Abstract

We present a novel method to realize a multi-target-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling n − 1 target microwave photonic qubits. This gate is implemented with n microwave cavities coupled to a superconducting flux qutrit. Each cavity hosts a microwave photonic qubit, whose two logic states are represented by the vacuum state and the single photon state of a single cavity mode, respectively. During the gate operation, the qutrit remains in the ground state and thus decoherence from the qutrit is greatly suppressed. This proposal requires only a single-step operation and thus the gate implementation is quite simple. The gate operation time is independent of the number of the qubits. In addition, this proposal does not need applying classical pulse or any measurement. Numerical simulations demonstrate that high-fidelity realization of a controlled phase gate with one microwave photonic qubit simultaneously controlling two target microwave photonic qubits is feasible with current circuit QED technology. The proposal is quite general and can be applied to implement the proposed gate in a wide range of physical systems, such as multiple microwave or optical cavities coupled to a natural or artificial Λ-type three-level atom.

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

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2018 (1)

B. Ye, Z. F. Zheng, and C. P. Yang, “Multiplex-controlled phase gate with qubits distributed in a multicavity system,” Phys. Rev. A 97, 062336 (2018).
[Crossref]

2017 (5)

Q. Wei, 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]

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]

Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96, 052317 (2017).
[Crossref]

X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718, 1–102 (2017).
[Crossref]

Qi-Ping Su, H. H. Zhu, L. Yu, Y. Zhang, S. J. Xiong, J. M. Liu, and C. P. Yang, “Generating double NOON states of photons in circuit QED,” Phys. Rev. A 95, 022339 (2017).
[Crossref]

2016 (7)

C. P. Yang, Q. P. Su, S. B. Zheng, and F. Nori, “Crosstalk-insensitive method for simultaneously coupling multiple pairs of resonators,” Phys. Rev. A 93, 042307 (2016).
[Crossref]

T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multicavity system,” Sci. Rep. 6, 21562 (2016).
[Crossref]

F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, 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)
[Crossref]

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, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
[Crossref]

C. H. Bai, D. Y. Wang, S. Hu, W. X. Cui, X. X. Jiang, and H. F. Wang, “Scheme for implementing multitarget qubit controlled-NOT gate of photons and controlled-phase gate of electron spins via quantum dot-microcavity coupled system,” Quantum. Inf. Process 15, 1485–1498 (2016).
[Crossref]

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 Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

M. Hua, M. J. Tao, and F. G. Deng, “Quantum state transfer and controlled-phase gate on one-dimensional superconducting resonators assisted by a quantum bus,” Sci. Rep. 6, 22037 (2016).
[Crossref] [PubMed]

2015 (4)

M. Hua, M. J. Tao, and F. G. Deng, “Fast universal quantum gates on microwave photons with all-resonance operations in circuit QED,” Sci. Rep. 5, 9274 (2015).
[Crossref] [PubMed]

E. A. Sete, E. Mlinar, and A. N. Korotkov, “Robust quantum state transfer using tunable couplers,” Phys. Rev. B 91, 144509 (2015).
[Crossref]

M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015).
[Crossref] [PubMed]

S. J. Xiong, Z. Sun, J. M. Liu, T. Liu, and C. P. Yang, “Efficient scheme for generation of photonic NOON states in circuit QED,” Opt. Lett. 40, 2221–2224 (2015).
[Crossref] [PubMed]

2014 (13)

S. E. Nigg, “Deterministic hadamard gate for microwave cat-state qubits in circuit QED,” Phys. Rev. A 89, 022340 (2014).
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H. F. Wang, A. D. Zhu, and S. Zhang, “One-step implementation of multiqubit phase gate with one control qubit and multiple target qubits in coupled cavities,” Opt. Lett. 39, 1489–1492 (2014).
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C. P. Yang, Q. P. Su, F. Y. Zhang, and S. B. Zheng, “Single-step implementation of a multiple-target-qubit controlled phase gate without need of classical pulses,” Opt. Lett. 39, 3312–3315 (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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M. Hua, M. J. Tao, and F. G. Deng, “Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics,” Phys. Rev. A 90, 18824 (2014).
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Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, “Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,” Nature (London) 508, 369–372 (2014).
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S. J. Srinivasan, N. M. Sundaresan, D. Sadri, Y. Liu, J. M. Gambetta, T. Yu, S. M. Girvin, and A. A. Houck, “Time-reversal symmetrization of spontaneous emission for quantum state transfer,” Phys. Rev. A 89, 033857 (2014).
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J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
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J. R. Johansson, N. Lambert, I. Mahboob, H. Yamaguchi, and F. Nori, “Entangled-state generation and Bell inequality violations in nanomechanical resonators,” Phys. Rev. B 90, 174307 (2014).
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H. W. Wei and F. G. Deng, “Scalable quantum computing based on stationary spin qubits in coupled quantum dots inside double-sided optical microcavities,” Sci. Rep. 4, 7551 (2014).
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M. Hua, M. J. Tao, and F. G. Deng, “Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics,” Phys. Rev. A 90, 012328 (2014).
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2013 (8)

