Introduction

It has been scrutinized that entanglement1,2 in multi-partite quantum systems, plays a crucial role in quantum technologies applications such as quantum information processing3,4,5, quantum computation5,6, and quantum simulation7, respectively. Entanglement as a quantum resource can lead to a speed-up in the running time of quantum algorithms8,9,10,11. Furthermore, the entanglement characterization in many-body systems may provide helpful information concerning whether it is possible to find underlying features of the low-lying energy spectrum with accurate numerical methods12,13. Likewise, entanglement discontinuity is an excellent indicator to characterize phase transitions on quantum systems14,15,16,17. Besides, in the context of quantum metrology and sensing, using entangled states has achieved quantum-enhanced precision measurements near the shoot-noise limit18,19,20, among other applications.

A fundamental condition to generate such multi-partite entangled states relies on the capability of our quantum platform to access a set of interactions or controlled quantum gates that communicate all the system components (all-to-all). Feasibly controllable trapped-ion platform21 gives a step forward in this direction with the implementation of the so-called Sørensen-Mølmer quantum gate22,23,24, where external lasers applied to a confined array of alkali atoms permit access to the red and blue sidebands so that the effective interaction between the ions is of the form \({\sigma }_{\ell }^{x}{\sigma }_{{\ell }^{{\prime} }}^{x}\)25,26.

On the other hand, it is possible to engineer similar interactions in superconducting quantum circuit and circuit quantum electrodynamics (cQED)27,28,29,30,31,32,33,34,35,36,37,38,39,40,41, where Josephson junction-based electrical circuits having discrete energy spectrum mimics artificial atoms42,43,44,45,46,47,48,49,50,51,52,53,54,55, whereas quantized field modes correspond to either LC resonators, coplanar waveguide or stripline resonator56,57,58,59,60. In this architecture, it is possible to engineer multi-qubit interactions by coupling all of them to a common resonator bus such that in the dispersive regime, we obtain a like-Sørensen-Mølmer interaction61,62,63,64 as a result of a second-order interaction which in principle are slower than the resonant case make them fragile against the unavoidable action with environment65,66,67,68.

Enhanced performance of these protocols may require access to higher coupling strength values where the system operates in the so-called ultrastrong or deep strong coupling regime69,70,71,72. In such regimes, however, the high hybridization of the energy levels makes it difficult to distinguish between the light and matter degree of freedom. An alternative approach relies on engineering either a pulse sequence or the coupling strength following the ubiquitous methods of shortcut to adiabaticity (STA)73,74 that allow us to control a quantum system to accelerate an adiabatic evolution overcoming preparation errors and minimizing the action of the environment74. STA has received renewed interest in the context of cQED since it has been generalized to open quantum system75,76 permitting to design of counter-diabatic and optimal pulses to speed up a dissipative evolution77,78. Motivated by this, we propose a reverse-engineering method to accelerate the generation of multi-partite entangled states. We design a modulated longitudinal coupling strength that accelerate the generation of multi-partite photnonic/qubit states within the nanosecond scale that does not scale with the number of systems (field modes and qubits). Because of the short time, we observe no detrimental effect produced by the action of the environment. Finally, we propose the possible implementation in cQED architecture.

Results

Longitudinal Interaction

We start describing the fundamental ingredient in generating multi-partite entangled states. To do so, let us consider a set of N two-levels systems of frequency Ωn coupled to M quantized field modes of frequency ωm through time-dependent longitudinal coupling \({g}_{n}^{m}(t)\) governed by the following Hamiltonian (ℏ ≡ 1)

$${{{{{{{{\mathcal{H}}}}}}}}}_{n}^{m}= \mathop{\sum}\limits_{n}\frac{{\Omega }_{n}}{2}{\sigma }_{n}^{x}+\mathop{\sum}\limits_{m}{\omega }_{m}{a}_{m}^{{{{\dagger}}} }{a}_{m}\\ +\mathop{\sum}\limits_{n,m}{g}_{n}^{m}(t){\sigma }_{n}^{x}({a}_{m}^{{{{\dagger}}} }+{a}_{m}),$$
(1)

where \({\sigma }_{n}^{x}\) corresponds to the x-component Pauli matrix characterizing the n-th two-level system, and am (\({a}_{m}^{{{{\dagger}}} }\)) stands for the annihilation (creation) bosonic operator of each field mode, respectively. In cQED systems, we can engineer the longitudinal interaction in an artificial atom coupled to a resonator sharing the common external magnetic flux79,80, or couple both subsystems through a superconducting quantum interference device (SQUID)81,82.

On the other hand, the structure of the longitudinal coupling between two-level systems with a quantized field mode is suitable for implementing reversed engineering protocols since both interaction and free energy term commutes. Such relation allows us to find an exact adiabatic transformation that accelerates quantum processes such as the qubit readout78 or implementing faster two-qubit gates83, and now we will use it to accelerate the generation of multi-partite entangled states of either photonic or qubit states, respectively.

