Abstract
The principle of superposition is a key ingredient for quantum mechanics. A recent work [Phys. Rev. Lett. 116, 110403 (2016) [CrossRef] ] has shown that a quantum adder that deterministically generates a superposition of two unknown states is forbidden. Here we consider the implementation of the probabilistic quantum adder in the 3D cavity-transmon system. Our implementation is based on a three-level superconducting transmon qubit dispersively coupled to two cavities. Numerical simulations show that high-fidelity generation of the superposition of two coherent states is feasible with current circuit QED technology. Our method also works for other physical systems such as two optical cavities coupled to a three-level atom or two nitrogen-vacancy center ensembles interacted with one three-level superconducting flux qubit.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The superpositions of quantum states is at the heart of the basic postulates and theorems of quantum mechanics. Quantum superposition studies the application of quantum theory or phenomena, that is different from the classical world, leads to many other intriguing quantum phenomena such as quantum entanglement [1] and quantum coherence [2, 3]. It is a vital physical resource and has many important applications in quantum information processing (QIP) and quantum computation such as quantum algorithms [4, 5], quantum metrology [6], and quantum cryptography [7].
A quantum adder is a quantum machine adding two arbitrary unknown quantum states of two different systems onto a single system [8, 9]. How to generate a superposition of two arbitrary states has recently aroused great interest in the field of quantum optics and quantum information. For example, Refs. [8, 9] have proved that it is impossible to generate a superposition of two unknown states, but Ref. [9] proposed a method to probabilistically creating the superposition of two known pure states with the fixed overlaps. Reference [10] has shown that superpositions of orthogonal qubit states can be produced with unit probability, and Ref. [11] has demonstrated that the state transfer can be protected via an approximate quantum adder. Recently, the probabilistic creation of superposition of two unknown quantum states has been demonstrated experimentally in linear optics [12] and nuclear magnetic resonance (NMR) [13].
Circuit quantum electrodynamics (QED) consisting of superconducting qubits and microwave cavities are now moving toward multiple superconducting qubits, multiple three-dimensional (3D) cavities with greatly enhanced coherence time, making them particularly appealing for large-scale quantum computing [14–16]. For example, a 3D microwave cavity with the photon lifetime up to 2 s [17] and a transmon with a coherence time 0.1 ms [18] have been recently reported in 3D circuit QED. Hence, 3D cavities are good memory elements, which can have coherence time at least four orders of magnitude longer than the transmons. By encoding quantum information in microwave cavities, many schemes have been proposed for synthesizing Bell states [19], NOON states [20–26], and entangled coherent states [27–29] of multiple cavities, implementing multi-qubit gates in a mult-cavity system [30–32], and realizing cross-Kerr nonlinearity interaction between two cavities [33, 34].
Three-dimensional circuit QED has emerged as a well-established platform for QIP and quantum computation [35–40], including creation of a Schrödinger cat state of a microwave cavity [36], preparation and control of a five-level transmon qudit [37], demonstration of a quantum error correction [38], realization of a two-mode cat state of two microwave cavities [39], and implementation of a controlled-NOT gate between multiphoton qubits encoded in two cavities [40]. Considering these advancements in 3D circuit QED, it is quite meaningful and necessary to implement a quantum adder in such systems.
In this paper, we present a probabilistic scheme to realize a quantum adder that creates a superposition of two unknown states by using a superconducting transmon qubit dispersively coupled to two 3D cavities. This circuit architecture has been experimentally demonstrated recently in [39]. Our protocol has the following features and advantages: (i) The superposition of two states are encoded in two three-dimensional cavities which have long coherence time rather than encoded in qubits. (ii) The states of cavities can be in arbitrary states, e.g., discrete-variable states or continuous-variable states. (iii) Our proposal can also be applied to other physical systems such as quantum dot-cavity system [41], natural atom-cavity system [42], superconducting circuits with other types of superconducting qubits (e.g., phase qubit [43], Xmon qubit [44], flux qubit [45]), hybrid circuits [46] for two nitrogen-vacancy center ensembles coupled to a single flux qubit [47].
