Efficient quantum simulation of nonlinear interactions using SNAP and Rabi gates

Quantum simulations provide means to probe challenging problems within controllable quantum systems. However, implementing or simulating deep-strong nonlinear couplings between bosonic oscillators on physical platforms remains a challenge. We present a deterministic simulation technique that efficiently and accurately models nonlinear bosonic dynamics. This technique alternates between tunable Rabi and SNAP gates, both of which are available on experimental platforms such as trapped ions and superconducting circuits. Our proposed simulation method facilitates high-fidelity modeling of phenomena that emerge from higher-order bosonic interactions, with an exponential reduction in resource usage compared to other techniques. We demonstrate the potential of our technique by accurately reproducing key phenomena and other distinctive characteristics of ideal nonlinear optomechanical systems. Our technique serves as a valuable tool for simulating complex quantum interactions, simultaneously paving the way for new capabilities in quantum computing through the use of hybrid qubit-oscillator systems.

A prototypical example is the strong nonlinear radiation pressure interaction between a movable mirror in a Fabry-Perot cavity and the intracavity field [34,35] in nonlinear optomechanics.This interaction is fundamentally nonlinear, involving third power of ladder operators in the Hamiltonian [34].At the quantum level, such a resulting pressurelike interaction enables, in principle, challenging single-photon-scale displacements of the mirror.Exploring this highly nonlinear coupling, which can generate nonclassical superposition states from phase-insensitive noise [36][37][38][39], will substantially advance current quantum technologies [40][41][42][43][44][45][46][47] and therefore, it is attractive for hybrid quantum simulations [48].Nonlinear pressure-like phenomena are predicted to enable breakthrough applications in quantum information processing, metrology, communication, fundamental tests of quantum mechanics [49][50][51].
However, the inherent weakness of nonlinear radiation pressure interactions presents a challenge in directly observing quantum nonlinear phenomena without the introduction of additional noise.Although strong optical pumping can enhance effective coupling strengths, it inevitably leads to linearization of the coupling [34], removing the underlying nonlinear dynamics.Experimental strategies like amplification have also been employed to access the single-photon scale nonlinear optomechanical phenomena [52][53][54][55][56][57][58].The membrane-in-the-middle configuration provides enhanced nonlinear optomechanical interactions [59], making it an attractive resource for quantum simulation of the hybrid system.Nevertheless, proof-of-principle demonstrations and analysis of these higher-order nonlinearities remain challenging, and finding new routes to harness these interactions has been an attractive prospect for quantum technologies.
An effective strategy for exploring deep-strong and nonlinear quantum phenomena involves the use of quantum simulation based on engineered interactions [60].There are several methods available for engineering continuousvariable gates [61].One such method involves simulating a conditional form of optomechanical interaction [62] using photonic ancillas and homodyne detectors [63].Despite its potential, the achievable coupling strength is weak, and the implementation is probabilistic.Previous methods [23,64,65] to engineer operations using an auxiliary qubit mainly focused on higher-order single-mode Hamiltonians.However, extending these to two-mode simulations presented significant complexities, particularly the necessity of an exotic three-body interaction for [64].Recently, schemes utilizing the optical Fredkin gate and cross-Kerr interaction induced by a strong drive have been proposed [66][67][68].However, these methods involve linearizing a naturally weak, higher-order optical interaction into a lower-order interaction, leading to decreased accuracy and resource efficiency.Extending these methods to simulate various nonlinear couplings, especially between hybrid modes at the different platforms, is challenging.
In this work, we introduce a highly efficient constructive approach to simulate the dynamics of deep-strong quantum multimode bosonic coupling.We show how alternating a selective dispersive interaction conditioned on the individual Fock state in a cavity mode (SNAP gate) [69,70] with Rabi gates can effectively reproduce arbitrary-order radiationpressure interaction.Our approach does not rely on existing nonlinearities between oscillators and offers an exponential enhancement in the resource usage over other methods, reproducing distinctive effects of radiation pressure and membrane-in-the-middle optomechanical systems.Superconducting circuits are a promising platform for realizing this approach.Recent advancements have demonstrated the feasibility of coupling a transmon qubit to multiple cavity modes and achieving switchable dispersive and resonant interactions [25][26][27][28].Our work not only facilitates the exploration of experimentally challenging nonlinear coupling regimes but also significantly enhances the capabilities of quantum simulation, opening up new capabilities for hybrid quantum information processing using qubits and oscillators and tests of new quantum nonlinear phenomena.