To generate entangled photonic states, we assume that the field modes couple to a single two-level system, i.e., N = 1. In such a case, the Hamiltonian reads

$${{{{{{{{\mathcal{H}}}}}}}}}_{1}^{m}=\frac{\Omega }{2}{\sigma }^{x}+\mathop{\sum}\limits_{m}{\omega }_{m}{a}_{m}^{{{{\dagger}}} }{a}_{m}+\mathop{\sum}\limits_{m}{g}_{1}^{m}(t){\sigma }^{x}({a}_{m}^{{{{\dagger}}} }+{a}_{m}).$$
(2)

The Hamiltonian dynamics \({{{{{{{{\mathcal{H}}}}}}}}}_{1}^{m}\), in the interaction picture, corresponds to a state-dependent cavity drive represented by the following time-evolution operator

$${U}_{1}^{m}(t)=\mathop{\prod}\limits_{m}{\hat{{{{{{{{\mathcal{D}}}}}}}}}}_{m}({\alpha }_{m}(t){\sigma }^{x}),$$
(3)

where \({\hat{{{{{{{{\mathcal{D}}}}}}}}}}_{m}({\sigma }^{x}{\alpha }_{m}(t))=\exp ({\sigma }^{x}({\alpha }_{m}(t){a}_{m}^{{{{\dagger}}} }+{\alpha }_{m}^{* }(t){a}_{m}))\) is the displacement operator of the mth field mode with \({\alpha }_{m}(t)=-i\int\nolimits_{0}^{t}{g}_{m}(s){e}^{i{\omega }_{m}s}ds\) as the cavity displacement. For a time-independent coupling strength, we obtain that at time Tk = (2k + 1)π/ωm the field mode reaches its maximum displacement \({\alpha }_{\max }=\pm 2{g}_{1}^{m}/{\omega }_{m}\) that depends on the qubit state. Thus, for the system prepared in the initial state \(\left\vert \Psi (0)\right\rangle ={\left\vert g\right\rangle \bigotimes }_{m = 1}^{M}{\left\vert 0\right\rangle }_{m}\) (\(\left\vert g\right\rangle\) is the ground state of the two-level system, and \({\left\vert 0\right\rangle }_{m}\) is the vacuum state of the mth field mode) the system evolves to the so-called Greenberger-Horne-Zeilinger cat state84,85,86,87

$$\left\vert \Psi (T)\right\rangle =\frac{1}{\sqrt{2}}\left[\left\vert e\right\rangle \mathop{\bigotimes}\limits_{m = 1}^{M}\left\vert {\alpha}_{\max }\right\rangle_{m}+\left\vert g\right\rangle \mathop{\bigotimes}\limits_{m = 1}^{M}\left\vert -{\alpha }_{\max }\right\rangle_{m}\right],$$
(4)

by rotating the qubit state along the y axis, we obtain

$$\left\vert \Psi \right\rangle =\frac{1}{\sqrt{2}}\left[\left\vert e\right\rangle \mathop{\bigotimes}\limits_{m = 1}^{M}{\left\vert {\alpha }_{+}\right\rangle }_{m}+\left\vert g\right\rangle \mathop{\bigotimes}\limits_{m = 1}^{M}{\left\vert {\alpha }_{-}\right\rangle }_{m}\right],$$
(5)

where \({\vert {\alpha }_{\pm }\rangle }_{m}=({\vert {\alpha }_{\max }\rangle }_{m}\pm {\vert -{\alpha }_{\max }\rangle }_{m})/\sqrt{2}\) is the even/odd coherent state superposition88. Thus we are lead with a multi-partite hybrid light-matter state embedding maximal entanglement, not being affected by local operations and classical communication (LOCC)89.

On the other hand, it is possible to generate multi-partite entangled states of qubits when we consider that N two-level systems are coupled to a single field mode described by the Hamiltonian

$${{{{{{{{\mathcal{H}}}}}}}}}_{n}^{1}=\omega {a}^{{{{\dagger}}} }a+\mathop{\sum}\limits_{n}\frac{{\Omega }_{n}}{2}{\sigma }_{n}^{x}+\mathop{\sum}\limits_{n}{g}_{n}^{1}(t){\sigma }_{n}^{x}({a}^{{{{\dagger}}} }+a).$$
(6)

In the interaction picture, the time-evolution operator of \({{{{{{{{\mathcal{H}}}}}}}}}_{n}^{1}\) can be written in a factorized form using the Baker-Campbell-Hausdorff (BCH) formula90

$${U}_{n}^{1}(t)=\mathop{\prod}\limits_{n}{e}^{-i{A}_{n}{\sigma }_{x}^{x}a}\mathop{\prod}\limits_{n}{e}^{-i{A}_{n}^{* }{\sigma }_{x}^{x}{a}^{{{{\dagger}}} }}\mathop{\prod}\limits_{n,{n}^{{\prime} }}{e}^{-i{B}_{n{n}^{{\prime} }}{\sigma }_{n}^{x}{\sigma }_{{n}^{{\prime} }}^{x}}.$$
(7)

The coefficients An(t) (\({A}_{n}^{* }(t)\)) and \({B}_{n{n}^{{\prime} }}(t)\) are obtained calculating the Schrödinger equation for the time-evolution operator \(i\,{\dot{U}}_{n}^{1}(t)={{{{{{{{\mathcal{H}}}}}}}}}_{n}^{1}{U}_{n}^{1}(t)\) (we refer to the reader to the Supplementary Note 1 for the detailed calculation), we obtain that these coefficients must satisfy the following differential equations

$$\frac{d{A}_{n}(t)}{dt}={g}_{n}^{1}(t){e}^{-i\omega t},\quad \frac{d{B}_{n{n}^{{\prime} }}(t)}{dt}=i{A}_{{n}^{{\prime} }}(t)\frac{d{A}_{n}^{* }(t)}{dt}.$$
(8)

