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**|**Scientific Reports**|**PMC5599512

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- Abstract
- Introduction
- Short-time oscillator interaction transduced by a qubit
- Near-unitarity of short-time realistic interaction
- Example of self-Kerr quantum interaction
- Example of cross-Kerr quantum interaction
- Applications and outlook
- Electronic supplementary material
- References

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Sci Rep. 2017; 7: 11536.

Published online 2017 September 14. doi: 10.1038/s41598-017-11353-3

PMCID: PMC5599512

0000 0001 1245 3953grid.10979.36Department of Optics, Palacký University, 17. listopadu 1192/12, 77146 Olomouc, Czech Republic

Kimin Park, Email: zc.lopu.scitpo@krap.

Received 2017 May 12; Accepted 2017 August 22.

Copyright © The Author(s) 2017

Quantum nonlinear operations for harmonic oscillator systems play a key role in the development of analog quantum simulators and computers. Since strong highly nonlinear operations are often unavailable in the existing physical systems, it is a common practice to approximate them by using conditional measurement-induced methods. The conditional approach has several drawbacks, the most severe of which is the exponentially decreasing success rate of the strong and complex nonlinear operations. We show that by using a suitable two level system sequentially interacting with the oscillator, it is possible to resolve these issues and implement a nonlinear operation both nearly deterministically and nearly perfectly. We explicitly demonstrate the approach by constructing self-Kerr and cross-Kerr couplings in a realistic situation, which require a feasible dispersive coupling between the two-level system and the oscillator.

Quantum computers or quantum Turing machines^{1, 2} take advantage of their quantum mechanical architecture and are capable of solving tasks which are exponentially hard for their classical counterparts^{3–5}. Their predecessors are quantum simulators^{6–8, 9}, which seek to emulate specific quantum dynamics of particular quantum systems in place of general processing. The fundamental principle of the simulations relies on mapping the complex quantum systems onto other more accessible and better controllable ones, such as trapped ions^{10–13}, photons^{14, 15}, atomic lattices^{16, 17} and superconducting circuit^{18, 19}. The analog simulators are dedicated to continuous variables (CV) systems with infinite dimensional Hilbert space^{20}. These systems allow for simulations of unexplored highly nonlinear open quantum dynamics^{21–27}. Some CV nonlinear operations naturally appear in other physical systems, such as Bose-Einstein condensates^{28}, cold ions^{29}, or circuit quantum electrodynamics^{30}. The spectrum of nonlinear operations is however limited and typically determined by the unique physics of specific experimental platforms.

A broader set of nonlinear operations for quantum harmonic oscillator can be elegantly realized by coupling them to suitable two-level systems (qubits)^{31–35}. This realization is possible because the two-level systems are naturally nonlinear due to their saturability and offer a wide variety of qubit-oscillator couplings. The nonlinear nature in turn leads to dynamics of the oscillator which can be used for deterministic generation of nonclassical states^{36} or for conditional realization of nonlinear quantum potentials^{37, 38}. The two level systems are also beneficial from a technical standpoint, allowing for a significantly larger number of individual interactions^{39} than what is allowed for purely optical ancillary single photon states^{40, 41}. The conditional nature of these hybrid operations, however, limits them in their suitability for practical applications as well as quantum simulations, which ultimately leads to success rate exponentially decreasing with the number of operations involved.

In this report we propose a method for deterministic implementation of nonlinear unitary operations for quantum harmonic oscillators sequentially coupled to single qubits. This method relies on employing a sequence of available non-commuting qubit-oscillator interactions, similarly as in^{23, 42–44}. The qubits act only as mediators rather than for control unlike the conceptually similar quantum Zeno gates^{45}, starting and finishing the operation in a factorized state. The repeated gates incrementally create a Zeno-like nonlinear unitary dynamics deterministically and with a nearly unit fidelity. We illustrate the quality of the proposed method by explicitly analyzing realization of the self-Kerr and cross-Kerr nonlinearities done with help of a qubit sequentially coupled to the oscillator by dispersive interactions^{46–52} under photon losses.