The remaining of this paper is organized as follows. In Sec. 2, we review some basic theory of quantum adder that creates the superposition of two arbitrary states. Our experimental system and Hamiltonian are introduced in Sec. 3. In Sec. 4, we show the method to implementing a quantum adder in 3D circuit QED system. In Sec. 5, we give a brief discussion on the experimental implementation of a quantum adder with state-of-the-art circuit QED technology. Finally, Sec. 6 gives a brief concluding summary.
2. Basic theory of quantum adder
We now review some basic theory of quantum adder (Fig. 1) which can generate the superposition of two pure states. Assume that particles A and B are respectively initially in arbitrary pure states and , and an ancilla particle T is prepared in an arbitrary superposition state with normalized coefficients αT and βT. We first implement a three-qubit controlled-SWAP gate such that the initial state of three qubits evolves into
where the qubit T is a control qubit and qubits A and B are two target qubits. Equation (1) means that the states of the target qubits are swapped only if the the control qubit is in the state and unchanged otherwise.Then we make projective measurements on the states and of qubits T and A, respectively. Here, and is the referential state which satisfies and . Accordingly, one obtains the following superposition state of qubit A
where , , and the normalization constant . Here, the sign or of the output state depending on the measurement or of ancilla qubit T. It can be seen that the performance of the above superposition state is possible with prior knowledge of the overlaps of and . As shown in above operations, the superposition state of Eq. (2) can be produced under the knowledge about the overlaps and . We find that if we vary the coefficients αT and βT, then the arbitrary superposition state of qubit A is prepared with prior knowledge of the overlaps.3. System and Hamiltonian
Superconducting qubits based on Josephson junctions are mesoscopic element circuits like artificial atoms, with multiple discrete energy levels whose spacings can be rapidly adjusted by varying external control parameters [44, 48–50]. Typically, a transmon [51] has weakly anharmonic multilevel structure and the transition between non-adjacent levels is forbidden or very weak.
Motivated by the experimental advances in 3D circuit QED, we here consider a circuit system consisting of a transmon qutrit (with three states , and ) capacitively coupled to two separate 3D superconducting cavities as shown in Fig. 2. The Hamiltonian to describe the microwave cavities coupled to the transmon reads . The free Hamiltonian H0 is given by
with () denoting the () transition frequency of transmon qubit, ωA (ωB) denoting the frequency of cavity A (B), and and a ( and b) representing the creation and annihilation operators for cavity A (B).The coupling Hamiltonian HI of the whole system is given by
where (), () represent the raising (lowering) operators respectively corresponding to the and the transitions in the transmon, gA (gB) and () denote the coupling strengths. Here we would like to emphasize that both the transitions in the transmon should be considered due to the weak anharmonicity [51]. We take into account the influence of the coupling of the cavity with the transition of transmon, when the cavity is off-resonant with the transition of transmon. However, the transition of the transmon is a forbidden dipole transition [51], so the transition can be neglected.To proceed, let’s turn to the the interaction picture with respect to . Then the Hamiltonian can be given by
where the detunings and , with the transmon anharmonicity (). The first and second terms describe the off-resonant coupling between cavity j and the transition of transmon [Fig. 2(b)], while the third and fourth terms denote the off-resonant coupling between cavity j and the transition of transmon [Fig. 2(b)], respectively. For a transmon, a ratio of the anharmonicity between the transition frequency and the transition frequency is readily achieved in experiments. In the following, we choose to derive the following effective Hamiltonian by using the method [52].Applying the large-detuning conditions and (i.e., and ), the Hamiltonian (5) changes to [52]
where , , , and . Notice that the first, second, and third lines of Eq. (6) describe Stark shifts of the levels , , and of the transmon, respectively. If we eliminate the degrees of freedom of the transmon, the fourth or fifth line of Eq. (6) is exactly the standard Hamiltonian which describes the Jaynes-Cummings interaction between the cavities A and B. For simplicity, we set and . Therefore, the Hamiltonian (6) can be rewritten as where the dispersive shifts , , and , and we have used and . Here the dispersive shifts χ, λ, and Λ can be adjusted by the couplings strength g and the detuning Δ. One can obtain the dispersive shifts , and with the detuning . On the contrary, if the detuning , we will find that the dispersive shifts , and .4. Experimental implementation of a quantum adder in 3D circuit QED