For time-independent coupling strength \({g}_{n}^{1}\) with the initial conditions \({A}_{n}(0)={B}_{n{n}^{{\prime} }}=0\), we obtain62

$${A}_{n}(t)=\frac{i{g}_{n}^{1}}{\omega }({e}^{-i\omega t}-1),$$
(9a)
$${B}_{n{n}^{{\prime} }}(t)=\frac{{g}_{n}^{1}{g}_{{n}^{{\prime} }}^{1}}{\omega }[-i({e}^{i\omega t}-1)-t].$$
(9b)

By choosing τ = 2π/ω, the coefficient An(τ) vanishes and \({B}_{n{n}^{{\prime} }}(\tau )\equiv {\theta }_{n{n}^{{\prime} }}=2\pi {g}_{n}^{1}{g}_{{n}^{{\prime} }}^{1}/{\omega }^{2}\). Consequently, the time evolution operator reduces to

$${U}_{n}^{1}(\tau ) = \mathop{\prod}\limits_{n,{n}^{{\prime} }}\exp \left[-i{\theta }_{n{n}^{{\prime} }}{\sigma }_{n}^{x}{\sigma }_{{n}^{{\prime} }}^{x}\right]\\ = \mathop{\prod}\limits_{n\ne {n}^{{\prime} }}\left[\cos ({\theta }_{n{n}^{{\prime} }}){\mathbb{I}}+i\sin ({\theta }_{n{n}^{{\prime} }}){\sigma }_{n}^{x}{\sigma }_{{n}^{{\prime} }}^{x}\right],$$
(10)

corresponding to the Sørensen-Mølmer quantum gate22,23 (SMG). It is worthy to mention that the SMG has been widely used in the context of quantum simulation to codify fermionic system in a set of coupled two-level system91,92. If we initialize the system in the state \(\left\vert \Phi (0)\right\rangle { = \bigotimes }_{n = 1}^{N}{\left\vert g\right\rangle }_{n}\otimes \left\vert 0\right\rangle\), and choose \({\theta }_{n{n}^{{\prime} }}=\pi /4\), the state evolves to a GHZ state93

$$\left|{\Phi}(\tau)\right\rangle=\frac{1}{{\sqrt{2}}}\left[\mathop{\bigotimes}\limits_{n=1}^{N}\left|g\right\rangle_{n}\, +\, e^{i\pi(N+1)/2}\mathop{\bigotimes}\limits_{n=1}^{N}\left|e\right\rangle_{n}\right]\left|0\right\rangle,$$
(11)

Notice that in those derivations, the time generation is constrained by the ratios \({g}_{1}^{m}/{\omega }_{m}\), and \({g}_{n}^{1}/{\omega }_{m}\), respectively. In particular, it is possible to implement the multi-partite gate for photons in the timescale T = 47.61 (ns), whereas for the multi-partite operation with qubits, the gate time is T = 19 (ns)62. Thus, to obtain faster time generation, we must achieve larger coupling strength values, meaning that the system will operate in the ultra-strong or deep-strong coupling regime69,70,71,72. A way to accelerate the generation time without demanding larger coupling strength relies upon the technique of STA74, where the modulation of system parameters or the addition of an additional term on the system dynamics leads to a speedup of the quantum processes. In what follows that we will explore the former alternative, the reverse-engineering method73,78, to design the coupling strength \({g}_{1}^{n}\) and \({g}_{n}^{1}\) such that the gating time is shortened significantly.

reverse engineering protocol

Afterwards defining the model, we will discuss how to design the coupling strengths \({g}_{1}^{n}\) and \({g}_{n}^{1}\) in the frame of reverse-engineering approach73,78. We will develop the theory considering a single two-level system (N = 1) coupled to a single field mode (M = 1) with coupling strength g(t), the extension to many qubits and field modes is easily derivable. In this case, we propose the solution of the Schrödinger equation of the form \(\left\vert \Psi (x,t)\right\rangle =\exp (-it\epsilon ){{{{{{{\mathcal{U}}}}}}}}(t)\left\vert \varphi (x,t)\right\rangle\), where ϵ = ω(n + 1/2) and \(\left\vert \varphi (x,t)\right\rangle =\root 4 \of {m{\omega }_{r}/\pi }\exp (-m{\omega }_{r}{x}^{2}/2)\) corresponds to the eigenfunction for the uncoupled cavity with Hamiltonian ωra†a and \(\left\vert \xi \right\rangle \equiv \left\vert g\right\rangle\) describes the ground state of the qubit. Additionally, \({{{{{{{\mathcal{U}}}}}}}}(t)\) stands for a unitary transformation that eliminates the qubit-field mode coupling94,95,96

$${{{{{{{\mathcal{U}}}}}}}}(t)={e}^{i\beta (t)}{e}^{-i{\dot{g}}_{c}(t){\sigma }^{x}({a}^{{{{\dagger}}} }+a)/{\omega }^{2}}{e}^{-{g}_{c}(t){\sigma }^{x}({a}^{{{{\dagger}}} }-a)/\omega }.$$
(12)

