PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of scirepAboutEditorial BoardFor AuthorsScientific Reports
 
Sci Rep. 2017; 7: 11536.
Published online 2017 September 14. doi:  10.1038/s41598-017-11353-3
PMCID: PMC5599512

Qubit-mediated deterministic nonlinear gates for quantum oscillators

Abstract

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.

Introduction

Quantum computers or quantum Turing machines1, 2 take advantage of their quantum mechanical architecture and are capable of solving tasks which are exponentially hard for their classical counterparts35. Their predecessors are quantum simulators68, 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 ions1013, photons14, 15, atomic lattices16, 17 and superconducting circuit18, 19. The analog simulators are dedicated to continuous variables (CV) systems with infinite dimensional Hilbert space20. These systems allow for simulations of unexplored highly nonlinear open quantum dynamics2127. Some CV nonlinear operations naturally appear in other physical systems, such as Bose-Einstein condensates28, cold ions29, or circuit quantum electrodynamics30. 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)3135. 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 states36 or for conditional realization of nonlinear quantum potentials37, 38. The two level systems are also beneficial from a technical standpoint, allowing for a significantly larger number of individual interactions39 than what is allowed for purely optical ancillary single photon states40, 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 in23, 4244. The qubits act only as mediators rather than for control unlike the conceptually similar quantum Zeno gates45, 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 interactions4652 under photon losses.

Short-time oscillator interaction transduced by a qubit

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq1.gif
, where
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq2.gif
with j = x, y, z relates to the qubit system and stands for one of Pauli matrices, and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq3.gif
is an operator acting on the oscillator. To achieve the desired gate on the oscillator, we can consider a pair of non-commuting unitary operators
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq4.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq5.gif
where the oscillator operators
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq6.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq7.gif
commute
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq8.gif
. As depicted in Fig. 1a, we can join them into a sequence
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq9.gif
following the idea of geometric phase effect53. In a manner similar to23, 54, 55, this operator can be simplified to

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ1.gif
1

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq11.gif
and coupled to the qubit by
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq12.gif
. The qubit degree of freedom can be straightforwardly eliminated by preparing and measuring the qubit system in one of the relevant eigenstates, such as |gright angle bracket. The measurement then substitutes the discarding of qubit depicted in Fig. 1a. The whole sequence
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq13.gif
then realizes a conditional operator

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ2.gif
2

which approximates unitary operation

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ3.gif
3

in the limit of small τ. The commutativity of

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq14.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq15.gif
restricts the generality of the scheme, but still allows for many interesting cases. The base operators
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq16.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq17.gif
can be compatible operators on a single oscillator (as in Fig. 1a), or different operations on two separate oscillators (illustrated in Fig. 1b). The most apparent scenarios in which the product of two operators is highly nontrivial and practically useful operation are the self-Kerr and cross-Kerr evolutions, which we will address in detail later.

Figure 1
Concept of deterministic gates with oscillators mediated by a qubit where the interactions H  σ x,y A, H  σ x,y A 1 and H  σ x,y B 2 between optical mode ...

Near-unitarity of short-time realistic interaction

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq19.gif
which approximates the ideal operation
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq20.gif
. Interestingly enough, in the limit of sufficiently small τ the re-initialization of qubit is not needed, as the approximate operator can be also obtained as
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq21.gif
.

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq23.gif
and fidelity
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq24.gif
. These metrics inherently depend on the chosen state |ψ, but we can also directly analyze the sandwiched operators
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq26.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq27.gif
. In the ideal case of
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq28.gif
, both of these operators
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq29.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq30.gif
reduce to the identity operator 1. We can therefore discern the quality of the operation by looking at how far we are from this ideal scenario. This analysis is best accomplished by considering the joint eigenbasis of the commuting operators
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq31.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq32.gif
consisting of states |m with the respective eigenvalues m A and m B. Note that the basis does not need to be discrete. We can write the diagonal elements of
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq34.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq35.gif
as

