Qubit-mediated deterministic nonlinear gates for quantum oscillators.

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 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.

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 , where with j = x, y, z relates to the qubit system and stands for one of Pauli matrices, and 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 and where the oscillator operators and commute . As depicted in Fig. 1a, we can join them into a sequence following the idea of geometric phase effect[53]. In a manner similar to[23, 54, 55], this operator can be simplified towhere the last line corresponds to a weak strength limit [56]. The resulting oscillator dynamics is driven by the product of operators and coupled to the qubit by . The qubit degree of freedom can be straightforwardly eliminated by preparing and measuring the qubit system in one of the relevant eigenstates, such as |g〉. The measurement then substitutes the discarding of qubit depicted in Fig. 1a. The whole sequence then realizes a conditional operatorwhich approximates unitary operationin the limit of small τ. The commutativity of and restricts the generality of the scheme, but still allows for many interesting cases. The base operators and 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 A, H ≈ σ A 1 and H ≈ σ B 2 between optical mode and ancillary mode are arranged to achieve a high-order nonlinearity: (a) scheme for single-mode optical interaction operator, and (b) scheme for a two-mode optical evolution operator. Each box with a written interaction Hamiltonian H represents the evolution exp[iH] for a unit of time, and the different colored boxes represent different operators acting on the ancillas. The ancillas prepared in a chosen state are discarded after each set of interactions. Repetition of these unit approximate operators represented by dashed boxes makes high-strength operators.

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 which approximates the ideal operation . 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 .

For a specific test state , the performance of the operation can be quantified by looking at its successful implementation probability and fidelity . These metrics inherently depend on the chosen state , but we can also directly analyze the sandwiched operators and . In the ideal case of , both of these operators and 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 and consisting of states with the respective eigenvalues m and m . Note that the basis does not need to be discrete. We can write the diagonal elements of and aswhere the unitarity of the operator and the commutativity between , and 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

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 |ψ〉 = ∑ c |m〉, and for a strictly bounded state, we can write where m  = max(|m |, |m |) is the dimension(s) of the Hilbert space(s). The bounds for success probability and fidelity can be found aswhere we used the fact that is a decreasing function of m and the error bound is a function . Now as for any T and m , the error can be made arbitrarily small. For an arbitrarily chosen error bound ε and desired strength of the interaction T, a number of repetitionsimplements 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 any ε 2. With help of (6) we can now always lower bound the success probability and the fidelity by P , F > 1 − ε − ε 2, and we can again find R and 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 , but by a trace preserving map which deterministically transforms any input state intowhere is the successful operation and is the erroneous operation. When the individual operation is repeated R times, the final output state can be expressed aswhere P denotes the success probability of the probabilistic scheme with otherwise identical parameters and the density matrix groups together all the realizations which would be in the probabilistic scenario disqualified by measurements. For states from Hilbert space limited by m the fidelity is lower bounded by

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 .

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 operation[23, 57] is realized by a unitary operator and in our approach it can be straightforwardly achieved by setting . The implementation requires coupling with Hamiltonian , where σ are Pauli matrices. Such operations can be obtained from the Jaynes-Cummings Hamiltonian by diagonalizing it into the dispersive form [50] and eliminating the commuting local Hamiltonians by either by suitable strengths of the Hamiltonian constants , , or applying suitable local operations. In the dispersive limit of the Jaynes-Cummings model, only a form of is available, but other operations can be achieved by performing suitable local rotations of the qubit: and . 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 setup[49]. The circular Rydberg states of Rb atoms and the Ramsey cavity field are coupled in this regime in cavity QED systems[46, 47], and the cooper pair box qubit and the resonator field in 1D transmission line resonator are coupled in circuit QED systems[48, 50].

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 . 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 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 state[59]. Such states are necessary for advanced application of quantum information processing such as quantum computation[60], and can be recognized by negative regions of their Wigner functions[61, 62]. In relation to the self-Kerr effect a larger Kerr interaction strength T produces more complex structures of negative Wigner function[63, 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  = 2.06, while only around 73% of the photons in the coherent state 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 have an average photon number with an unclear maximum photon number. The fidelity for these states scales as for the lowest order expansion. Therefore we notice that these states have a smaller error in fidelity than the bound of errors scaling as .

Figure 2

Negative regions of Wigner functions for coherent state 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 (fourth column). The top row shows the ideal realization of the operation, the middle row shows simulations with single step strength of τ = 0.02, and the bottom row shows realistic lossy simulation with repeated single step transmittance η = 1–5.6 × 10−4. Insets show fidelities of the states with the ideal versions. We can see that the simulations faithfully recreate the ideal Wigner functions, even under the effects of moderate loss.

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 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 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 , 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 and . The two dispersive interactions and should be applied alternatingly being turned on and off by drive laser beams[47, 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 gates[65–67, 71]. In an example of the control-Z gate[71], a separable state of two oscillators is changed to entangled state 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 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 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 , where ρ 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 with ideal cross-Kerr operator and the one achieved by our method with τ = 0.05. We can see that for T > 0.8, the entanglement is purely non-Gaussian. When both oscillators suffer from loss with η = 1–3.5 × 10−3, we observe both reduction of overall entanglement and increase of Gaussian entanglement. This is the consequence of the loss drawing both states towards the pointer Gaussian vacuum state and Gaussifying them in the process. However, even under the effects of loss, purely non-Gaussian entanglement can still be obtained.

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 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 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.