. Liu, “Engineering entangled microwave photon states through multiphoton interactions between two cavity fields and a superconducting qubit,” Sci. Rep. 6, 23646 (2016).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schröinger cat living in two boxes,” Science 352, 1087–1091 (2016).
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2015 (3)

L. C. Song, Y. Xia, and J. Song, “Experimentally optimized implementation of the Fredkin gate with atoms in cavity QED,” Quantum Inf. Process. 14, 511–529 (2015).
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H. R. Wei and G. L. Long, “Hybrid quantum gates between flying photon and diamond nitrogen-vacancy centers assisted by optical microcavities,” Sci. Rep. 5, 12918 (2015).
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B. C. Ren, G. Y. Wang, and F. G. Deng, “Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime,” Phys. Rev. A 91, 032328 (2015).
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2014 (6)

X. Q. Shao, T. Y. Zheng, X. L. Feng, C. H. Oh, and S. Zhang, “One-step implementation of the genuine Fredkin gate in high-Q coupled three-cavity arrays,” J. Opt. Soc. Am. B 31, 697–703 (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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Q. P. Su, C. P. Yang, and S. B. Zheng, “Fast and simple scheme for generating NOON states of photons in circuit QED,” Sci. Rep. 4, 3898 (2014).
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S. L. Ma, Z. Li, A. P. Fang, P. B. Li, S. Y. Gao, and F. L. Li, “Controllable generation of two-mode-entangled states in two-resonator circuit QED with a single gap-tunable superconducting qubit,” Phys. Rev. A 90, 062342 (2014).
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F. Yoshihara, Y. Nakamura, F. Yan, S. Gustavsson, J. Bylander, W. D. Oliver, and J. S. Tsai, “Flux qubit noise spectroscopy using Rabi oscillations under strong driving conditions,” Phys. Rev. B 89, 020503 (2014).
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I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devore, “Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,” Nature 508, 369–372 (2014).
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2013 (6)

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 three metres,” Nature 497, 86–90 (2013).
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B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Deterministically encoding quantum information using 100-Photon Schröinger cat states,” Science 342, 607–610 (2013).
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M. Reagor, H. Paik, G. Catelani, L. Sun, C. Axline, E. Holland, I. M. Pop, N. A. Masluk, T. Brecht, L. Frunzio, M. H. Devoret, L. Glazman, and R. J. Schoelkopf, “Reaching 10 ms single photon lifetimes for superconducting aluminum cavities,” Appl. Phys. Lett. 102, 192604 (2013).
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N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, “Solid-state electronic spin coherence time approaching one second,” Nat. Commun. 4, 1743 (2013).
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S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
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Z. L. Xiang, X. Y. Lü, T. F. Li, J. Q. You, and F. Nori, “Hybrid quantum circuit consisting of a superconducting flux qubit coupled to a spin ensemble and a transmission-line resonator,” Phys. Rev. B 87, 144516 (2013).
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2012 (7)

T. Hümmer, G. M. Reuther, P. Hänggi, and D. Zueco, “Nonequilibrium phases in hybrid arrays with flux qubits and nitrogen-vacancy centers,” Phys. Rev. A 85, 052320 (2012).
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X. C. Yao, T. X. Wang, P. Xu, H. Lu, G. S. Pan, X. H. Bao, C. Z. Peng, C. Y. Lu, Y. A. Chen, and J. W. Pan, “Observation of eight-photon entanglement,” Nat. Photonics 6, 225–228 (2012).
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F. W. Strauch, D. Onyango, K. Jacobs, and R. W. Simmonds, “Entangled-state synthesis for superconducting resonators,” Phys. Rev. A 85, 022335 (2012).
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X. Q. Shao, T. Y. Zheng, and S. Zhang, “Fast synthesis of the Fredkin gate via quantum Zeno dynamics,” Quantum Inf. Process. 11, 1797–1808 (2012).
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Q. Chen, W. L. Yang, and M. Feng, “Controllable quantum state transfer and entanglement generation between distant nitrogen-vacancy-center ensembles coupled to superconducting flux qubits,” Phys. Rev. A 86, 022327 (2012).
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H. F. Hofmann, “Weak values emerge in joint measurements on cloned quantum systems,” Phys. Rev. Lett. 109, 020408 (2012).
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C. S. Yu, J. Zhang, and H. Fan, “Quantum dissonance is rejected in an overlap measurement scheme,” Phys. Rev. A 86, 052317 (2012).
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2011 (10)