Here, the overdot notation represents time-derivative, and \(\beta (t)=-\int\nolimits_{0}^{t}{{{{{{{{\mathcal{L}}}}}}}}}_{g}(s)ds\) corresponds to a phase factor that relates the coupling strength g(t) with the auxiliary variable gc(t) through the classical Lagrangian

$${{{{{{{{\mathcal{L}}}}}}}}}_{g}(t)=\frac{{\dot{g}}_{c}^{2}(t)}{{\omega }^{3}}-\frac{{g}_{c}^{2}(t)}{\omega }-\frac{2{g}_{c}(t)g(t)}{\omega }.$$
(13)

To guarantee that \(\left\vert \varphi (x,t)\right\rangle\) corresponds to the exact solution of the time-dependent Schödinger equation, the classical variable must obey the following equation of motion

$${\ddot{g}}_{c}(t)+{\omega }^{2}[\,{g}_{c}(t)+g(t)]=0,$$
(14)

which is nothing but the Euler-Lagrange equation from the classical Lagrangian. The auxiliary variable gc(t) needs to satisfy the following boundary conditions to guarantee that at the initial/final time, the auxiliary variable does not participate in the system

$${g}_{c}({t}_{0})={\dot{g}}_{c}({t}_{0})={\ddot{g}}_{c}({t}_{0})=0,$$
(15a)
$${g}_{c}({t}_{f})={\dot{g}}_{c}({t}_{f})={\ddot{g}}_{c}({t}_{f})=0,$$
(15b)

Furthermore, we can add more conditions depending on the problem to be solved. For the photonic GHZ, we require that the final cavity displacement be larger as we can at the final time tf the final cavity displacement is larger than we can. In this scenario, together with the boundary conditions in Eq. (15a) and Eq. (15b), we add the following constrains

$$\alpha ({t}_{f})=-i\int\nolimits_{0}^{{t}_{f}}g(s){e}^{i\omega s}ds={d}_{\max },$$
(16)

corresponding to an arbitrary cavity displacement. On the other hand, for the generation of the GHZ state for qubits, apart from the boundary conditions given in Eq. (15a) and Eq. (15b) we require that

$${A}_{n}({t}_{f})=0,\quad {B}_{n{n}^{{\prime} }}({t}_{f})=\frac{\pi }{4}.$$
(17)

In light of the experimental implementation of our model provided in Section Physical implementation, the modulation of the longitudinal coupling strength g(t) is through an adequate modulation of the external flux ϕx(t). Therefore, we propose a Fourier decomposition of the form \({\phi }_{x}(t)={\sum }_{k}{c}_{k}\sin (\pi k\,t/{t}_{f})\) and plug this ansätze in Eq. (??) to compute the reversed engineering g(t) (see Section Physical implementation for the detailed derivation). In that way, the problem reduces to calculate the coefficient ck such gc(t) fulfills the boundary condition given in Eq. (15a), Eq. (15b) together with Eq. (16) for the photonic case, and Eqs. (17) for the qubits, respectively.

We numerically calculate the coefficients ck using the gradient descent minimization package of python97. Figure 1 shows the auxiliary variable gc(t) and the coupling strength g(t) obtained through Eq. (14) as a function of the dimensionless time t/tf for three different maximal cavity displacement \({d}_{\max }=\{1,3,5\}\), respectively. It manifests that the maximal value of the coupling strength g(t) strongly depends on the maximal value of the cavity displacement constraining the coherent state’s size, unlike the qubit case, where we do not observe such behavior in the resulting gc(t) and g(t), respectively. Moreover, it is shown that the modulations obtained in Fig. 1 satisfy the additional conditions stated in Eq. (16) and Eqs. (17) for the photonic and qubit case, respectively. We plot these quantities as a function of the dimensionless time t/tf depicted in Fig. 2. For the coupling strength g/2π = 21 MHz79 for the photonic system, we obtain a generation time tf = π/(40g) ≡ 3.74 (ns) achieving \(\alpha ({t}_{f})={d}_{\max }\). Whereas, the qubit-based GHZ state can be generated within the timescale tf = πω/(80g2) ≡ 1.89 (ns) for a coupling strength g/2π = 114 MHz62 where the conditions An(tf) = 0 and \({B}_{n,{n}^{{\prime} }}({t}_{f})=\pi /4\) are fulfilled, respectively.

Fig. 1: Coupling strength modulation.
figure 1

Reversely engineered coupling strength gc(t) (continuous line) and g(t) (dotted) as a function of the dimensionless time t/tf for a the GHZ for photonic systems, blue, orange and green data corresponds to the pulse considering α(tf) = 1, 3, 5, respectively. Whereas b stands for the GHZ in qubits. The numerical simulations are preformed with the following system parameters a ω/2π = 6.6 GHz, g/2π = 21 MHz, and b Ω/2π = 10 GHz, ω/2π = 1 GHz, and g/2π = 144 MHz.