In this section we will show how to prepare a superposition state for arbitrary input states in 3D circuit QED with the above Hamiltonian with the weak anharmonicity. Let and be two arbitrary input pure states of microwave cavities A and B, respectively. Here, the Fock states of the cavities are expressed as and . We assume that the level of transmon is not occupied, thus the initial state of the transmon is
Accordingly, the effective Hamiltonian Eq. (7) reduces to
According to the Hamiltonian (9), the initial state of the total system at time t becomes
where , , , and . We should mention that here we assume that the detuning and coupling strength of cavity A and B are equal. Thus, we can obtain and . The time evolution operators can be defined as and for and components.By solving the Heisenberg equations for and , the dynamics of the operators and can be derived as
for component, and for component. We then discuss how to use the Eqs. (11) and (12) to realize a superposition state of two cavities.4.1. as the control state
By waiting for the evolution time , one has and . That corresponds to for the level , an exchange of quantum states between two microwave cavities except for a phase shift conditioned on the photon number in cavities. When the condition is satisfied, we also have and for the level . Here, the sign or of the expression t depends on the detuning or , and k1 and k2 are non-negative integers. Accordingly, we can obtain the following relationship between the Δ and anharmonicity α
When the conditions and are satisfied, the Eq. (10) changes to (see Eq. (26) of Appendix A)
where we have used , , , and . It should be noted here that and have the same Hilbert space while they are arbitrary asymmetric states.We perform a rotation on the transmon qubit that realizes the conversions and with and Now we perform a projective measurement onto the state or of transmon qubit, the state (14) becomes
Then we perform another measurement on the cavity B in the referential state that satisfies and . Thus, one can obtain the following superposition state of cavity A
where , , and . Here, the sign or of the output state conditioning on the measurement or of transmon qubit. It can be seen that the performance of the above superposition state is possible with prior knowledge of the overlaps of and .4.2. as the control state
By waiting for the evolution time , one has and . That corresponds to for the level , an exchange of quantum states between two microwave cavities except for a phase shift conditioned on the photon number in cavities. Here the sign of the expression t corresponds to while the sign corresponds to . When the condition is satisfied, we also have and for the level . Accordingly, one has the following relationship between the detuning Δ
and anharmonicity α
When the conditions and are satisfied, the Eq. (10) changes to (see Eq. (28) of Appendix A)
where we have used , , , and .We perform a rotation on the transmon qubit that realizes the conversions and with and Now we perform a projective measurement onto the state or of transmon, the state (18) becomes
Then we perform another measurement on the cavity B in the referential state that satisfies and . Thus, one can obtain the following superposition state of cavity A
where , , and . Here, the sign of the output state depending on the measurement or of transmon.Equation (16) or (20) shows that a superposition of two states is prepared in the cavity-transmon system. In order to maintain the prepared state, the level spacings of the transmon need to be rapidly (within 1-3 ns [44, 48–50]) adjusted so that the transmon is decoupled from cavities A and B after the desired superposition state is generated. Alternatively, to have the cavities coupled or decoupled from the transmon, one also can tune the frequencies of cavities that because the rapid (with a few nanoseconds) tuning of cavity frequency has been reported in experiments [53–55].
It should be mentioned that when the level of transmon is not occupied, i.e., , the above superposition state (16) or (20) can also be prepared for or as the control state.
5. Possible experimental implementation
Recent experimental results for the 3D circuit system demonstrate the great promise of quantum computation and QIP. For an experimental implementation, our setup of two superconducting 3D cavities coupled to a transmon has been demonstrated recently by [39].