II. METHOD
Our objective is to implement general deep-strong cavity optomechanical Hamiltonians, denoted as Ĥk,l = Ω k,l nk Here Ω k,l represents the coupling strength, and k, l ∈ Z. Here, the number operator nL = â † â and a quadrature position operator XM = b+ b † √ 2 where â ( b) is the annihilation operator for the optical (mechanical) mode.The mode index L, M will be set as 1, 2 in the simulation.The unitary evolution operator by this Hamiltonian is given as Ûk where T k,l is the dimensionless strength.The challenge lies in both nonlinear regimes at the level of a few energy quanta and the deep-strong coupling that largely dominates over free oscillations of both optical and mechanical modes.We focus on the lowest orders to demonstrate feasibility with k = 1 and l = 1 representing deep-strong coupling in movable cavity mirror systems causing radiation-pressure-like coupling linear in the mechanical operator and the l = 2 representing the membrane-in-the-middle optomechanics (Fig. 1a).

A. Dispersive-Rabi gate method
Our first approach, which serves a benchmark to a new method in Sec.II.B, considers a qubit that interacts with two oscillators in the different regimes of resonance alternatingly between dispersive and Rabi qubit-oscillator gates, mediating a virtual effective interactions between the oscillators deterministically without the need for any measurement, as in Fig. 1b.The basic premise is that by switching between these two different coupling regimes and leveraging the non-commuting Pauli operators σj with j = x, y, z, the resulting operator transformations can progressively build up higher-order nonlinear interactions.This circuit takes advantage of the Baker-Campbell-Hausdorff (BCH) formula [71], which allows the multiplication of non-commuting operators to be approximated as a series of commutators.This formula possesses the power to systematically generate higher order nonlinearity when applied recursively [23,61,64].It first builds a tripartite interaction, where a qubit non-demolitionally mediates the indirect effective interactions between the two oscillators.A previous version of this incremental approach exploiting a mediating qubit has been employed for various purposes [65,72,73].The BCH formula can approximate the method in [64] for small parameters, reducing it to the dispersive-Rabi gate method.
For example, alternating two different kinds of unitary interactions such as dispersive gates Ûz = exp[iϵσ z n1 ] [25] and Rabi gates Ûx = exp[iϵσ x X2 ] [1,2] as in the following sequence makes a weak bosonic interaction: for a very small ϵ ≪ 1.The qubit degree of freedom can be ignored if we initially prepare the eigenstate of the engineered Pauli operator σy , such as |± i ⟩.The dispersive interaction can arise from the Rabi coupling far from resonance [19], and can be linearized to Rabi intereaction by applying strong displacement operations [74].Recent advancements in the experimental realization of these interactions have been made in cavity [75,76] and circuit QED [25,[77][78][79] and trapped ion systems [7,29].However, this incremental approach has limited validity at a high strength, requiring a large number M ≫ 1 of repetitions to achieve a high strength target gate [23,61,71].The total resource usage in terms of total coupling strength is counted as 4M ϵ.We can improve the approximation using concatenated FIG. 1.Quantum simulation circuits of optomechanical interactions switching between two types of qubit-oscillator gates.a Target nonlinear interactions involving radiation pressure coupling in movable cavity mirror Ĥ1,1 or membrane-in-the-middle coupling Ĥ1,2 between an optical cavity mode and a mechanical mode.We aim to simulate these systems using physical experimental setups such as trapped ion or circuit QED by mapping cavity and mechanical modes to two harmonic oscillators.b Circuit diagram illustrating the incremental dispersive-Rabi gate approach serving as a benchmark, alternating dispersive (off-resonant) and Rabi (on-resonant) qubit-oscillator interactions to induce an effective radiation-pressure coupling between the modes.This sequence should be repeated multiple times M ≫ 1 to simulate a strong target coupling accurately.c Circuit diagram showing the non-incremental SNAP-Rabi gate approach using N photon-number selective dispersive gates (SNAP gates) and Rabi gates to accurately reproduce the movable cavity mirror Hamiltonians.In both methods, the membrane-inthe-middle coupling can be simulated by substituting X by X2 .
operations as in Appendix A, and online squeezing [80].A more complex circuit using concatenated BCH operations can create Ôk,l [T k,l ] that simulates ideal operations Ûk,l [T k,l ] for higher k and l as theoretically shown in [72].