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Fig. 2: Reversed enginnered modulation.
figure 2

Reversely engineered a absolute cavity displacement ∣α(t)∣ as a function of the dimensionless time t/tf, blue, orange and green lines correspond to the maximal cavity displacement of \({d}_{\max }=\{1,3,5\}\), respectively. b, c stands for the real (blue) and imaginary (green) parts of the reversely engineered ∣An(t)∣ and \(| {B}_{n{n}^{{\prime} }}(t)|\) coefficients, respectively. The numerical simulations are performed with the same parameters as those in Fig. 1.

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Multi-partite photonic state

We analyze the performance of the reversely engineered coupling strength g(t) to generate photonic GHZ states for M = 3 modes, see Hamiltonian of Eq. (2). In such a case, we assume that \({g}_{1}^{m}(t)\equiv g(t)\) ∀ m. We quantify the Wigner function W(x, p) of the reduced density matrix of the first field modes given by \({\rho }_{{f}_{1}}(t)={{{{{{{{\rm{Tr}}}}}}}}}_{\{{f}_{2},...,{f}_{M},q\}}[\vert \Psi (t)\rangle \langle \Psi (t)\vert ]\), where the sub-indexes fk, and q corresponds to the partial trace over the k-th field mode and the qubit, respectively. Given that W(x, p) for a coherent state \(\left\vert \alpha \right\rangle\) corresponds to a Gaussian centered in \(({{{{{{{\rm{Re}}}}}}}}(\alpha ),{{{{{{{\rm{Im}}}}}}}}(\alpha ))\)88, at the final STA time tf, where \({\rho }_{{f}_{1}}({t}_{f})=(1/2)(\vert {d}_{\max }\rangle \langle {d}_{\max }\vert +\vert -{d}_{\max }\rangle \langle -{d}_{\max }\vert )\), the Wigner function represents two Gaussians centered at \(\pm {d}_{\max }\). Figure 3a shows the Wigner function on the phase space corresponding to two points far apart from each other from a distance \(2{d}_{\max }\). Likewise, Fig. 3b, c shows the one-dimensional projection of W(x, p) in the coordinate and momentum space, respectively. In concordance with Fig. 3, we appreciate that on the coordinate projection, we should expect to see a single Gaussian centered at zero (no imaginary component on the coherent state) placed at \(p=\pm {d}_{\max }\). In contrast, in the momentum projection, we have to see two overlapping Gaussian centered at \(x=\pm {d}_{\max }\), respectively. From Fig. 3c we see a wide region where W(x, p) takes negative values, showing the non-local nature of the multi-partite entangled state88.

Fig. 3: Multi-partite photonic state.
figure 3

a Wigner representation of the reduced density matrix of the first field mode at final time tf for a system consisting into three field mode coupled to a single qubit. Projection of the Wigner function to the b coordinate space and c the momentum space, corresponding to a Gaussian function centered in \(\pm {d}_{\max }=3\). d Fidelity \({{{{{{{\mathcal{F}}}}}}}}\) between the target Greenberger-Horne-Zeilinger state in Eq. (5) with the state \(\left\vert \Psi (t)\right\rangle\) for two different maximal cavity displacement \({d}_{\max }=1\) to the blue line, and \({d}_{\max }=3\) for the green line, respectively. The numerical simulations are carried out with the same parameters as those in Fig. 1.

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Furthermore, we also calculate the Fidelity \({{{{{{{\mathcal{F}}}}}}}}(\rho ,\sigma )={({{{{{{{\rm{Tr}}}}}}}}[\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}])}^{2}\) (ρ and σ are arbitrary density matrices) between the state \(\left\vert \Psi (t)\right\rangle\) obtained by numerically solve the Schödinger equation of Eq. (2) with the target GHZ state defined in Eq. (5) for M = 3 field modes for two different maximal field displacement \({d}_{\max }=\{1,3\}\) as depicted in Fig. 3d. We observe that the reversed engineering protocol is insensitive to the final displacement \({d}_{\max }\), where we appreciate fidelities close to one for both modulations.

Thus far, we have demonstrated that our reverse-engineering protocol provides a flexible way to engineer the longitudinal coupling strength that allows us to generate a multi-partite photonic states at a short timescale (tf ≡ 3.2 (ns)), which is one order of magnitude shorter than the same protocol without the use of the reverse engineering approach (T = 47.61 (ns)). Moreover, our reverse engineering approach permit us fixing the final cavity displacement beyond the maximal distplacement \({\alpha }_{\max }=\pm 2{g}_{1}^{m}/{\omega }_{m}\). Nevertheless, nothing is said about the scaling of the protocol in terms of the number of field modes. We should expect that the gating time tf does not scale from our calculations. To assure that, we solve the dynamics by increasing the number of field modes from M = 1 to M = 4 and compute the fidelity \({{{{{{{\mathcal{F}}}}}}}}({t}_{f})\) at tf for each case. Those results are depicted in Fig. 4a, where we observe a high-fidelity generation of the cat Schödinger states (the maximal fidelities obtained with the protocol are around 98–99% for each case). Furthermore, we also see minor variations in the achieved maximal fidelity for the maximal number of modes simulated here. The exploration regarding a higher amount of field modes may be helpful to probe our conjecture. However, we are contained by the maximal displacement that bound the size of the whole Hilbert space, i.e., the maximal size of each field mode is chosen such \({N}_{\max }=| {d}_{\max }| +1\). In this sense, there exists a trade-off between the macroscopic size of the coherent state (related to the maximal displacement) with the maximal number of field modes to be simulated.