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ4.gif
4

where the unitarity of the operator

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq36.gif
and the commutativity between
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq37.gif
,
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq38.gif
and |m〉〈m| is utilized. From (4), we may notice an interesting behavior: the fidelity and the success probability are not complementary and can approach unity simultaneously. This near-unitarity is the characteristic of schemes utilizing the qubit in the eigenstate of the realized operator as in (1). In the limit of small τ, the probability of success derived from (2) is expanded up to the lowest order as

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ5.gif
5

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq40.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq41.gif
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 |ψright angle bracket = m c m|mright angle bracket, and for a strictly bounded state, we can write
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq42.gif
where m max = max(|m A|, |m B|) is the dimension(s) of the Hilbert space(s). The bounds for success probability and fidelity can be found as

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ6.gif
6

where we used the fact that

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq43.gif
is a decreasing function of m and the error bound is a function
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq44.gif
. Now as limRε = 0 for any T and m max, the error can be made arbitrarily small. For an arbitrarily chosen error bound ε and desired strength of the interaction T, a number of repetitions

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ7.gif
7

implements the desired operation with an error lower than ε.

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq46.gif
such that
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq47.gif
for any ε 2. With help of (6) we can now always lower bound the success probability and the fidelity by P S, F > 1  ε  ε 2, and we can again find R and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq48.gif
such that the joint error ε + ε 2 is made arbitrarily small. We emphasize that the obtained bound is derived from the worst case scenario, and its main purpose lies in proving conceptual viability. In practical scenarios in which the approached quantum states are not centered at the boundary of the Hilbert space, the number of required repetitions can be significantly smaller.

The 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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq49.gif
, but by a trace preserving map which deterministically transforms any input state
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq50.gif
into

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ8.gif
8

where

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq51.gif
is the successful operation and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq52.gif
is the erroneous operation. When the individual operation is repeated R times, the final output state can be expressed as

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ9.gif
9

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq53.gif
groups together all the realizations which would be in the probabilistic scenario disqualified by measurements. For states from Hilbert space limited by m max the fidelity is lower bounded by

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_Equ10.gif
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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq54.gif
.

Example of self-Kerr quantum interaction

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 operation23, 57 is realized by a unitary operator

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq55.gif
and in our approach it can be straightforwardly achieved by setting
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq56.gif
. The implementation requires coupling with Hamiltonian
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq57.gif
, where σ j are Pauli matrices. Such operations can be obtained from the Jaynes-Cummings Hamiltonian by diagonalizing it into the dispersive form
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq58.gif
50 and eliminating the commuting local Hamiltonians by either by suitable strengths of the Hamiltonian constants g′ ≫ ω, g′ ≫ Ω, or applying suitable local operations. In the dispersive limit of the Jaynes-Cummings model, only a form of
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq61.gif
is available, but other operations can be achieved by performing suitable local rotations of the qubit:
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq62.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq63.gif
. The realistic implementation in these systems therefore can be achieved by a qubit interacting with an oscillator in the dispersive regime, with intermittent qubit rotations in Bloch sphere and re-initialization back to ground state after each round. The operation can be also found in other physical systems: it can be obtained as a part of the dispersive interaction available between two-level systems and oscillators. The cavity field of a high finesse mirrors and the motional energy eigenstate of a thin dielectric membrane was used in the optomechanical setup49. The circular Rydberg states of Rb atoms and the Ramsey cavity field are coupled in this regime in cavity QED systems46, 47, and the cooper pair box qubit and the resonator field in 1D transmission line resonator are coupled in circuit QED systems48, 50.

In contrast to the approach of circuit QED58, 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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq64.gif
with β = 1. The self-Kerr operation is non-classical and non-Gaussian operation, and produces a non-classical and non-Gaussian state when applied to a coherent state59. Such states are necessary for advanced application of quantum information processing such as quantum computation60, and can be recognized by negative regions of their Wigner functions61, 62. In relation to the self-Kerr effect a larger Kerr interaction strength T produces more complex structures of negative Wigner function63, 64.