R. Amsüss, Ch. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, “Cavity QED with magnetically coupled collective spin states,” Phys. Rev. Lett. 107, 060502 (2011).
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Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J. F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. Karimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, “Coherent coupling of a superconducting flux qubit to an electron spin ensemble in diamond,” Nature 478, 221–224 (2011).
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J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589–597 (2011).
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M. Mariantoni, H. Wang, T. Yamamoto, M. Neeley, Radoslaw C. Bialczak, Y. Chen, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, Y. Yin, J. Zhao, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Implementing the quantum von Neumann architecture with superconducting circuits,” Science 334, 61–65 (2011).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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W. L. Yang, Y. Hu, Z. Q. Yin, Z. J. Deng, and M. Feng, “Entanglement of nitrogen-vacancy-center ensembles using transmission line resonators and a superconducting phase qubit,” Phys. Rev. A 83, 022302 (2011).
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J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107, 083601 (2011).
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Y. F. Huang, B. H. Liu, L. Peng, Y. H. Li, L. Li, C. F. Li, and G. C. Guo, “Experimental generation of an eight-photon Greenberger-Horne-Zeilinger state,” Nat. Commun. 2, 546 (2011).
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T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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2010 (9)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
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S. T. Merkel and F. K. Wilhelm, “Generation and detection of NOON states in superconducting circuits,” New J. Phys. 12, 093036 (2010).
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F. W. Strauch, K. Jacobs, and R. W. Simmonds, “Arbitrary control of entanglement between two superconducting resonators,” Phys. Rev. Lett. 105, 050501 (2010).
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C. Simon, M. Afzelius, J. Appel, A. Boyer de la Giroday, S. J. Dewhurst, N. Gisin, C. Y. Hu, F. Jelezko, S. Kröll, J. H. Müller, J. Nunn, E. S. Polzik, J. G. Rarity, H. De Riedmatten, W. Rosenfeld, A. J. Shields, N. Sköld, R. M. Stevenson, R. Thew, I. A. Walmsley, M. C. Weber, H. Weinfurter, J. Wrachtrup, and R. J. Young, “Quantum memories,” Eur. Phys. J. D 58, 1–22 (2010).
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T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nat. Physics 6, 772–776 (2010).
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P. Forn-Díaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, “Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime,” Phys. Rev. Lett. 105, 237001 (2010).
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M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, M. Mariantoni, A. D. O’Connell, D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, T. Yamamoto, A. N. Cleland, and J. M. Martinis, “Generation of three-qubit entangled states using superconducting phase qubits,” Nature 467, 570–573 (2010).
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L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Preparation and measurement of three-qubitentanglement in a superconducting circuit,” Nature 467, 574–578 (2010).
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P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen, S. Filipp, and A. Wallraff, “Cavity quantum electrodynamics with separate photon storage and qubit readout modes,” Phys. Rev. Lett. 104, 100504 (2010).
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2009 (4)

M. Baur, S. Filipp, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, P. J. Leek, A. Blais, and A. Wallraff, “Measurement of Autler-Townes and Mollow transitions in a strongly driven superconducting qubit,” Phys. Rev. Lett. 102, 243602 (2009).
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M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, John M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature 459, 546–549 (2009).
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P. Neumann, R. Kolesov, V. Jacques, J. Beck, J. Tisler, A. Batalov, L. Rogers, N. B. Manson, G. Balasubramanian, F. Jelezko, and J. Wrachtrup, “Excited-state spectroscopy of single NV defects in diamond using optically detected magnetic resonance,” New J. Phys. 11, 013017 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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2008 (7)