Let us start by considering a short time evolution of a quantum oscillator mediated by a single qubit. The unitary oscillator-qubit interaction that enables the desired dynamics is governed by Hamiltonians of the type

, where with1

where the last line corresponds to a weak strength limit *τ* ≪ 1
^{56}. The resulting oscillator dynamics is driven by the product of operators

2

which approximates unitary operation

3

in the limit of small *τ*. The commutativity of

The perfect operation (3) is realized only in the limit of short time *τ* → 0. However, we can increase the strength by repeating the individual operations. In each step, the ancillary qubit is initialized in the ground state, led to interact with the oscillator systems, and finally projected onto the ground state again. It does not matter whether a single physical qubit is used repetitively or if a number of different systems is employed. In any case, *R* repetitions realize quantum operation

For a specific test state |*ψ*〉, the performance of the operation can be quantified by looking at its successful implementation probability

4

where the unitarity of the operator

and the commutativity between , and |5

We can now use this expression to lower bound both the fidelity and the success probability for arbitrary quantum states. The operators

and typically represent position, momentum, or number of quanta of the oscillators whose statistical distribution are asymptotically vanishing outside a certain range, and therefore are reasonably bounded in realistic physical systems. Any state can be expressed as the superposition |6

where we used the fact that

is a decreasing function of7

implements the desired operation with an error lower than *ε*.

Even for quantum states which are not sharply bounded, we can always find

such that for anyThe prominent aspect of our scheme is that its success probability can approach one even for many repetitions, implying that the measurement can be removed from the setup. We therefore follow the deterministic scheme depicted in Fig. 1. Formally, a single step of the operation is no longer represented by an operator

, but by a trace preserving map which8

where

is the successful operation and is the erroneous operation. When the individual operation is repeated9

where *P*
_{s} denotes the success probability of the probabilistic scheme with otherwise identical parameters and the density matrix

10

This result shows that the performance of the deterministic scheme is comparable to the probabilistic regime. Considering the respective fidelities, the deterministic scheme achieves the performance of the probabilistic one when the number of repetitions *R* is increased by a factor of

Let us explicitly demonstrate the performance of the proposed gate by realizing some of the nonlinear gates prevalent in quantum information theory and quantum technology. The self-Kerr operation^{23, 57} is realized by a unitary operator

In contrast to the approach of circuit QED^{58}, which employs suitable time-dependent driving of the qubit-oscillator, our method employs a set of identical elementary gates, which can be repeated in order to obtain strong interaction. As a consequence, the whole operation is less demanding from the point of view of the ability to control the employed quantum systems. The performance of the gate can be generally estimated from the parameters and from the available dimension given by *m*
_{max}. However, such a bound may be too loose, and actual performance depends on the specific choice of the states. Let us apply the self-Kerr operation to a sample coherent state

In Fig. 2, we display the negative regions of Wigner function of self-Kerr transformed coherent states with various coupling parameters *T*=0.2, 0.4, 0.6, 0.8. Apparently, a birth of highly nonclassical quantum interference in phase space can be observed. It is manifested by three separated regions of negativity. The figures show practically no difference between the ideal operation (above) and the deterministic approximate realization with *τ*=0.02 (middle). This observation is reinforced by a near unit fidelity *F*=1–0.8×10^{−4} for *T*=0.8. Interestingly, based on (10) and the parameters of the operation, the maximal Fock number corresponding to such a high bound of fidelity should be as small as *n*
_{max}=2.06, while only around 73% of the photons in the coherent state |*β* = 1〉 live in the subspace under 2. This again shows that the actual fidelity for general states can be higher than the bound given in (10). Coherent states |*β*〉 have an average photon number

Negative regions of Wigner functions for coherent state |*β* = 1〉 subjected to self-Kerr interaction with total strengths *T*=0.2 (first column), *T*=0.4 (second column), *T*=0.6 (third column), and *T*=0.8 **...**