S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (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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H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Deterministic CNOT gate and entanglement swapping for photonic qubits using a quantum-dot spin in a double-sided optical microcavity,” Phys. Lett. A 377, 2870 (2013).
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Z. L. Wang, Y. P. Zhong, L. J. He, H. Wang, J. M. Martinis, A. N. Cleland, and Q. W. Xie, “Quantum state characterization of a fast tunable superconducting resonator,” Appl. Phys. Lett. 102, 163503 (2013).
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C. P. Yang, Q. P. Su, S. B. Zheng, and S. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
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R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
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M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: an outlook,” Science 339, 1169–1174 (2013).
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2012 (2)

C. Rigetti, S. Poletto, J. M. Gambetta, B. L. T. Plourde, J. M. Chow, A. D. Corcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in waveguide cavity with coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506(R) (2012).
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C. P. Yang, Q. P. Su, and S. Y. Han, “Generation of Greenberger-Horne-Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction,” Phys. Rev. A 86, 022329 (2012).
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2011 (6)

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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A. N. Korotkov, “Flying microwave qubits with nearly perfect transfer efficiency,” Phys. Rev. B 84, 014510 (2011).
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J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. F. David, G. Cory, Y. Nakamura, J. S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys. 7, 565–570 (2011).
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H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature (London) 474, 589–597 (2011).
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2010 (6)

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. Phys. 6, 772–776 (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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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. P. Yang, Y. X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
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C. P. Yang, S. B. Zheng, and F. Nori, “Multiqubit tunable phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 82, 062326 (2010).
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W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One step implementation of multi-qubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silicamicro sphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
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2009 (4)

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the Quantum Toffoli Gate with Trapped Ions,” Phys. Rev. Lett. 102, 040501 (2009).
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P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
[Crossref]

M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009).
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H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009).
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2008 (5)

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 (London) 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).
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J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
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M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4, 523–526 (2008).
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M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon life time,” Appl. Phys. Lett. 92, 203501 (2008).
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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 (2)

X. Zou, Y. Dong, and G. C. Guo, “Implementing a conditional z gate by a combination of resonant interaction and quantum interference,” Phys. Rev. A 74, 032325 (2006).
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C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
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2005 (3)

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58, 42–47 (2005).
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C. P. Yang and S. Han, “n-qubit-controlled phase gate with superconducting quantum-interference devices coupled to a resonator,” Phys. Rev. A 72, 032311 (2005).
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L. M. Duan, B. Wang, and H. J. Kimble, “Robust quantum gates on neutral atoms with cavity-assisted photonscattering,” Phys. Rev. A 72, 032333 (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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Y. X. Liu, L. F. Wei, and F. Nori, “Generation of nonclassical photon states using a supercon ducting qubit in a microcavity,” Europhys. Lett. 67, 941–947 (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 (London) 431, 162–167 (2004).
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2003 (2)

C. P. Yang, S. I. Chu, and S. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
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J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
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2001 (3)

X. Wang, A. Søensen, and K. Mømeret, “Multibit Gates for Quantum Computing,” Phys. Rev. Lett. 86, 3907 (2001).
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M. Šašura and V. Buzek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
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S. L. Braunstein, V. Bužek, and M. Hillery, “Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit,” Phys. Rev. A 63, 052313 (2001).
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Ansmann, M.

M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009).
[Crossref]

H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009).
[Crossref]

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 (London) 454, 310–314 (2008).
[Crossref]

M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4, 523–526 (2008).
[Crossref]

Ashhab, S.

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
[Crossref]

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Axline, C.

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, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (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 Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
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Bader, S. J.

M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015).
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Bai, C. H.