Taking into account the effect of transmon weak anharmonicity, the dissipation and the dephasing, the dynamics of the lossy system is governed by the Markovian master equation
where ρ is the density matrix of the whole system, HI is given by Eq. (5), , and is the dissipator. Here, is the decay rate of cavity A (B). In addition, , , and are the energy relaxation rates from state to , to , and to of transmon qubit, respectively. () is the dephasing rate of the level () of transmon qubit.The generation efficiency can be evaluated by fidelity , where is the ideal target state. It is obvious that how to realize the controlled-SWAP gate is the key of our proposal. So we mainly consider the impurity introduced in this process. In fact, the initial state preparation and measurement of transmon and cavities can be relatively accurately performed [39, 40, 56, 57], thus it is also reasonable for us not to consider the initial state preparation, the rotations and measurement impurites on the fidelity. Accordingly, the ideal target state is given by Eq. (14) or Eq. (18) for the case of (i) the state acts as a control state or (ii) the state acts as a control state. The corresponding parameters used are: (i) i.e., GHz, and (ii) i.e., GHz, respectively. In addition, the input state of the transmon-cavity system is . The initial state of cavities and can be arbitrary states such as discrete-variable states or continuous-variable states. In the following, we choose the cavities are initially in the coherent states, i.e., and with . Thus, the input state of the transmon-cavity system is .
We numerically simulate the fidelity of the operation by solving the master Eq. (21). Since the transmon qubit relaxation time and the transmon qubit dephasing time have been achieved in similar 3D circuit system [39], we set s, s, s, s, s, In addition, we set s, s [39]. For cavities with frequency GHz [39] and dissipation times and used in the numerical simulation, the quality factors of the cavity A and B are and . Note that three-dimensional cavities with a loaded quality factor have been reported in experiments [18, 35]. According to Ref. [39], we choose the transmon anharmonicity MHz in our numerical simulation. In the following, we choose the dispersive shift χ = 1 MHz that because this value of χ is readily available in experiments [39]. Accordingly, one can obtain the coupling strength 32 MHz (easy available in experiments [18, 35]).
Figures 3(a) and 3(b) display the effect of the unwanted inter-cavity crosstalk on the fidelity, respectively. The effect of the inter-cavity crosstalk can be taken into account by adding a Hamiltonian of the form in HI, where gAB is the inter-cavity crosstalk coupling strength. Figures 3(a) and 3(b) show fidelity versus θ with for , which consider the case of (i) the state acts as a control state and (ii) the state acts as a control state, respectively. From the Figs. 3(a) and 3(b) we can see that the effect of the crosstalk on the fidelity is negligibly small for . Moreover, we calculates the average fidelities are approximately (i) , , and , (ii) , , and for , respectively. In the following, we choose the inter-cavity crosstalk coupling strength . This coupling strength condition is easily satisfied by the present circuit QED technology [32, 58].
Figures 4(a) and 4(b) take into account the inhomogeneity in transmon-cavity interaction for the case of (i) the state acts as a control state and (ii) the state acts as a control state, respectively. Figure 4 displays the fidelity versus c and θ, where we set and with . Figure 4(a) shows that for and , the effect of the inhomogeneity on the fidelity is very small. Moreover, one can see that for , the fidelity can be greater than 95.61%. Figure 4(b) displays that the fidelity is almost unaffected by the inhomogeneity for .
In a realistic situation, it may be a challenge to obtain identical cavity-transmon frequency detuning. Thus, we consider the the effect of the unequal detuning on the fidelity in Figs. 5(a) and 5(b) for the case of (i) the state acts as a control state and (ii) the state acts as a control state, respectively. We set with . Here, we choose . One can find that the fidelities in Fig. 5(a) and Fig. 5(b) are, respectively greater than 94.18% and 95.83%, and the optimal fidelities can reach 97.98% and 97.68%.
The operational time is estimated ass or s for the case of (i) or (ii). To investigate the effect of the operation time errors on the fidelity, we consider a small operation time error for the case of (i) and (ii) in Fig. 6(a) and 6(b). Here, t is the optimal operation time and () is the actual operation time. Figure 6(a) or 6(b) shows that the effect of the operation time error on the fidelity is small for and .
We also consider the effect of the anharmonicity α or the detuning Δ errors on the fidelity. We consider a small deviation for the case of (i) and (ii). Accordingly, with . Figures 7(a) and 7(b) indicate that the fidelity is negligibly affected by the anharmonicity or the detuning errors for .
The simulations above show that the quantum adder that generates a superposition of two unknown states with high fidelity can be achieved for small errors in the undesired the weak anharmonicity, inter-cavity crosstalk, the inhomogeneity and detuning, etc.