Fig. 4: Fidelity as function of the number of subsystems.
figure 4

Fidelity \({{{{{{{\mathcal{F}}}}}}}}\) at the shortcut to adiabaticity time tf as a function of the number of system constituents for a system corresponding to a one qubit coupled to M = {1, 2, 3, 4} field modes with maximal cavity displacement \({d}_{\max }=3\), and b one field mode coupled to N = {2, 4, 6, 8, 10, 12}, respectively. The numerical simulations are carried out with the same parameters as those in Fig. 1.

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Multi-partite qubit state

Alternatively, we will analyze the performance of the reverse-engineering protocol to generate GHZ states in a qubit-based system. For doing so, we study the dynamics of the Hamiltonian in Eq. (6) by considering N = 2 qubits assuming homogeneous coupling strength between the qubits with the field mode, i.e., \({g}_{n}^{1}(t)=g(t)\) ∀ n. Then, we can extend the result to a multi-qubit system up to 12 of them.

Similar to the previous section, we look at a reduced part of the Hilbert space corresponding to the reduced density matrix of the N qubits defined as \({\rho }_{q}(t)={{{{{{{{\rm{Tr}}}}}}}}}_{f}[\vert \Phi (t)\rangle \langle \Phi (t)\vert ]\), where the sub-index f corresponds to the trace over the field mode, where \(\left\vert \Phi (t)\right\rangle\) is the solution of the Schödinger equation for the Hamiltonian given in Eq. (6). Figure 5a shows the population evolution for a system composing by N = 2 qubits. We see that the occupation probabilities of the symmetric subspace \(\{\vert ge\rangle ,\vert eg\rangle \}\) does not be equal to zero during all the dynamics because the terms related to their evolution in the time evolution operator given in Eq. (7) is not zero at all the times. In fact, the quantity An(t) only vanishes at t = 0, and t = tf, respectively. Likewise, the population of the state \(\left\vert ee\right\rangle\) keeps changing until tf achieves its optimal value. In Fig. 5b we plot the fidelity \({{{{{{{\mathcal{F}}}}}}}}\) as a function of time with the target state given in Eq. (11). Since the symmetric state contributes to the whole state’s probability, fidelity tends to decrease until it reaches its maximal value around \({{{{{{{\mathcal{F}}}}}}}}=99 \%\). Concerning the scaling of the generation time for the GHZ state, we obtain that the gating time does not scale with the number of qubits. In order to prove that, we calculate the fidelity \({{{{{{{\mathcal{F}}}}}}}}\) evaluated at t = tf by increasing the number of qubits from N = {2 − 12}. Figure 4b shows the maximal fidelity as a function of the number of qubits. It is observed that the fidelities \({{{{{{{\mathcal{F}}}}}}}}\) are bounded between 98% and 99%, thus demonstrating that the gating time does not scale with the number of qubits for achieving a large fidelity.

Fig. 5: Revsersed engineering multi-qubit state generation.
figure 5

a Population evolution of the two-qubit states \(\left\vert gg\right\rangle\) (continuous blue), \(\left\vert ge\right\rangle\) (continuous orange), \(\left\vert eg\right\rangle\) (dotted green), and \(\left\vert ee\right\rangle\) (continuous red) computed from the Hamiltonian in Eq. (6) through the reverse- engineering protocol as a function of the dimensionless time t/tf, we observe that at the final time the state evolves to a GHZ state. b Fidelity \({{{{{{{\mathcal{F}}}}}}}}\) as a function of t/tf between the GHZ state of two qubits with the state \(\left\vert \Phi (t)\right\rangle\). The numerical simulations are carried out with the same parameters as those in Fig. 1.

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Notice that in the derivation of the unitary transformation provided in Eq. (10), no approximation has been made, which makes this protocol faster than other proposals based on dispersive interaction corresponding to second-order processes61,63,64. In this sense, the reverse-engineering protocol of generating these multi-parties entangled states allows us to accelerate its generation leading to generation times shorter than the tenth nanosecond scale, which mitigates the error produced by the unavoidable interaction with the environment. In the next section, we will quantify how the dissipation affects the performance of our protocols.

On the other hand, more quantum control techniques are available such as DRAG98,99, FAQUAD100, and GRAPE101,102. The latter corresponds to a gradient-based protocol that found the optimal modulation to control a system with decimated dynamics.

The working principle of this algorithm relies upon assuming constant control fields for time windows. Under this approximation, we write the time-evolution operator without the time-ordered operator corresponding only to Hamiltonian exponentiation, and we update the control fields through their gradients. The complexity scales with (a) the number of time steps of the control field, the Hamiltonian size, and its exponentiation. In our particular problem, a GRAPE-based implementation requires computing the exponentiation and their gradients for each time step at each minimization iteration, making them computationally challenging even for a modest number of cavities/qubits. Therefore, for the photonic GHZ, we must exponentiate matrices of dimension \(2\otimes \dim {(a)}^{M}\), whereas, for the qubit multi-partite entangled state, the dimension scales as \({2}^{N}\otimes \dim (a)\). Contrary, the STA reversed engineering approach is less computationally demanding since we are only required to solve a set of differential equations that can be efficiently solved using standard numerical/analytical techniques.