C. H. Bai, D. Y. Wang, S. Hu, W. X. Cui, X. X. Jiang, and H. F. Wang, “Scheme for implementing multitarget qubit controlled-NOT gate of photons and controlled-phase gate of electron spins via quantum dot-microcavity coupled system,” Quantum. Inf. Process 15, 1485–1498 (2016).
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Barends, R.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
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R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
[Crossref] [PubMed]

Bauch, T.

M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon life time,” Appl. Phys. Lett. 92, 203501 (2008).
[Crossref]

Baur, M.

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).
[Crossref] [PubMed]

P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
[Crossref]

Bennett, D. A.

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).
[Crossref]

Bertet, P.

M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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Beth, T.

T. Beth and M. Rötteler, Quantum Information (Springer, Berlin, 2001), Vol. 173, Ch. 4, p. 96.
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Bialczak, R. C.

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).
[Crossref] [PubMed]

H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009).
[Crossref]

M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009).
[Crossref]

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 (London) 454, 310–314 (2008).
[Crossref]

M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4, 523–526 (2008).
[Crossref]

Bianchetti, R.

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).
[Crossref] [PubMed]

P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
[Crossref]

Bienfait, A.

M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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Birenbaum, J.

F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, 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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Bishop, L. S.

H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
[Crossref]

Blais, A.

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 (London) 431, 162–167 (2004).
[Crossref]

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).
[Crossref]

Blatt, R.

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the Quantum Toffoli Gate with Trapped Ions,” Phys. Rev. Lett. 102, 040501 (2009).
[Crossref] [PubMed]

Blumoff, J.

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 Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

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, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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[Crossref]

Rep. Prog. Phys. (1)

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Rev. Mod. Phys. (1)

Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
[Crossref]

Sci. Rep. (5)

T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multicavity system,” Sci. Rep. 6, 21562 (2016).
[Crossref]

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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A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,” arXiv:1807.11636.

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F. Gaitan, Quantum Error Correction and Fault Tolerant Quantum Computing (CRC Press, USA, 2008).
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[Crossref]

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Figures (7)

Fig. 1
Fig. 1 (a) Diagram of n cavities (1, 2, ..., n) coupled to a superconducting flux qutrit A. A square represents a cavity, which can be a one-dimensional or three-dimensional cavity. The qutrit is capacitively or inductively coupled to each cavity. (b) Level configuration of the flux qutrit, for which the transition between the two lowest levels can be made weak by increasing the barrier between two potential wells. (c) Diagram of a flux qutrit, which consists of three Josephson junctions and a superconducting loop.
Fig. 2
Fig. 2 Cavity 1 is dispersively coupled to the |g〉 ↔ |f〉 transition of the qutrit with coupling strength g1 and detuning δ1, while cavity l (l = 2, 3, ..., n) is dispersively coupled to the |e〉 ↔ |f〉 transition of the qutrit with coupling strength gl and detuning δl. The purple vertical line represents the frequency ωc1 of cavity 1, while the blue, green, ..., and red vertical lines represent the frequency ωc2 of cavity 2, the frequency ωc3 of cavity 3,..., and the frequency ωcn of cavity n, respectively.
Fig. 3
Fig. 3 Setup for three one-dimensional transmission line resonators capacitively coupled to a superconducting flux qutrit.
Fig. 4
Fig. 4 Illustration of the unwanted coupling between cavity 1 and the |e〉 ↔ |f〉 transition of the qutrit (with coupling strength 1 and detuning δ̃1) as well as the unwanted coupling between cavity l and the |g〉 ↔ |f〉 transition of the qutrit (with coupling strength l and detuning δ̃l) (l = 2, 3). Note that the coupling of each cavity with the |g〉 ↔ |e〉 transition of the qutrit is negligible because of the weak |g〉 ↔ |f〉 transition.
Fig. 5
Fig. 5 Fidelity versus T and κ−1. The parameters used in the numerical simulation are referred to the text.
Fig. 6
Fig. 6 Fidelity versus . Here, is the detuning error, which applies to each of detunings δ1, δ2, and δ3. The figure is plotted for T = 5 μs and κ−1 = 10 μs. Other parameters used in the numerical simulation are the same as those used in Fig. 5.
Fig. 7
Fig. 7 Schematic diagram for n cavities coupled by a superconducting flux qutrit. Each cavity here is a one-dimensional transmission line resonator, which is coupled to the qutrit via a capacitor.