In realistic scenarios, the operation will have to endure the effects of imperfections, mainly the loss as the dominant decoherence model for quantum oscillators. The loss can be modeled by passively coupling the evolving system to a set of zero temperature oscillators. In our model, we consider a sequence of discrete couplings, one after each cycle of the elementary sequence (1). Each of these couplings transforms annihilation operator of the system as

, where is annihilation operator of the auxiliary zero temperature oscillator which is immediately discarded. The single step transmittance parameterAnother example of quantum nonlinear interactions is the cross-Kerr coupling between two harmonic oscillators. This gate is a key component in building important two-qubit single photon gates in linear optical quantum computation such as controlled NOT gates and Fredkin gates^{65–67}, and nondestructive photon detection^{68, 69}. It also enables direct photon-photon interaction used for many quantum information processing such as a one-way computation^{70}. The cross-Kerr interaction, represented by a unitary operator

An elementary application is altering phase of a single photon based on the presence or absence of another, which is the basis for many discrete computation gates^{65–67, 71}. In an example of the control-Z gate^{71}, a separable state of two oscillators

However, there are other applications in which larger photon numbers are significant^{68, 69}. To test for this scenario, we consider the cross-Kerr coupling between two coherent states with amplitudes *α*=*β*=1. Considering again interaction strength *T*=*π*, the operation can be implemented with fidelity *F*=0.989 for *R*=1000 and *F*=1–5×10^{−4} with *R*=2500 repetitions. A higher number of individual operations is demanded by the larger Hilbert space of the states for a fidelity comparable with the previous example. We can also analyze the operation from the point of view of entanglement it generates. There are several measures of entanglement^{72}, and here we adopt the negativity due to the ease of its evaluation^{73}. The negativity of a bipartite state given by a density operator *ρ* can be obtained as

Entanglement generated by cross-Kerr gates with different strength *T* on a pair of coherent states |*α*〉_{1}|*β*〉_{2} with ideal cross-Kerr operator and the one achieved by our method with *τ*=0.05. We can see that for *T*>0.8, **...**

To assess an impact of the decoherence on the cross Kerr interaction, we introduce an equal loss in the both oscillators. Simulations with a realistic loss with *η*=1–3.5×10^{−3}, corresponding to the same level of noise as in a previous section, show results conceptually similar to the self-Kerr case. Again, the loss limits the achievable number of elementary gates and the corresponding total interaction strength. State with dominantly non-Gaussian entanglement can be still achieved, but the maximal difference between non-Gaussian and Gaussian entanglement is limited. For our simulation, this difference max_{ρ}{*N*[*ρ*]−*N*_{G}[*ρ*]} was 0.31 at the energy loss of about 40% for a single arm. There is, however, another interesting effect. In addition to reducing the overall correlations, the loss also drives the quantum state towards Gaussianity. As a consequence, there is less of entanglement, but higher portion of it is Gaussian. In fact, for certain values of parameters the lossy scenario produces more Gaussian entanglement than the ideal one, while non-Gaussian nature is still accessible. It supports previous statements about a sufficient robustness of the method to the loss in oscillator.

In summary, using a single qubit as a recyclable mediator allows for synthesis of high order nonlinear operations on quantum oscillators. These operations can be realized at an arbitrary strength with both fidelity and probability of success approaching one. The only cost is represented by the required number of repetitions of the basic building block, which may be mitigated by using an optimized architecture. Operations which can be implemented depend on the available qubit-oscillator couplings. With the feasible dispersive coupling^{46–52, 76} it is possible to realize self-Kerr and cross-Kerr operations, which play a significant role in quantum information processing, with high quality under a moderate level of environmental effects. The extension of the scheme ranges from engineering high order quadrature nonlinear operators, such as cubic-phase gate operator by Rabi interactions^{77–81}, to hybrid interaction operator such as principally nonlinear optomechanical interactions^{82–93} by combination of the dispersive and Rabi interactions. The higher-order versions of both dispersive and Rabi interactions open a broad class of CV nonlinear interactions. The involved harmonic oscillators can be physically varied (optical, mechanical, electrical, collective spins), and therefore this method can potentially provide wide class of nonlinear gates between these platforms. All of these potential applications open up a possibility of deterministic quantum simulators.