6. Conclusion
We have devised a quantum adder based on 3D circuit QED that is a good candidate for quantum information processing, computation and simulation. The 3D microwave cavities dispersively interacting with the transmon to effectively create a superposition state of two arbitrary states encoded in two cavities. The initial state of each cavity is arbitrarily that selected as the discrete-variable state or the continuous-variable state. Our proposal also can be applied to other types of qubits such as natural atoms [42] and artificial atoms (other superconducting qubits (e.g., phase qubits [43], Xmon qubits [44], flux qubit [45]), NV centers [47], and quantum dots [41]). The Numerical simulations imply that the high-fidelity generation of a superposition state of two cavities is feasible with the current circuit QED technology. Finally, our proposal provides a way for realizing a quantum adder in a 3D circuit QED system, and such quantum machine may find other applications in quantum information processing and computation. We hope this work would stimulate experimental activities in the near future.
Appendix A: derivation of the Eqs. (14) and (18)
By solving the Heisenberg equations for
the dynamics of the operators and can be derived as for component, and for component.(i) acts as a control state. By waiting for the interaction time and the condition , we have and for Eq. (23) and and for Eq. (24), respectively. Here, the sign or of the expression t depends on the detuning or , and k1 and k2 are non-negative integers. Under the conditions of and , the state (10)
changes to where we have used , and . Here, the states and have the same Hilbert space while they are arbitrary asymmetric states.(ii) acts as a control state. By waiting for the interaction time and the condition , one has and for Eq. (23), and and for Eq. (24), respectively. Here the sign of the expression t corresponds to while the sign corresponds to . When the conditions and are satisfied, the Eq. (10)
changes to where we have used , and . Here, the states and have the same Hilbert space while they are arbitrary asymmetric states.Funding
National Natural Science Foundation of China (NSFC) (11775040,1375036, 11847128); Fundamental Research Fund for the Central Universities (DUT18LK45); Key R&D Program of Guangdong Province (2018B030326001).
References
1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865 (2009). [CrossRef]
2. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014). [CrossRef] [PubMed]
3. C. S. Yu, “Quantum coherence via skew information and its polygamy,” Phys. Rev. A 95, 042337 (2017). [CrossRef]
4. P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM Rev. 41, 303–332 (1999). [CrossRef]
5. L. K. Grover, “A fast quantum mechanical algorithm for database search,” https://arxiv.org/abs/quant-ph/9605043.
6. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011). [CrossRef]
7. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 745 (2002). [CrossRef]
8. Y. Alvarez-Rodriguez, M. Sanz, L. Lamata, and E. Solano, “The forbidden quantum adder,” Sci. Rep. 5, 11983 (2015). [CrossRef] [PubMed]
9. M. Oszmaniec, A. Grudka, M. Horodecki, and A. Wójcik, “Creating a superposition of unknown quantum states,” Phys. Rev. Lett. 116, 110403 (2016). [CrossRef] [PubMed]
10. M. Doosti, F. Kianvash, and V. Karimipour, “Universal superposition of orthogonal states,” Phys. Rev. A 96, 052318 (2017). [CrossRef]
11. G. Gatti, D. Barberena, M. Sanz, and E. Solano, “Protected state transfer via an approximate quantum adder,” Sci. Rep. 7, 6964 (2017). [CrossRef] [PubMed]
12. X. M. Hu, M. J. Hu, J. S. Chen, B. H. Liu, Y. F. Huang, C. F. Li, G. C. Guo, and Y. S. Zhang, “Experimental creation of superposition of unknown photonic quantum states,” Phys. Rev. A 94, 033844 (2016). [CrossRef]
13. K. Li, G. Long, H. Katiyar, T. Xin, G. Feng, D. Lu, and R. Laflamme, “Experimentally superposing two pure states with partial prior knowledge,” Phys. Rev. A 95, 022334 (2017). [CrossRef]
14. J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589–597 (2011). [CrossRef] [PubMed]
15. M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013). [CrossRef] [PubMed]
16. 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]
17. A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Posen, and A. Grassellino, “Three-dimensional superconducting resonators at T < 20 mK with the photon lifetime up to τ = 2 seconds,” https://arxiv.org/abs/1810.03703.