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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq67.gif
with an unclear maximum photon number. The fidelity for these states scales as
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq68.gif
for the lowest order expansion. Therefore we notice that these states have a smaller error in fidelity than the bound of errors scaling as
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq69.gif
.

Figure 2
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

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq71.gif
, where
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq72.gif
is annihilation operator of the auxiliary zero temperature oscillator which is immediately discarded. The single step transmittance parameter η strongly impacts the performance of the method. In order to see how large loss can the system actually tolerate, we have simulated the imperfect operation for η = 1–5.6 × 10−4, which corresponds to η/2τ 2 = 0.3. The loss counteracts the effects of the nonlinear operation. As time of the interaction increases, the state is continuously becoming more and more non-classical, which is witnessed by the appearance of negative areas in its Wigner function. This is actually as high as the resilience of the setup goes, because when the loss is larger, the negativities in Wigner function are not observed. However, even in this case the loss is accumulated with time and at some point so much of the energy is lost that the non-classical features vanish. This can be seen in the bottom row of Fig. 2. We can see that while the loss of 13% of the energy for T = 0.2 did not severely affect the non-classicality, 40% loss for T = 0.8 removed one area of negativity. We therefore conclude that proposed method is not critically sensitive to basic decoherence caused by a loss in the oscillator.

Example of cross-Kerr quantum interaction

Another 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 gates6567, and nondestructive photon detection68, 69. It also enables direct photon-photon interaction used for many quantum information processing such as a one-way computation70. The cross-Kerr interaction, represented by a unitary operator

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq73.gif
, can be engineered from the same fundamental component as the self-Kerr operation: the dispersive coupling between an oscillator and a qubit, only this time the qubit is coupled to two separate oscillators (as in Fig. 1b) so
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq74.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq75.gif
. The two dispersive interactions
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq76.gif
and
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq77.gif
should be applied alternatingly being turned on and off by drive laser beams47, 50.

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 gates6567, 71. In an example of the control-Z gate71, a separable state of two oscillators

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq78.gif
is changed to entangled state
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq79.gif
by the cross-Kerr gate with a strength T = π. Within our approach, the deterministic cross-Kerr gate with fidelity F = 1–10−5 can be achieved from R = 1000 instances of the basic block. This scenario suits the approximation well due to a limited number of photons in the systems.

However, there are other applications in which larger photon numbers are significant68, 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 entanglement72, and here we adopt the negativity due to the ease of its evaluation73. The negativity of a bipartite state given by a density operator ρ can be obtained as

An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq80.gif
as the measure of entanglement, where ρ PT is the partial transposed density matrix and Tr[| · |] is the trace norm. The analysis should also clearly show that the cross-Kerr gate is non-Gaussian and the created entanglement should therefore be of the non-Gaussian nature. To that end we also look at the Gaussian negativity
An external file that holds a picture, illustration, etc.
Object name is 41598_2017_11353_Article_IEq81.gif
, where ρ G is the density matrix of a Gaussian state which has all first and second moments of quadrature operators identical with ρ 74, 75. Both the Gaussian and the non-Gaussian entanglement of the state generated by the cross-Kerr gate are plotted in Fig. 3 for various values of the interaction strength T. The interaction strength of dispersive interactions was chosen as τ = 0.05. We can see that the entanglement created for larger values of T is practically completely non-Gaussian, as expected, and that the simulated process closely follows the ideal scenario.

Figure 3
Entanglement generated by cross-Kerr gates with different strength T on a pair of coherent states |α1|β2 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 maxρ{N[ρ]−NG[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.

Applications and outlook

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 coupling4652, 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 interactions7781, to hybrid interaction operator such as principally nonlinear optomechanical interactions8293 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.

Electronic supplementary material

Acknowledgements

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

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.

Notes

Competing Interests

The authors declare that they have no competing interests.

Footnotes

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.

References

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. A92, 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. A65, 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]

Articles from Scientific Reports are provided here courtesy of Nature Publishing Group