We acknowledge Project GB14-36681G of the Czech Science Foundation. K.P. acknowledges support by the Development Project of Faculty of Science, Palacký University.

Author Contributions

K.P. conceived the theory. P.M. and R.F. conceived the quantification, interpreted the implications and extended the scope. P.M. and R.F. led the project. All authors analyzed the results, wrote the article, and reviewed the manuscript.

The authors declare that they have no competing interests.

**Electronic supplementary material**

**Supplementary information** accompanies this paper at doi:10.1038/s41598-017-11353-3

**Publisher's note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

1. Benioff P. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 1980;22:563. doi: 10.1007/BF01011339. [Cross Ref]

2. Deutsch D. Quantum theory, the Church-Turing principle and the universal quantum computer. P. Roy. Soc. Lond. A. 1985;400:97. doi: 10.1098/rspa.1985.0070. [Cross Ref]

3. Shor, P. W. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. *SIAM J. Comput*., **26**, 1484 (1997).

4. Simon, D. R. On the power of quantum computation. *Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on:* 116 (1994).

5. Deutsch D, Jozsa R. Rapid solution of problems by quantum computation. P. Roy. Soc. Lond. A. 1992;439:553. doi: 10.1098/rspa.1992.0167. [Cross Ref]

6. Kendon VM, Nemoto K, Munro WJ. Quantum analogue computing. Philos. Trans. R. Soc. Lond. A. 2010;368:3609. doi: 10.1098/rsta.2010.0017. [PubMed] [Cross Ref]

7. Cirac JI, Zoller P. Quantum computations with cold trapped ions. Phys. Rev. Lett. 1995;74:4091. doi: 10.1103/PhysRevLett.74.4091. [PubMed] [Cross Ref]

8. Georgescu IM, Ashhab S, Nori F. Quantum simulation. Rev. Mod. Phys. 2014;86:153. doi: 10.1103/RevModPhys.86.153. [Cross Ref]

9. Feynman RP. Simulating physics with computers. Int. J. Theor. Phys. 1982;21:467. doi: 10.1007/BF02650179. [Cross Ref]

10. Kim K, et al. Quantum simulation of the transverse Ising model with trapped ions. Nature. 2010;465:590. doi: 10.1038/nature09071. [PubMed] [Cross Ref]

11. Gerritsma R, et al. Quantum simulation of the Dirac equation. Nature. 2010;463:68. doi: 10.1038/nature08688. [PubMed] [Cross Ref]

12. Lanyon BP, et al. Universal digital quantum simulation with trapped ions. Science. 2011;334:57. doi: 10.1126/science.1208001. [PubMed] [Cross Ref]

13. Blatt R, Roos CF. Quantum simulations with trapped ions. Nat. Phys. 2012;8:277. doi: 10.1038/nphys2252. [Cross Ref]

14. Peruzzo A, et al. Quantum walks of correlated photons. Science. 2010;329:1500. doi: 10.1126/science.1193515. [PubMed] [Cross Ref]

15. Aspuru-Guzik A, Walther P. Photonic quantum simulators. Nat. Phys. 2012;8:285. doi: 10.1038/nphys2253. [Cross Ref]

16. Simon J, et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature. 2011;472:307. doi: 10.1038/nature09994. [PubMed] [Cross Ref]

17. Bloch I, Dalibard J, Nascimbéne S. Quantum simulations with ultracold quantum gases. Nat. Phys. 2012;8:267. doi: 10.1038/nphys2259. [Cross Ref]

18. Houck AA, Türeci HE, Koch J. On-chip quantum simulation with superconducting circuits. Nat. Phys. 2012;8:292. doi: 10.1038/nphys2251. [Cross Ref]