B. SNAP-Rabi gate method
To overcome the limitations of the incremental method, we introduce a non-incremental approach that leverages SNAP gates [69,70] to address each Fock state independently.We utilize a circuit that bears resemblance with [33].For approximate simulation, we use SNAP gates that flip the qubit selectively for a specific Fock state |n⟩.Combined with Rabi gates, the following sequence constructs a selective optomechanical coupling on n-th Fock subspace Πn ≡ |n⟩ ⟨n|: where |±⟩ are σx eigenstates.Repeating this sequence for all n upto a finite number N yields an N -th order approximation of the optomechanical interaction without incremental limitations: In the qubit |−⟩ subspace, the erroneous operation is eliminated by orthogonality.An improved SNAP-Rabi gate sequence in terms of simulation accuracy and resource efficiency is given as (Fig. 1c): where the term exp[−i π 2 N −1 n=0 Πn σ z ] can be omitted for resource efficiency by sacrificing the accuracy.A visual depiction of how the SNAP-Rabi gate sequence constructs the simulated optomechanical interaction in a progressive, selective manner is provided in Appendix C.This gate uses only total strength T 1,1 N/2 of the Rabi gates, as opposed to the T 1,1 N (N + 1)/4 in the previous approach in (3), and thus is even more resource-efficient for a large N .The total resource usage of total coupling strength is given as T 1,1 N/2 + πN .It also yields better fidelities under qubit errors, due to the absence of an large erroneous displacement depending on N (N + 1) in ( 3) and a better behavior in the high Fock subspace beyond ΠN .
It is noteworthy that by using a Rabi interaction of the form exp[itF (n) 1+σx 2 X2 ] with a modulated strength function , instead of a standard Rabi interaction in (2), we can simulate an arbitrary-order optomechanical interaction.If we modulate the Rabi gate strength to implement a second order optomechanical interaction, we obtain: The total Rabi gate usage is given as ].The relative resource scaling advantages of the SNAP-Rabi approach compared to other methods is analyzed in more depth in Appendix E.
To engineer a higher optomechanical interaction of the form Ŝ1,2 [T ] in a similar manner, we require access to a second order Rabi interaction of the form exp[itσ x X2 ] with a varied strength t.The standard Rabi gate can be transformed into the second order Rabi gate of the form σx X2 at large detuning, with a noise-controlled squeezing occurring concurrently, as observed in [81].Alternatively, the second and higher order Rabi interaction can be efficiently engineered using a decomposition method [23].By employing higher-order Rabi interactions engineered via decomposition methods or derived from dispersive interactions at large detuning, our technique can simulate the membrane-in-themiddle interaction Û1,2 and more generally Ûk,l , combining such interactions with SNAP gates as outlined in (2).This SNAP-Rabi technique can be extended further to multipartite gates such as exp[iχn k1 .], providing a versatile toolbox for multimode bosonic simulations.The following sections compare these simulated dynamics by engineered optomechanical interactions with the benchmark Kerr-based method.The novel SNAP-Rabi gate method provides exponential improvements in both accuracy and resource usage beyond those offered by benchmarks.This method can be also adapted to engineer a cross-Kerr gate by simultaneously using SNAP gates on both modes.

III. RESULTS AND DISCUSSION
The discussion in this section focuses on assessing the non-Gaussian properties of the simulated nonlinear dynamics that are not accessible by the linearized Gaussian one.Additionally, we compare our methods with an approach based on naturally existing higher-order Kerr nonlinear interactions, which is briefly described in Appendix.B. This Kerr-based method also needs many repetitions by M times, due to its incremental limitation.