P. van Loock, N. Lütkenhaus, W. J. Munro, and K. Nemoto, “Quantum repeaters using coherent-state communication,” Phys. Rev. A 78, 062319 (2008).
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P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, and J. Wrachtrup, “Multipartite entanglement among single spins in diamond,” Science 320, 1326–1329 (2008).
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Y. X. Gong, G. C. Guo, and T. C. Ralph, “Methods for a linear optical quantum Fredkin gate,” Phys. Rev. A 78, 012305 (2008).
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J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
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M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78, 104508 (2008).
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M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in a superconducting quantum circuit,” Nature 454, 310–314 (2008).
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W. Chen, D. A. Bennett, V. Patel, and J. E. Lukens, “Substrate and process dependent losses in superconducting thin film resonators, Supercond,” Sci. Technol. 21, 075013 (2008).

2007 (1)

D. F. James and J. Jerke, “Effective Hamiltonian theory and its applications in quantum information,” Can. J. Phys. 85, 625–632 (2007).
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2006 (1)

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 (1)

R. T. Horn, S. A. Babichev, K. P. Marzlin, A. I. Lvovsky, and B. C. Sanders, “Single-qubit optical quantum fingerprinting,” Phys. Rev. Lett. 95, 150502 (2005).
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2004 (3)

A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
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I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, “Coherent dynamics of a flux qubit coupled to a harmonic oscillator,” Nature 431, 159–162 (2004).
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2003 (2)

G. K. Brennen, “An observable measure of entanglement for pure states of multi-qubit systems,” Quantum Inf. Comput. 3, 619–626 (2003).

P. Horodecki, “Measuring Quantum Entanglement without Prior State Reconstruction,” Phys. Rev. Lett. 90, 167901 (2003).
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2002 (3)

R. Filip, “Overlap and entanglement-witness measurements,” Phys. Rev. A 65, 062320 (2002).
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F. Grosshans and P. Grangier, “Continuous variable quantum cryptography using coherent states,” Phys. Rev. Lett. 88, 057902 (2002).
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P. Kok, H. Lee, and J. P. Dowling, “Creation of large-photon-number path entanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
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2001 (3)

S. J. van Enk and O. Hirota, “Entangled coherent states: Teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
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S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, “Coherent control of an atomic collision in a cavity,” Phys. Rev. Lett. 87, 037902 (2001).
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H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, “Quantum fingerprinting,” Phys. Rev. Lett. 87, 167902 (2001).
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2000 (3)

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Step-by-step engineered multiparticle entanglement,” Science 288, 2024–2028 (2000).
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C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, “Experimental entanglement of four particles,” Nature 404, 256–259 (2000).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
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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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W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245 (1998).
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1996 (3)

J. A. Smolin and D. P. DiVincenzo, “Five two-bit quantum gates are sufficient to implement the quantum Fredkin gate,” Phys. Rev. A 53, 2855 (1996).
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1995 (1)