19. Devoret MH, Schoelkopf RJ. Superconducting circuits for quantum information: an outlook. Science. 2013;339:1169–1174. doi: 10.1126/science.1231930. [PubMed] [Cross Ref]

20. Braunstein SL, van Loock P. Quantum information with continuous variables. Rev. Mod. Phys. 2005;77:513. doi: 10.1103/RevModPhys.77.513. [Cross Ref]

21. Filip R, Marek P, Andersen UL. Measurement-induced continuous-variable quantum interactions. Phys. Rev. A. 2005;71:042308. doi: 10.1103/PhysRevA.71.042308. [Cross Ref]

22. Miwa Y, et al. Exploring a new regime for processing optical qubits: squeezing and unsqueezing single photons. Phys. Rev. Lett. 2014;113:013601. doi: 10.1103/PhysRevLett.113.013601. [PubMed] [Cross Ref]

23. Lloyd S, Braunstein SL. Quantum computation over continuous variables. Phys. Rev. Lett. 1999;82:1784. doi: 10.1103/PhysRevLett.82.1784. [Cross Ref]

24. Spiller TP, et al. Quantum computation by communication. New J. Phys. 2006;8:30. doi: 10.1088/1367-2630/8/2/030. [Cross Ref]

25. Gottesman D, Kitaev A, Preskill J. Encoding a qubit in an oscillator. Phys. Rev. A. 2001;64:012310. doi: 10.1103/PhysRevA.64.012310. [Cross Ref]

26. Marek P, Filip R, Furusawa A. Deterministic implementation of weak quantum cubic nonlinearity. Phys. Rev. A. 2011;84:053802. doi: 10.1103/PhysRevA.84.053802. [Cross Ref]

27. Miyata K, et al. Implementation of a quantum cubic gate by an adaptive non-Gaussian measurement. Phys. Rev. A. 2016;93:022301. doi: 10.1103/PhysRevA.93.022301. [Cross Ref]

28. Sefi S, van Loock P. How to decompose arbitrary continuous-variable quantum operations. Phy. Rev. Lett. 2011;107:170501. doi: 10.1103/PhysRevLett.107.170501. [PubMed] [Cross Ref]

29. Sefi S, Vaibhav V, van Loock P. Measurement-induced optical Kerr interaction. Phys. Rev. A. 2013;88:012303. doi: 10.1103/PhysRevA.88.012303. [Cross Ref]

30. Yukawa M, et al. Emulating quantum cubic nonlinearity. Phys. Rev. A. 2013;88:053816. doi: 10.1103/PhysRevA.88.053816. [Cross Ref]

31. Greiner M, Mandel O, Hänsch TW, Bloch I. Collapse and revival of the matter wave field of a Bose-Einstein condensate. Nature. 2002;419:51. doi: 10.1038/nature00968. [PubMed] [Cross Ref]

32. Roos CF, et al. Nonlinear coupling of continuous variables at the single quantum level. Phys. Rev. A. 2008;77:040302(R). doi: 10.1103/PhysRevA.77.040302. [Cross Ref]

33. Kirchmair G, et al. Observation of quantum state collapse and revival due to the single-photon Kerr effect. Nature. 2013;495:205. doi: 10.1038/nature11902. [PubMed] [Cross Ref]

34. Leibfried D, Blatt R, Monroe C, Wineland D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 2003;75:281. doi: 10.1103/RevModPhys.75.281. [Cross Ref]

35. Xiang ZL, Ashhab S, You JQ, Nori F. Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems. Rev. Mod. Phys. 2013;85:623. doi: 10.1103/RevModPhys.85.623. [Cross Ref]

36. Aspelmeyer M, Kippenberg TJ, Marquardt F. Cavity optomechanics. Rev. Mod. Phys. 2014;86:1391. doi: 10.1103/RevModPhys.86.1391. [Cross Ref]