In our system, the implemented optomechanical gates Ôk,l [T k,l ] from incremental dispersive-Rabi method (1) and Ŝk,l [T k,l ] from non-incremental SNAP-Rabi methods (3) are not identical to the ideal optomechanical interactions Ûk,l [T k,l ], and therefore, it is crucial to investigate the extent to which the observable effects can reproduce the target nonlinear dynamics for various input states, especially at deep-strong coupling regime T k,l ≈ 1. Nonlinear optomechanical processes allow the phase-insensitive generation of coherent phase-sensitive effects (displacement and squeezing) on the mechanical mode, which is not possible with linearized versions.Therefore, phase-insensitive coherent states ρ prc [β] =  n+1 n /(n + 1) with Bose-Einstein statistics are chosen as input states, whose mean photon number n can be experimentally adjusted.A thermal state represents the maximum entropy average of arbitrary pure states, subject to the constraint of a given average number of quanta.These classical states can be easily prepared for any oscillators, providing experimentally straightforward initial states to test quantum effects from the simulations.For entanglement analysis of simulated interaction, we use phase-sensitive coherent states to understand the impact of amplitude and phase noise.The state in the on-resonant mode simulating mechanical mode is chosen to be in a ground state or in thermal states with the same average quanta from realistic experimental considerations.
We now assess the accuracy of the simulated optomechanical interactions based on three important criteria: i) fidelity between output states of ideal and simulated interactions, ii) ability to reproduce optomechanical phenomena like noise-driven displacement and squeezing, iii) accuracy in generating target entanglement.The rough evaluation is given by the fidelity, defined as , between the output states by the target ideal interactions ρ id and the approximate interactions ρ re .This quantifies how close the output states from the simulated dynamics are to the ideal target dynamics and serves as the average measure of closeness of the states.Higher fidelity indicates the simulation more accurately reproduces the desired nonlinear effects.In Fig. 2, the SNAP-Rabi gate method on Poissonian and Bose-Einstein noise shows a linear scaling in the resource usage for the infidelity drop, corresponding to an exponential improvement over the incremental dispersive-Rabi and Kerr-based methods that show exponential scaling in the resource usage for the infidelity drop.For example, at N = 10 of the SNAP-Rabi gate, the total resource gate strength consumed is about 0.5% of that by the dispersive-Rabi with similar fidelity.This exponential enhancement in resource usage is maintained even for simulating the higher-order membrane-in-the-middle interaction.The results demonstrate the significantly higher accuracy and resource-efficiency of the SNAP-Rabi gate approach in reproducing the target nonlinear dynamics.The fidelity scaling trends and analysis of how the different simulation methods react to increased resources are further detailed in Appendix E. In our stability analysis, we considered the effects of losses and noise.Each Rabi and SNAP gate was subjected to a constant level of qubit dephasing (dq), boson loss (dη), and qubit loss (dr), although in practical scenarios these values could depend on gate strengths.These losses led to a saturation in accuracy for large N , a consequence of the interplay between enhanced approximations and accumulating errors.Notably, these losses had a more substantial impact on the benchmark methods (dispersive-Rabi method and Kerr-based method) due to their significantly higher resource usage compared to the SNAP-Rabi gate method.A detailed summary of the effect of boson loss is provided in Appendix G. Secondly, we investigate optical noise-induced mechanical quantum coherence, which is a basic nonlinear effect at a low number of quanta.The nonlinear optomechanical interaction can induce displacement in the movable cavity mirror.It is quantified by the signal-to-noise ratio (SNR) of displacement comparing the mean induced displacement to the initial uncertainty quantifying how well the simulated interaction displaces the mechanical state beyond Gaussian noise (see Appendix D for its definition).The signal-to-noise ratio (SNR) compares the induced displacement of the mechanical state to its initial uncertainty.A higher SNR indicates the nonlinear optomechanical interaction can displace the mechanical state beyond the original noise levels.Squeezing defined by the smallest eigenvalue of a covariance matrix can be induced in the mechanical mode by membrane-in-the-middle interaction from phaseinsensitive states in the optical mode depending on the phase-insensitive photon number.The mechanical quantum Gaussian coherence arising from the noise in the optical mode serves as a key effective witness of the genuine nonlinear optomechanical interaction at the quantum level.The noise-induced squeezing of the membrane-in-the middle also appears qualitatively different from many other methods used to induce squeezing [82].The accuracy of the SNR and induced squeezing has a similar resource scaling as the fidelity.The details about these metrics is explained in Appendix D.