Dissipative dynamics

The next step is to analyze the performance of the reversed engineering approach to generate entangled states under dissipative and dephasing dynamics on qubits described by the following master equation

$$\frac{d\rho (t)}{dt} = -i[{{{{{{{{\mathcal{H}}}}}}}}}_{n}^{m},\rho (t)]+\mathop{\sum }\limits_{m=1}^{M}{\kappa }_{m}{{{{{{{\mathcal{L}}}}}}}}[{a}_{m},\rho (t)]\\ +\mathop{\sum }\limits_{N=1}^{N}({\gamma }_{n}{{{{{{{\mathcal{L}}}}}}}}[{\sigma }_{n}^{-},\rho (t)]+{\gamma }_{\phi ,n}{{{{{{{\mathcal{L}}}}}}}}[{\sigma }_{n}^{z},\rho (t)]),$$
(18)

where \({{{{{{{{\mathcal{H}}}}}}}}}_{n}^{m}\) is the Hamiltonian given in Eq. (1), \({{{{{{{\mathcal{L}}}}}}}}[{{{{{{{\mathcal{O}}}}}}}},\rho (t)]={{{{{{{\mathcal{O}}}}}}}}\rho (t){{{{{{{{\mathcal{O}}}}}}}}}^{{{{\dagger}}} }-\{{{{{{{{{\mathcal{O}}}}}}}}}^{{{{\dagger}}} }{{{{{{{\mathcal{O}}}}}}}},\rho (t)\}/2\) is the Lindbladian operator describing the dynamics of an open quantum system103. κm and γn stand for the relaxation ratio of the m-th cavity and the n-th qubits, respectively. Finally, γϕ,n is the dephasing ratio of the n-th qubit.

In superconducting circuits and cQED, energy relaxation appears as the consequence of the coupling between the quantum system with its electronic environment from the external circuitry, which we interpret as the interaction through an effective resistance. This noise profile is known as Nyquist noise or ohmic noise54,104. Another source of relaxation relies upon quasiparticle decay105,106, generated by the unpairing of Cooper-pairs due to electronic/thermal fluctuations. These electrons tunnel through the device, and we interpret them as an ohmic current leading to random spin flips. On the other hand, one of the dephasing sources corresponds to random spin flips in the superconducting metal forming the device due to the modulation of the external magnetic flux in the artificial atom107. Another source of error is produced by the longitudinal coupling with the quasiparticle bath that randomly changes the energy of the artificial atom108,109.

In light of our experimental implementation that requires manipulating the external flux for the reversed engineering protocol, a way to mitigate errors associated with the tuning relies upon accessing a flat energy spectrum in the artificial atom for achieving reduced flux sensitivity ∂[Ωn]/∂[ϕx] ≈ 0, which can be achieved with larger Josephson junctions105.

Here, we will focus on the previous cases, where we consider a single-qubit interacting with M modes described by \({{{{{{{{\mathcal{H}}}}}}}}}_{1}^{m}\) as in Eq. (2), and N qubits interacting with a single cavity characterized by \({{{{{{{{\mathcal{H}}}}}}}}}_{n}^{1}\) in Eq. (6). Without loss of generality, we assume homogeneous decay rates for the energy relaxation of the field mode, qubit relaxation, and dephasing rate, i.e., κm = κ, γn = γ, and γϕ,n = γϕ, respectively. We start by preparing the systems in the following state \({\rho }_{1}^{m}(0)=\left\vert g\right\rangle {\left\langle g\right\vert \bigotimes }_{m = 1}^{M}\left\vert {0}_{m}\right\rangle \left\langle {0}_{m}\right\vert\) for the photonic case, and \({\rho }_{n}^{1}(0){ \,= \,\bigotimes }_{n = 1}^{N}\left\vert {g}_{n}\right\rangle \left\langle {g}_{n}\right\vert \otimes \left\vert 0\right\rangle \left\langle 0\right\vert\) for the pqubit based, respectively. We let them evolve until tf, and compute the fidelity \({{{{{{{\mathcal{F}}}}}}}}\) at that time. For the GHZ cat state, we obtain that the fidelities for M = 1 and M = 2 is equal to \({{{{{{{\mathcal{F}}}}}}}}=\{0.998,0.996\}\). Due to the constraints imposed by the size of the Hilbert space, we cannot explore the system, including more field modes. For the qubit-based GHZ state, we obtain that the fidelities are given by \({{{{{{{\mathcal{F}}}}}}}}=\{0.999,0.998\}\) for N = {2, 4}, respectively. Since the coherence times are longer than the generation time, we do not observe an appreciable change in the optimal fidelity. We have performed our numerical simulation considering the same physical parameters as Fig. 1. For the decay rates we have chosen κ/2π = 1 MHz79 for the field modes, and γ/2π = 1/T1, with T1 = 40 μs and γϕ/2π = 1/T2 with T2 = 40 μs110 for the two-level system, respectively.