18. C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Córcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506 (2012). [CrossRef]
19. 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). [CrossRef]
20. S. T. Merkel and F. K. Wilhelm, “Generation and detection of NOON states in superconducting circuits,” New J. Phys. 12, 093036 (2010). [CrossRef]
21. F. W. Strauch, K. Jacobs, and R. W. Simmonds, “Arbitrary control of entanglement between two superconducting resonators,” Phys. Rev. Lett. 105, 050501 (2010). [CrossRef] [PubMed]
22. 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]
23. R. Sharma and F. W. Strauch, “Quantum state synthesis of superconducting resonators,” Phys. Rev. A 93, 012342 (2016). [CrossRef]
24. 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). [CrossRef] [PubMed]
25. Q. P. 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]
26. 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]
27. C. P. Yang, Q. P. Su, and S. 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). [CrossRef]
28. T. Liu, Q. P. Su, S. J. Xiong, J. M. Liu, C. P. Yang, and F. Nori, “Generation of a macroscopic entangled coherent state using quantum memories in circuit QED,” Sci. Rep. 6, 32004 (2016). [CrossRef] [PubMed]
29. 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]
30. S. B. Zheng, C. P. Yang, and F. Nori, “Arbitrary control of coherent dynamics for distant qubits in a quantum network,” Phys. Rev. A 82, 042327 (2010). [CrossRef]
31. C. P. Yang, Q. P. Su, S. B. Zheng, and F. Nori, “Entangling superconducting qubits in a multi-cavity system,” New J. Phys. 18, 013025 (2016). [CrossRef]
32. T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multi-cavity system,” Sci. Rep. 6, 21562 (2016). [CrossRef] [PubMed]
33. 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). [CrossRef]
34. H. Zhang, A. Alsaedi, T. Hayat, and F. G. Deng, “Entanglement concentration and purification of two-mode squeezed microwave photons in circuit QED,” Ann. Phys. 391, 112–119 (2018). [CrossRef]
35. 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]
36. 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 Schrödinger cat states,” Science 342, 607–610 (2013). [CrossRef] [PubMed]
37. 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]
38. N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “Extending the lifetime of a quantum bit with error correction in superconducting circuits,” Nature 536, 441–445 (2016). [CrossRef] [PubMed]
39. 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 Schrödinger cat living in two boxes,” Science 352, 1087–1091 (2016). [CrossRef] [PubMed]
40. S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A CNOT gate between multiphoton qubits encoded in two cavities,” Nature Comm. 9, 652 (2018). [CrossRef]
41. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef] [PubMed]
42. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature 436, 87–90 (2005). [CrossRef] [PubMed]
43. 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]
44. 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]
45. 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,” Nature Commun. 7, 12964 (2016). [CrossRef]
46. 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 (2013). [CrossRef]
47. 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). [CrossRef] [PubMed]
48. 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. Physics 4, 523–526 (2008). [CrossRef]
49. 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]
50. J. D. Strand, M. Ware, F. Beaudoin, T. A. Ohki, B. R. Johnson, A. Blais, and B. L. T. Plourde, “First-order sideband transitions with flux-driven asymmetric transmon qubits,” Phys. Rev. B 87, 220505 (2013). [CrossRef]
51. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A 76, 042319 (2007). [CrossRef]
52. D. F. James and J. Jerke, “Effective Hamiltonian theory and its applications in quantum information,” Can. J. Phys. 85, 625–632 (2007). [CrossRef]
53. 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 lifetime,” Appl. Phys. Lett. 92, 203501 (2008). [CrossRef]
54. 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). [CrossRef]
55. F. Souris, H. Christiani, and J. P. Davis, “Tuning a 3D microwave cavity via superfluid helium at millikelvin temperatures,” Appl. Phys. Lett. 111, 172601 (2017). [CrossRef]
56. M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, “High-fidelity readout in circuit quantum electrodynamics using the Jaynes-Cummings nonlinearity,” Phys. Rev. Lett. 105, 173601 (2010). [CrossRef]
57. K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “Deterministic teleportation of a quantum gate between two logical qubits,” Nature 561, 368–373 (2018). [CrossRef] [PubMed]
58. 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]