In Fig 3 a), we demonstrate that the SNRs for the Poissonian and Bose-Einstein noise with different n values in the simulated optical mode, obtained from the simulated movable cavity mirror interactions Ô1,1 [T 1,1 ] and Ŝ1,1 [T 1,1 ], approach those from the ideal optomechanical interaction Û1,1 [T 1,1 ] for various T 1,1 , especially for a low n.In Fig. 3 b), we illustrate the minimum variance of quadratures by various processes simulating Û1,2 [T 1,2 ].These results demonstrate the squeezing generated beyond a Gaussian approximation of linearized interaction, as indicated by the minimum variance falling below shot-noise limit ∆X 2 p < 1/2, which implies the existence of a nonclassical phasesensitive state.For both the Poissonian and Bose-Einstein noises in the simulated optical mode with a ground state in the simulated mechanical mode at the initial time, the general trends of the SNRs and induced squeezing by the SNAP-Rabi gate are highly overlapping with that by the ideal processes even at a deep-strong strength beyond incremental regime, suggesting that our approach effectively achieves the desired nonlinear effect with a higher accuracy than the Kerr-based approach.
The third criterion characterizes the non-local aspect of various dynamics via entanglement negativity.We quantify the entanglement of a bipartite state density operator ρ of simulated optical and mechanical modes by the entanglement negativity , where ρ PT is the partial transpose of ρ and Tr[| • |] denotes the trace norm [83,84].Entanglement negativity quantifies the amount of entanglement between the optical and mechanical modes.Higher negativity indicates greater quantum entanglement.Note that bipartite qubit entanglement of the form |ψ qub ⟩ ≡ 2 −1/2 (|1⟩ 1 |0⟩ 2 ±|0⟩ 1 |1⟩ 2 ) has a maximal negativity of N[|ψ qub ⟩ ⟨ψ qub |] = 1/2 or 1 ebit.This exceeds the entanglement predicted by the covariance matrix under Gaussian approximation.On the other hand, continuous variable states such as Gaussian states can have more than 1 ebit of entanglement, but they lack quantum non-Gaussian properties.Entanglement exceeding 1 ebit with non-Gaussian characteristics beyond the covariance matrix belongs to a completely distinct class.An ideal optomechanical interaction has a specific entangling nature due to the product of two different operators, nL and XM , which have discrete and continuous spectra of eigenvalues.Generating such hybrid entanglement is a key signature of nonlinear quantum effects.
In Fig. 4, the entanglement negativity is compared to Gaussian entanglement negativity, which refers to the entanglement predicted by the covariance matrix under Gaussian approximation, produced by the ideal and simulated dynamics.Both the simulated interactions and ideal optomechanical coupling generate entanglement that exceeds the Gaussian entanglement negativity, indicating the presence of authentic non-Gaussian entanglement beyond what is predicted within the Gaussian approximation framework.States diagonal in the Fock basis, such as a Bose-Einstein and a Poissonian noise in simulated optical mode, cannot generate entanglement and only result in classical correlations.In the simulated optical mode, therefore, we investigate input coherent states with various average photon number n that lacks quantum nonclassicality and non-Gaussianity.Both target ideal and simulated dynamics generate entanglement exceeding 1 ebit.This entanglement surpasses the entanglement of any Gaussian states with the same covariance matrix [85], demonstrating both authentic non-Gaussian, beyond-two-qubit entanglement.The output state by the membrane-in-the-middle dynamics has non-Gaussian entanglements without any entanglement predicted by the covariance matrix within Gaussian approximation.The agreement of the SNAP-Rabi gate with the target p induced by the ideal and approximate membrane-in-the-middle dynamics at T1,2 = 1.Repetition of M = 50 was used for incremental dispersive-Rabi method, while Kerr-based approach was without repetition.Variance below the shot noise limit (0.5) indicates the generation of squeezing and nonclassical mechanical states.In all cases, the SNAP-Rabi gate method shows excellent agreement with ideal results, validating its ability to accurately reproduce the target dynamics.FIG. 4. Entanglement negativity and Gaussian negativities generated by the ideal and simulated dynamics.The simulation parameters are the same as in Fig. 3. Entanglement above 1 ebit (Nc = 1/2) and Gaussian entanglement (gray shade) indicates presence of non-qubit and non-Gaussian entanglement.In b, the output state has purely non-Gaussian entanglements due to the inherent non-Gaussian nature of the interaction, as indicated by 0 Gaussian negativities.In all cases the output state has a Schmidt rank ∞ demonstrating the continuous variable nature of the entanglement.The SNAP-Rabi gate method closely reproduces both the Gaussian and non-Gaussian entanglement of the ideal interaction.This demonstrates its ability to simulate the complex hybrid continuous-discrete variable entanglement dynamics.