H. F. Chau and F. Wilczek, “Simple realization of the Fredkin gate using a series of two-body operators,” Phys. Rev. Lett. 75, 748 (1995).
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Y. Kubo, C. Grezes, A. Dewes, T. Umeda, J. Isoya, H. Sumiya, N. Morishita, H. Abe, S. Onoda, T. Ohshima, V. Jacques, A. Dréau, J. F. Roch, I. Diniz, A. Auffeves, D. Vion, D. Esteve, and P. Bertet, “Hybrid quantum circuit with a superconducting qubit coupled to a spin ensemble,” Phys. Rev. Lett. 107, 220501 (2011).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733 (2000).
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I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
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R. Amsüss, Ch. Koller, T. Nöbauer, S. Putz, S. Rotter, K. Sandner, S. Schneider, M. Schramböck, G. Steinhauser, H. Ritsch, J. Schmiedmayer, and J. Majer, “Cavity QED with magnetically coupled collective spin states,” Phys. Rev. Lett. 107, 060502 (2011).
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M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, John M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature 459, 546–549 (2009).
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C. Simon, M. Afzelius, J. Appel, A. Boyer de la Giroday, S. J. Dewhurst, N. Gisin, C. Y. Hu, F. Jelezko, S. Kröll, J. H. Müller, J. Nunn, E. S. Polzik, J. G. Rarity, H. De Riedmatten, W. Rosenfeld, A. J. Shields, N. Sköld, R. M. Stevenson, R. Thew, I. A. Walmsley, M. C. Weber, H. Weinfurter, J. Wrachtrup, and R. J. Young, “Quantum memories,” Eur. Phys. J. D 58, 1–22 (2010).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schröinger cat living in two boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

Other (2)

M. Hua, M. J. Tao, and F. G. Deng, “One-step implementation of entanglement generation on microwave photons in distant 1D superconducting resonators,” https://arxiv.org/abs/1508.00061 .

D. Gottesman and I. Chuang, “Quantum digital signatures,” https://arxiv.org/abs/quant-ph/0105032 .

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Figures (4)

Fig. 1
Fig. 1

(a) Setup of two superconducting resonators (i.e., 1 and 2) coupled to a flux qutrit (coupler) via capacitances C1 and C2. (b) Resonator 1 (2) is far-off resonant with |g〉 ↔ |a〉 transition of coupler with coupling strength g1 (g2) and detuning δ1 (δ). Here, the detunings δ 1 = ω a g ω a 1 and δ 2 = ω a g ω a 2, the ωag is the |g〉 ↔ |a〉 transition frequency of coupler and the ω a 1 ( ω a 2 ) is the frequency of resonator 1 (2).

Fig. 2
Fig. 2

(a) Energy level diagram of the flux and an NV center. By applying an external magnetic field along the crystalline axis of the NV center, an additional Zeeman splitting between |ms = ±1〉 sublevels occurs. Here, ωag is the |g〉 ↔ |a〉 transition frequency of flux qubit and ω0,+1 is the energy gap between the |ms = 0〉 and |ms = +1〉 levels of the NV center. (b) Illustration of the hybrid system consisted of a flux qubit and two NV ensembles. (c) The NV ensemble 1 (2) is far-off resonant with |g〉 ↔ |a〉 transition of qubit with the coupling strength µ1 (µ2) and the detuning Δ12).

Fig. 3
Fig. 3

Fidelity ℱ versus D = δ/g. (a) Fidelity versus D for NOON states. (b) Fidelity versus D for entangled coherent states. (c) Fidelity versus D for entangled cat states. The parameters used in the numerical simulation are referred in the text.

Fig. 4
Fig. 4

Fidelity versus c and d, which are plotted by choosing (a) D = 16, (b) D = 10, and (c) D = 22, respectively.

Equations (25)