37. Reiserer A, Rempe G. Cavity-based quantum networks with single atoms and optical photons. Rev. Mod. Phys. 2015;87:1379. doi: 10.1103/RevModPhys.87.1379. [Cross Ref]

38. Lodahl P, Mahmoodian S, Stobbe S. Interfacing single photons and single quantum dots with photonic nanostructures. Rev. Mod. Phys. 2015;87:347. doi: 10.1103/RevModPhys.87.347. [Cross Ref]

39. Marek P, Lachman L, Slodička L, Filip R. Deterministic nonclassicality for quantum-mechanical oscillators in thermal states. Phys. Rev. A. 2016;94:013850. doi: 10.1103/PhysRevA.94.013850. [Cross Ref]

40. Park K, Marek P, Filip R. Conditional nonlinear operations by sequential Jaynes-Cummings interactions. Phys. Rev. A. 2016;94:012332. doi: 10.1103/PhysRevA.94.012332. [Cross Ref]

41. Park K, Marek P, Filip R. Finite approximation of unitary operators for conditional analog simulators. Phys. Rev. A. 2016;94:062308. doi: 10.1103/PhysRevA.94.062308. [Cross Ref]

42. Sayrin C, et al. Real-time quantum feedback prepares and stabilizes photon number states. Nature. 2011;477:73. doi: 10.1038/nature10376. [PubMed] [Cross Ref]

43. Fiurášek J. Engineering quantum operations on traveling light beams by multiple photon addition and subtraction. Phys. Rev. A. 2009;80:053822. doi: 10.1103/PhysRevA.80.053822. [Cross Ref]

44. Park K, Marek P, Filip R. Nonlinear potential of a quantum oscillator induced by single photons. Phys. Rev. A. 2014;90:013804. doi: 10.1103/PhysRevA.90.013804. [Cross Ref]

45. Lloyd, S. Hybrid quantum computing. arXiv: quant-ph/0008057 (2000).

46. Huang YP, Moore MG. Interaction-and measurement-free quantum Zeno gates for universal computation with single-atom and single-photon qubits. Phys. Rev. A. 2008;77:062332. doi: 10.1103/PhysRevA.77.062332. [Cross Ref]

47. Gleyzes S, et al. Quantum jumps of light recording the birth and death of a photon in a cavity. Nature. 2007;446:297. doi: 10.1038/nature05589. [PubMed] [Cross Ref]

48. Guerlin C, et al. Progressive field-state collapse and quantum non-demolition photon counting. Nature. 2007;448:889. doi: 10.1038/nature06057. [PubMed] [Cross Ref]

49. Schuster DI, et al. Resolving photon number states in a superconducting circuit. Nature. 2007;445:515. doi: 10.1038/nature05461. [PubMed] [Cross Ref]

50. Thompson JD, et al. Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. Nature. 2008;452:72. doi: 10.1038/nature06715. [PubMed] [Cross Ref]

51. Blais A, Huang RS, Wallraff A, Girvin SM, Schoelkopf RJ. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A. 2004;69:062320. doi: 10.1103/PhysRevA.69.062320. [Cross Ref]

52. Wallraff A, et al. Circuit quantum electrodynamics: Coherent coupling of a single photon to a Cooper pair box. Nature. 2004;431:162. doi: 10.1038/nature02851. [PubMed] [Cross Ref]

53. Johnson BR, et al. Quantum non-demolition detection of single microwave photons in a circuit. Nat. Phys. 2010;6:663. doi: 10.1038/nphys1710. [Cross Ref]

54. Berry MV. Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. A. 1984;392:45. doi: 10.1098/rspa.1984.0023. [Cross Ref]

55. Aharonov Y, Anandan J. Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 1987;58:1593. doi: 10.1103/PhysRevLett.58.1593. [PubMed] [Cross Ref]

56. van Loock P. Optical hybrid approaches to quantum information. Laser Photonics Rev. 2010;5:167. doi: 10.1002/lpor.201000005. [Cross Ref]