ideal coupling is much closer than both the dispersive-Rabi gate and the Kerr-based simulation.The exponential enhancement in resource-scaling of the accuracy in the simulation of the entanglement negativity is maintained.This result demonstrates that SNAP-Rabi gate method reproduces the complex continuous-discrete variable entanglement arising from the nonlinear coupling more precisely than the Kerr-based method and the dispersive-Rabi gate method.The intensity-intensity correlation between the optical and mechanical modes is analyzed for the different simulation methods and summarized in Appendix F, providing further validation of the approaches.
Our work primarily operates within the interaction picture and omits the effective frequency.In current experimental optomechanics setups, the effective mechanical frequency (ω m ) typically falls within the MHz to GHz range.The actual experimental optomechanical interaction strength (g m ), however, is typically much smaller, with g m /ω m on the order of 10 −3 to 10 −6 .In our simulation, we can achieve an ultrastrong coupling where the resource coupling (denoted as T /2) is on the order of 1, such that T 1,1 /ω m reaching a value of approximately 2. This results in a simulated optomechanical interaction strength that is enhanced by several orders of magnitude relative to current state-of-the-art experiments.This significant enhancement of the interaction strength underscores the potential of our proposed method to advance research in this field.The SNAP-Rabi gate approach demonstrates a substantial efficiency advantage over conventional techniques such as the dispersive-Rabi and Kerr-based methods.This considerable gain in resource efficiency allows the SNAP-Rabi method to accurately simulate deep-strong nonlinear optomechanical interactions that remain difficult to directly probe and investigate in existing experimental optomechanical systems.It allows for the exploration of deep-strong nonlinear couplings between bosonic oscillators, a domain that remains challenging to access with current technology.

IV. CONCLUSIONS
Quantum simulation of nonlinear bosonic interactions in challenging deep-strong coupling can allow proof-ofprinciple tests of many new phenomena.We introduced an accurate and efficient method for simulating challenging nonlinear dynamics by alternating qubit-oscillator coupling regimes.Our technique does not rely on existing bosonic nonlinearities and provides an exponential improvement in accuracy resource efficiency over previous methods.Our simulations demonstrate that the simulated dynamics exhibit the essential nature of a new nonlinear generator of coherence and nonclassical states and hybrid entangler, validating the accuracy of our scheme in reproducing the target dynamics in a previously unexplored regime.The signal-to-noise ratio, and entanglement negativity analysis confirm the ability to reproduce the complex dynamics of ideal optomechanical systems beyond the accuracy of existing Kerr nonlinearity-based approaches.The SNAP-Rabi gate method based on SNAP gates [69,70], in particular, shows excellent agreement with ideal optomechanical interactions.The proposed approach can be extended to simulate higher-order nonlinearities by utilizing higher-order Rabi gates or dispersive gates, broadening the class of dynamics that can be simulated.
These first proposals focused on specific targets provide broad and valuable tools for hybrid continuous-discrete variable quantum information processing and simulations of nonlinear bosonic systems.However, implementing this method needs further experimental development, including switching a single qubit between resonant and dispersive coupling with multiple cavity modes, a capability never tested in current superconducting circuit QED and ion trap systems limited to fixed interaction regimes.While advancements in achieving switchable qubit-oscillator couplings on platforms like superconducting circuits suggest that an experimental demonstration may be feasible with continued progress [12,19,25,26,28], more research is required to fully explore their capabilities and overcome limitations in current systems regarding fixed interaction regimes.Our analysis serves as an important step toward accessing intriguing dynamics for future quantum technologies.arrangement of two types of interactions: and using a sequence with different signs as where the last line was obtained in a weak strength limit t 1 , t 2 ≪ 1.Here, a product form of Bf ( Â) with an infinite order polynomial f has been achieved in the exponent, which can become the sources of high-order nonlinear interactions.This approximation can be further improved by suppressing the errors by concatenating with similar operations with opposite signs by which the unwanted terms are cancelled out.More rigorously, we can use the Fourier series expansion to obtain the target interaction efficiently.Using exp Note that this is an exact equation, the validity of which is not limited by the parameters of t 1 , t 2 .Now these operations can be combined to make the target function exp For a finite order approximation with k max , we can get a fairly good approximation as the strength required is sufficiently weak for a high order contributions.