Equations (24)

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| 0 1 | i 2 | i 3 | i n | 0 1 | i 2 | i 3 | i n , | 1 1 | i 2 | i 3 | i n | 1 1 ( 1 ) i 2 ( 1 ) i 3 ( 1 ) i n | i 2 | i 3 | i n ,
H I = g 1 ( e i δ 1 t a ^ 1 + σ f g + h . c . ) + l = 2 n g l ( e i δ l t a ^ l + σ f e + h . c . ) ,
H e = λ 1 ( a ^ 1 + a ^ 1 | g g | a ^ 1 a ^ 1 + | f f | ) l = 2 n λ l ( a ^ l + a ^ l | e e | a ^ l a ^ l + | f f | ) l = 2 n λ 1 l ( e i Δ 1 l t a ^ 1 + a ^ l σ e g + h . c . ) + k l ; k , l = 2 n λ k l ( e i Δ k l t a ^ k + a ^ l + h . c . ) ( | f f | | e e | ) ,
H e = λ 1 ( a ^ 1 + a ^ 1 | g g | a ^ 1 a ^ 1 + | f f | ) l = 2 n λ l ( a ^ l + a ^ l | e e | a ^ l a ^ l + | f f | ) l = 2 n χ 1 l ( a ^ 1 + a ^ 1 a ^ l a ^ l + | g g | a ^ 1 a ^ 1 + a ^ l + a ^ l | e e | ) + k l ; k , l = 2 n λ k l ( e i Δ k l t a ^ k + a ^ l + h . c . ) ( | f f | | e e | ) ,
H e = λ 1 a ^ 1 + a ^ 1 | g g | l = 2 n χ 1 l a ^ 1 + a ^ 1 a ^ l a ^ l + | g g | .
H e = λ 1 n ^ 1 | g g | l = 2 n χ 1 l n ^ 1 | g g | l = 2 n χ 1 l n ^ 1 n ^ l | g g | ,
H ˜ e = η n ^ 1 χ l = 2 n n ^ 1 n ^ l ,
U = U 1 [ l = 2 n U 1 l ] ,
U 1 = exp ( i η n ^ 1 t ) ,
U 1 l = exp ( i χ n ^ 1 n ^ l t ) ,
U 1 l | 0 1 0 l = | 0 1 0 l , U 1 l | 0 1 1 l = | 0 1 1 l , U 1 l | 1 1 0 l = | 1 1 0 l , U 1 l | 1 1 1 l = | 1 1 1 l ,
U 1 l | 0 1 i l | g = | 0 1 i l | g U 1 l | 1 1 i l | g = ( 1 ) i 1 | 1 1 i l | g ,
l = 2 n U 1 l | 0 1 | i 2 | i 3 | i n = | 0 1 | i 2 | i 3 | i n , l = 2 n U 1 l | 1 1 | i 2 | i 3 | i n = | 1 1 ( 1 ) i 2 ( 1 ) i 3 ( 1 ) i n | i 2 | i 3 | i n .
U 1 | 0 1 = | 0 1 , U 1 | 1 1 = | 1 1 .
U 1 [ l = 2 n U 1 l ] | 0 1 | i 2 | i 3 | i n = | 0 1 | i 2 | i 3 | i n , U 1 [ l = 2 n U 1 l ] | 1 1 | i 2 | i 3 | i n = | 1 1 ( 1 ) i 2 ( 1 ) i 3 ( 1 ) i n | i 2 | i 3 | i n ,
g 1 2 g l 2 4 Δ 1 l ( 1 δ 1 + 1 δ l ) 2 = χ .
g 1 2 δ 1 = ( 2 m n + 1 ) χ .
H ˜ I = H I + δ H + ε ,
δ H = g ˜ 1 ( e i δ ˜ 1 t a ^ 1 + σ f e + h . c . ) + l = 2 3 g ˜ l ( e i δ ˜ l t a ^ l + σ f g + h . c . ) ,
ε = k l ; k , l = 1 3 g k l ( e i Δ ˜ k l t a ^ k + a ^ l + h . c . ) .
d ρ d t = i [ H ˜ I , ρ ] + l = 1 3 κ l [ a l ] + γ e g [ σ e g ] + γ f e [ σ f e ] + γ f g [ σ f g ] + j = e , f { γ φ j ( σ j j ρ σ j j σ j j ρ / 2 ρ σ j j / 2 ) } ,
= ψ id | ρ | ψ id ,
| ψ in = 1 2 2 ( | 000 + | 001 + | 010 + | 011 + | 100 + | 101 + | 110 + | 111 ) .
| ψ id = 1 2 2 ( | 000 + | 001 + | 010 + | 011 + | 100 | 101 | 110 + | 111 ) | g .

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