57. The details of derivation are presented in Supplementary material.

58. Turchette QA, Hood CJ, Lange W, Mabuchi H, Kimble HJ. Measurement of conditional phase shifts for quantum logic. Phys. Rev. Lett. 1995;75:4710. doi: 10.1103/PhysRevLett.75.4710. [PubMed] [Cross Ref]

59. Boca A, et al. Observation of the Vacuum Rabi Spectrum for One Trapped Atom. Phys. Rev. Lett. 2004;93:233603. doi: 10.1103/PhysRevLett.93.233603. [PubMed] [Cross Ref]

60. Krastanov, S. *et al*. Universal control of an oscillator with dispersive coupling to a qubit. *Phys. Rev. A***92**, 040303 (R) (2015).

61. Weedbrook C, et al. Gaussian quantum information. Rev. Mod. Phys. 2012;84:621. doi: 10.1103/RevModPhys.84.621. [Cross Ref]

62. Wang XB, Hiroshima T, Tomita A, Hayashi M. Quantum information with Gaussian states. Phys. Rep. 2007;448:1. doi: 10.1016/j.physrep.2007.04.005. [Cross Ref]

63. Jeong, H. & Kim, M. S. Efficient quantum computation using coherent states. *Phys. Rev. A***65**, 042305 (2002).

64. Ralph TC, Gilchrist A, Milburn GJ, Munro WJ, Glancy S. Quantum computation with optical coherent states. Phys. Rev. A. 2003;68:042319. doi: 10.1103/PhysRevA.68.042319. [Cross Ref]

65. Kok P. Effects of self-phase-modulation on weak nonlinear optical quantum gates. Phys. Rev. A. 2008;77:013808. doi: 10.1103/PhysRevA.77.013808. [Cross Ref]

66. Stobińska M, Milburn GJ, Wódkiewicz K. Wigner function evolution of quantum states in the presence of self-Kerr interaction. Phys. Rev. A. 2008;78:013810. doi: 10.1103/PhysRevA.78.013810. [Cross Ref]

67. Nemoto K, Munro WJ. Nearly deterministic linear optical controlled-NOT gate. Phys. Rev. Lett. 2004;93:250502. doi: 10.1103/PhysRevLett.93.250502. [PubMed] [Cross Ref]

68. Milburn GJ. Quantum optical Fredkin gate. Phys. Rev. Lett. 1989;62:2124. doi: 10.1103/PhysRevLett.62.2124. [PubMed] [Cross Ref]

69. Chuang IL, Yamamoto Y. Simple quantum computer. Phys. Rev. A. 1995;52:3489. doi: 10.1103/PhysRevA.52.3489. [PubMed] [Cross Ref]

70. Imoto N, Haus HA, Yamamoto Y. Quantum nondemolition measurement of the photon number via the optical Kerr effect. Phys. Rev. A. 1985;32:2287. doi: 10.1103/PhysRevA.32.2287. [PubMed] [Cross Ref]

71. Munro WJ, Nemoto K, Beausoleil RG, Spiller TP. High-efficiency quantum-nondemolition single-photon-number-resolving detector. Phys. Rev. A. 2005;71:033819. doi: 10.1103/PhysRevA.71.033819. [Cross Ref]

72. Hutchinson GD, Milburn GJ. Nonlinear quantum optical computing via measurement. J. Mod. Opt. 2004;51:1211. doi: 10.1080/09500340408230417. [Cross Ref]

73. Kok P, et al. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys. 2007;79:135. doi: 10.1103/RevModPhys.79.135. [Cross Ref]

74. Wootters WK. Entanglement of Formation of an Arbitrary State of Two Qubits. Phys. Rev. Lett. 1998;80:2245. doi: 10.1103/PhysRevLett.80.2245. [Cross Ref]

75. Hayden PM, Horodecki M, Terhal BM. The asymptotic entanglement cost of preparing a quantum state. J. Phys. A: Math. Gen. 2001;34:6891. doi: 10.1088/0305-4470/34/35/314. [Cross Ref]