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U ( γ | g + η | e ) | ψ 1 | φ 2 = γ | g | φ 1 | ψ 2 + η | e | ψ 1 | φ 2 ,
H I , 1 = g 1 ( e i δ 1 t a 1 σ a g + + h . c . ) + g 2 ( e i δ 2 t a 2 σ a g + + h . c . ) ,
H e = ( g 1 2 δ 1 a 1 a 1 + g 2 2 δ 2 a 2 a 2 ) | a a | ( g 1 2 δ 1 a 1 a 1 + g 2 2 δ 2 a 2 a 2 ) | g g | + λ ( e i δ t a 1 a 2 + e i δ t a 1 a 2 ) | a a | λ ( e i δ t a 1 a 2 + e i δ t a 1 a 2 ) | g g | ,
δ 1 = δ 2 = δ , g 1 = g 2 = g ,
H e = H 0 + H i
H 0 = ω ( a 1 a 1 + a 2 a 2 ) | g g | , H i = λ ( a 1 a 2 + a 1 a 2 ) | g g | ,
U | ϕ q | ψ 1 | φ 2 = e i H 0 t e i H i t ( γ | g + η | e ) | ψ 1 | φ 2 = e i ω ( a 1 a 1 + a 2 a 2 ) t | g g | e i λ ( a 1 a 2 + a 1 a 2 ) t | g g | ( γ | g + η | e ) | ψ 1 | φ 2 = e i ω ( a 1 a 1 + a 2 a 2 ) t e i λ ( a 1 a 2 + a 1 a 2 ) t γ | g | ψ 1 | φ 2 + η | e | ψ 1 | φ 2 = e i H 0   t e i H i   t γ | g | ψ 1 | φ 2 + η | e | ψ 1 | φ 2 ,
U | ϕ q | ψ 1 | φ 2 = e i H 0   t e i H i   t γ | g | ψ 1 | φ 2 + η | e | ψ 1 | φ 2 = e i H 0   t γ | g n = 0 m = 0 c n n ! d m m ! e i H i   t ( a 1 ) n ( a 2 ) m | 0 1 | 0 2 + η | e | ψ 1 | φ 2 = e i H 0   t γ | g n = 0 m = 0 c n n ! d m m ! [ e i H i   t ( a 1 ) n e i H i   t ] [ e i H i   t ( a 2 ) m e i H i   t ] e i H i   t | 0 1 | 0 2 + η | e | ψ 1 | φ 2 = e i H 0   t γ | g m = 0 d m m ! ( i a 1 ) m | 0 1 n = 0 c n n ! ( i a 2 ) n | 0 2 + η | e | ψ 1 | φ 2 = γ | g | φ 1 | ψ 2 + η | e | ψ 1 | φ 2 ,
H F N = k = 1 N μ k ( σ a g + τ k e i Δ t + σ a g τ k + e i Δ t ) ,
b = ( 1 N ) ( 1 μ ¯ ) k = 1 N μ k τ k + ,
H F N = μ ( e i Δ t b σ a g + + e i Δ t b σ a g ) ,
H I , 1 = μ 1 ( e i Δ 1 t b 1 σ a g + + h . c . ) + μ 2 ( e i Δ 2 t b 2 σ a g + + h . c . ) ,
H e = H 0 + H i
H 0 = ω ( b 1 b 1 + b 2 b 2 ) | g g | , H i = λ ( b 1 b 2 + b 1 b 2 ) | g g | ,
| ϕ q | ψ 1 | φ 2 1 2 | g ( γ | φ 1 | ψ 2 η | ψ 1 | φ 2 ) + 1 2 | e ( γ | φ 1 | ψ 2 η | ψ 1 | φ 2 ) .
p g = 1 2 [ 1 ( γ * η + γ η * ) F 2 ]
C ( | ψ ± ) = 2 [ 1 Tr ( ρ r 2 ) ]
C ( | ψ ± ) = 2 1 2 { | γ | 4 + | η | 4 + 2 [ 2 | γ | 2 | η | 2 ± ( γ η * + γ * η ) ] F 2 + ( γ 2 η * 2 + γ * 2 η 2 ) F 4 } .
( i ) | ϕ q | N 1 | 0 2 ,
( ii ) | ϕ q | α 1 | β 2 ,
( iii ) | ϕ q | ψ e 1 | φ o 2 ,
d ρ d t = i [ H I , k , ρ ] + i = 1 , 2 κ i [ a i ] + γ a g [ σ a g ] + γ e a [ σ e a ] + γ e g [ σ e g ] + j = a , e { γ φ j ( σ j j ρ σ j j σ j j ρ / 2 ρ σ j j / 2 ) } ,
( i ) | N 1 | 0 2 ± | 0 1 | N 2 ,
( ii ) | α 1 | β 2 ± | β 1 | α 2 ,
( iii ) | ψ e 1 | φ o 2 ± | φ o 1 | ψ e 2 .