A better approximation can be obtained using the following approximation: AB ≈ 0.6(−e i(−A−B) + e i(A−B) + e −i(A−B) − e i(A+B) ).Therefore, we obtain (A5) Here we used σ2 x = 1.Here, each term can be engineered from (A3) with substitution A → A ± B, B → 1.We can go a step further by using Fourier transform corresponding to a continuous limit of Fourier series expansions.We note that for any quantity x, dxx exp[ixp] = −iδ ′ (p) by the Fourier transform with a integration variable p where  In addition to fidelity, other metrics can also characterize how well the simulated interactions reproduce the target optomechanical dynamics.The first-order optomechanical interaction Û1,1 possesses phase sensitivity in the mechanical mode, which results in displacement beyond the noise that is proportional to the photon number in the optical mode.In contrast, the linearized interaction cannot generate net mechanical displacement for phase-insensitive states, such as PRC or thermal states.Therefore, the successful implementation of the optomechanical interaction Û1,1 [T ] can be demonstrated by producing a mechanical displacement beyond the linearized regime, which generates displacement beyond shot noise and results in a mechanical state with non-zero off-diagonal elements in the Fock state basis in the mechanical mode.These off-diagonal elements indicate the quantum coherence induced by the optical noise.
To evaluate the effectiveness of implementing the optomechanical interaction Û1,1 [T ], we can use the SNR to measures the displacement generated relative to the initial Gaussian uncertainty located at the phase space origin.In the Heisenberg picture, any operator Ô under a unitary evolution Û is described as Û † Ô Û .The momentum operator shifted by the adjoint action of an ideal Û1,1 interaction is described by In contrast, the momentum operator shifted by the linearized interaction is described by: P The average momentum shift in (D1) can be non-zero for an input Fock state in the optical mode 1, while that in (D2) is zero due to the phase insensitivity of the Fock states, or the states diagonal in the Fock basis used in the simulation.Therefore, a non-zero average momentum shift can serve as evidence of the nonlinear optomechanical interaction, but a sufficient strength interaction is required to have the shift beyond shot-noise limit.
We can calculate the SNR using the formula SNR =⟨δ P2 ⟩/ Var(δ P2 ) for various average photon number n in the optical mode, where the average ⟨•⟩ and the variance Var(•) is calculated for the initial states.Initially, the simulated mechanical mode is assumed to be in a vacuum state without quantum coherence in any basis, including the Fock state basis.For PRC in the optical mode, the SNR is calculated to be equal to SNR (PRC) = √ n = α for the ideal interaction Û1,1 [T ], which is the coherent amplitude of the state, and is independent of the strength T .For a thermal state in the optical mode, the SNR increases approximately as SNR (th) = n n+1 n≪1 → √ n, and asymptotically saturates to the value 1 when n ≫ 1.Again, the SNR is independent of the strength T .
On the other hand, a higher order interaction such as Û1,2 [T ] = exp[iT n1 X2 2 ] can induce both the quantum coherence and the nonclassical squeezing effects.The presence of a square term in X in the Hamiltonian provides a mechanism for quantum squeezing in the mechanical mode in general.By using this second order ideal optomechanical interaction, we can derive operator relations in Heisenberg picture as follows: These relations allow us to obtain the momentum variance: We observe that the uncertainty ∆P 2 2 monotonically increases with T due to the product of covariances of mechanical mode 2 and photon number moments of optical mode 1.This increased uncertainty in P implies the existence of squeezing along some direction of quadrature in phase space.Intensity-intensity correlation g (2) (0) between optical mode and mechanical mode vs the average photon number n in the optical mode for different simulation methods.Initial states are (a) an initial thermal state and (b) an initial phaserandomized coherent state in the optical mode.The mechanical mode is assumed to be in the vacuum state.The SNAP gate method provides better agreement with the ideal optomechanical correlation, particularly for larger n.
To evaluate the impact of bosonic losses on the accuracy of our simulation schemes, we calculate the infidelity of our simulation protocols under different loss rates per gate.Figure H.1 shows the logarithmic infidelity versus resource usage for the SNAP-Rabi gate method for simulating bosonic Gaussian operations using a movable cavity mirror and membrane-in-the-middle.Different lines correspond to various loss rates per gate, from 10 −4 to 10 −1 .For both systems, the infidelity increases with loss rate and decreases with more resources used.