76. Vidal G, Werner RF. Computable measure of entanglement. Phys. Rev. A. 2002;65:032314. doi: 10.1103/PhysRevA.65.032314. [Cross Ref]

77. Laurat J, et al. Entanglement of two-mode Gaussian states: characterization and experimental production and manipulation. J. Opt. B: Quantum Semiclassical Opt. 2005;7:S577. doi: 10.1088/1464-4266/7/12/021. [Cross Ref]

78. Casanova J, Romero G, Lizuain I, García-Ripoll JJ, Solano E. Deep strong coupling regime of the Jaynes-Cummings model. Phys. Rev. Lett. 2010;105:263603. doi: 10.1103/PhysRevLett.105.263603. [PubMed] [Cross Ref]

79. De Liberato S. Light-matter decoupling in the deep strong coupling regime: The breakdown of the Purcell effect. ibid. 2014;112:016401. doi: 10.1103/PhysRevLett.112.016401. [PubMed] [Cross Ref]

80. Mezzacapo A, et al. Digital Quantum Rabi and Dicke Models in Superconducting Circuits. Sci. Rep. 2014;4:7482. doi: 10.1038/srep07482. [PMC free article] [PubMed] [Cross Ref]

81. Kienzler D, et al. Observation of quantum interference between separated mechanical oscillator wave packets. Phys. Rev. Lett. 2016;116:140402. doi: 10.1103/PhysRevLett.116.140402. [PubMed] [Cross Ref]

82. Sillanpää MA, Park JI, Simmonds RW. Coherent quantum state storage and transfer between two phase qubits via a resonant cavity. Nature. 2007;449:438. doi: 10.1038/nature06124. [PubMed] [Cross Ref]

83. Günter G, et al. Sub-cycle switch-on of ultrastrong light-matter interaction. ibid. 2009;458:178. [PubMed]

84. Niemczyk T, et al. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. 2010;6:772. doi: 10.1038/nphys1730. [Cross Ref]

85. Forn-Díaz P, et al. Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime. Phys. Rev. Lett. 2010;105:237001. doi: 10.1103/PhysRevLett.105.237001. [PubMed] [Cross Ref]

86. Baust A, et al. Ultrastrong coupling in two-resonator circuit QED. Phys. Rev. B. 2016;93:214501. doi: 10.1103/PhysRevB.93.214501. [Cross Ref]

87. Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. *Cavity Optomechanics* (Berlin, Springer-Verlag, 2014).

88. Jayich AM, et al. Dispersive optomechanics: a membrane inside a cavity. New J. Phys. 2008;10:095008. doi: 10.1088/1367-2630/10/9/095008. [Cross Ref]

89. Nunnenkamp A, Børkje K, Harris JGE, Girvin SM. Cooling and squeezing via quadratic optomechanical coupling. Phys. Rev. A. 2010;82:021806(R). doi: 10.1103/PhysRevA.82.021806. [Cross Ref]

90. Sankey JC, Yang C, Zwickl BM, Jayich AM, Harris JGE. Strong and tunable nonlinear optomechanical coupling in a low-loss system. Nat. Phys. 2010;6:707. doi: 10.1038/nphys1707. [Cross Ref]

91. Liao J-Q, Nori F. Single-photon quadratic optomechanics. Sci. Rep. 2014;4:6302. doi: 10.1038/srep06302. [PMC free article] [PubMed] [Cross Ref]

92. Lee D, et al. Multimode optomechanical dynamics in a cavity with avoided crossings. Nat. Commun. 2014;6:6232. doi: 10.1038/ncomms7232. [PubMed] [Cross Ref]

93. Park K, Marek P, Filip R. All-optical simulations of nonclassical noise-induced effects in quantum optomechanics. Phys. Rev. A. 2015;92:033813. doi: 10.1103/PhysRevA.92.033813. [Cross Ref]

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