As for the question about dissipation in the referee report, we would like to point out that in realistic scenarios, dissipation can indeed impact the performance of our simulation protocols.However, our SNAP-Rabi gate method shows a certain level of resilience against bosonic losses.As can be seen from Fig. G.1, even under moderate loss rates, the method still maintains a relatively high fidelity, indicating that it could be robust against dissipation in practical implementations.Moreover, the rapid decrease of infidelity with increased resource usage suggests that the detrimental effects of the mild losses can be mitigated by using more resources, even though for a larger resource usage, the infidelity saturates due to the fixed per gate loss rate.Logarithmic infidelity versus resource usage by SNAP-Rabi gate method for simulating bosonic Gaussian operations using a movable cavity mirror and membrane-in-the-middle.Different lines correspond to various loss rates per gate, from − dη = 10 −4 to 10 −1 where dη is the loss parameter occurring at each gate.As resource usage increases, infidelity eventually saturates due to the constant loss rate per gate.

1 FIG. 2 .
FIG. 2.Simulation infidelity versus resource usage of various methods.Logarithmic infidelity vs. simulation order N proportional to the resource used in the SNAP-Rabi method and logarithmic repetition number log M used in the other methods for a,b Poissonian noise and c,d Bose-Einstein noise both with n = 1 at a,c movable cavity mirror dynamics at strength T1,1 = 1 and b,d simulation of membrane-in-the-middle dynamics at T1,2 = 1.Different curves show the impact of qubit loss (dashed) and dephasing errors (dot-dashed) during the Rabi gates. n

FIG. 3 .
FIG.3.Signal-to-noise ratio (SNR) of displacement and squeezing in the simulated mechanical mode induced by the ideal and simulated dynamics.a SNR of the displacement in the movable cavity mirror.The incremental dispersive-Rabi gate was made of M = 50 repetitions of the unit setups, Kerr-based gate was made of M = 10 repetitions, and SNAP-Rabi gate approach utilized N = 10 approximation order.The target strength was set as T1,1 = 1.b Minimum quadrature variance ∆X 2 p induced by the ideal and approximate membrane-in-the-middle dynamics at T1,2 = 1.Repetition of M = 50 was used for incremental dispersive-Rabi method, while Kerr-based approach was without repetition.Variance below the shot noise limit (0.5) indicates the generation of squeezing and nonclassical mechanical states.In all cases, the SNAP-Rabi gate method shows excellent agreement with ideal results, validating its ability to accurately reproduce the target dynamics.

2 √ πa 3 . 2 √Fock 1 ]
FIG. C.1.A diagram visualizing the SNAP-Rabi gate sequence on joint qubit-oscillator states.Each horizontal level shows the effect of the gate set S[n].It flips the qubit in X-basis for n-th Fock space, and displaces the mechanical mode conditioned by the higher Fock spaces.The upward arrows represent conditional displacement D[iT /√ 2] of the mechanical oscillator.This induces selective displacements that build up the simulated nonlinear optomechanical interaction in each Fock subspace.
Appendix D: Signal to noise ratio and induced mechanical squeezing

)FIG. E. 1 .
FIG. E.1.Scaling behavior against resource usage over various mean photon number of Poissonian and Bose-Einstein noise models.a Simulation of movable cavity mirror dynamics.b Simulation of Membrane-in-the-middle dynamics.
FIG.F.1.Intensity-intensity correlation g(2) (0) between optical mode and mechanical mode vs the average photon number n in the optical mode for different simulation methods.Initial states are (a) an initial thermal state and (b) an initial phaserandomized coherent state in the optical mode.The mechanical mode is assumed to be in the vacuum state.The SNAP gate method provides better agreement with the ideal optomechanical correlation, particularly for larger n.
FIG.G.1.Logarithmic infidelity versus resource usage by SNAP-Rabi gate method for simulating bosonic Gaussian operations using a movable cavity mirror and membrane-in-the-middle.Different lines correspond to various loss rates per gate, from − dη = 10 −4 to 10 −1 where dη is the loss parameter occurring at each gate.As resource usage increases, infidelity eventually saturates due to the constant loss rate per gate.