Physical implementation

Finally, we propose an implementation for \({{{{{{{{\mathcal{H}}}}}}}}}_{n}^{m}\) in Eq. (1) based on cQED architecture consisting of a coplanar waveguide resonator (CPWR)59 of length d with capacitance and inductance per unit length l and c, respectively. We couple N like-flux-qubit50,51,55 artificial atom formed by two Josephson junctions (JJs) capacitances \(\{{C}_{{J}_{1}}^{(n)},{C}_{{J}_{2}}^{(n)}\}\) and Josephson energies \(\{{E}_{{J}_{1}}^{(n)},{E}_{{J}_{2}}^{(n)}\}\) to the CPWR through an embedded junction of capacitance \({C}_{{J}_{E}}^{(n)}\) and Josephson energy \({E}_{{J}_{E}}^{(n)}\) uniformly distributed along it111 (see Fig. 6a). Moreover, we thread the artificial atom with an external magnetic flux \({\varphi }_{x}^{(n)}\) and driven by an external voltage \({V}_{g}^{(n)}\) through the capacitor \({C}_{g}^{(n)}\), as illustrated in Fig. 6b. The circuit Lagrangian reads

$${{{{{{{\mathcal{L}}}}}}}}=\mathop{\sum }\limits_{n=1}^{N}\left[{{{{{{{{\mathcal{L}}}}}}}}}_{{{{{{{{\rm{q}}}}}}}}}^{(n)}+{{{{{{{{\mathcal{L}}}}}}}}}_{{{{{{{{\rm{int}}}}}}}}}^{(n)}\right]+{{{{{{{{\mathcal{L}}}}}}}}}_{{{{{{{{\rm{CPWR}}}}}}}}},$$
(19)

where

$${{{{{{{{\mathcal{L}}}}}}}}}_{{{{{{{{\rm{q}}}}}}}}}^{(n)} = \frac{{C}_{{\Sigma }_{1}}^{(n)}}{2}{[{\dot{\phi }}_{1}^{(n)}]}^{2}-{q}_{g}^{(n)}{\dot{\phi }}_{1}^{(n)}+{E}_{{J}_{1}}^{(n)}\cos \left(\frac{{\phi }_{1}^{(n)}}{{\varphi }_{0}}\right)\\ +{E}_{{J}_{2}}^{(n)}\cos \left(\frac{{\phi }_{1}^{(n)}-{\phi }_{x}^{(n)}}{{\varphi }_{0}}\right),$$
(20)
$${{{{{{{{\mathcal{L}}}}}}}}}_{{{{{{{{\rm{int}}}}}}}}}^{(n)}= -{C}_{{J}_{1}}^{(n)}{\dot{\phi }}_{1}^{(n)}\dot{\psi }({x}_{n}^{-},t)-{C}_{{J}_{2}}^{(n)}{\dot{\phi }}_{1}^{(n)}\dot{\psi }({x}_{n}^{+},t)\\ -{C}_{{J}_{2}}^{(n)}{\dot{\phi }}_{1}^{(n)}{\dot{\phi }}_{x}^{(n)}-{C}_{{J}_{2}}^{(n)}\dot{\psi }({x}_{n}^{+},t){\dot{\phi }}_{x}^{(n)}\\ +{E}_{{J}_{1}}^{(n)}\sin \left(\frac{{\phi }_{1}^{(n)}}{{\varphi }_{0}}\right)\frac{\psi ({x}_{n}^{-},t)}{{\varphi }_{0}}\\ +{E}_{{J}_{2}}^{(n)}\sin \left(\frac{{\phi }_{1}^{(n)}-{\phi }_{x}^{(n)}}{{\varphi }_{0}}\right)\frac{\psi ({x}_{n}^{+},t)}{{\varphi }_{0}},$$
(21)
$${{{{{{{{\mathcal{L}}}}}}}}}_{{{{{{{{\rm{CPWR}}}}}}}}}= \int\nolimits_{0}^{d}\left[\frac{c}{2}{[{\partial }_{t}\psi (x,t)]}^{2}-\frac{1}{2l}{[{\partial }_{x}\psi (x,t)]}^{2}\right]dx\\ +\mathop{\sum }\limits_{n=1}^{N}\left[\frac{{C}_{{\Sigma }_{2}}^{(n)}}{2}{[{\partial }_{t}\psi ({x}_{n}^{-},t)]}^{2}+\frac{{C}_{{\Sigma }_{3}}^{(n)}}{2}{[{\partial }_{t}\psi ({x}_{n}^{+},t)]}^{2}-{C}_{{J}_{E}}^{(n)}{\partial }_{t}\psi ({x}_{n}^{-},t){\partial }_{t}\psi ({x}_{n}^{+},t)\right].$$
(22)

Here, \({\phi }_{1}^{(n)}\) and ψ(x, t) corresponds to the flux variables describing the artificial atom and the flux at the position x of the CPWR, moreover, \({C}_{{\Sigma }_{1}}^{(n)}={C}_{g}^{(n)}+{C}_{{J}_{1}}^{(n)}+{C}_{{J}_{2}}^{(n)}\), \({C}_{{\Sigma }_{2}}^{(n)}={C}_{{J}_{1}}^{(n)}+{C}_{{J}_{E}}^{(n)}\) and \({C}_{{\Sigma }_{3}}^{(n)}={C}_{{J}_{2}}^{(n)}+{C}_{{J}_{E}}^{(n)}\) are effective capacitances. Furthremore, \({q}_{g}^{(n)}={C}_{g}^{(n)}{V}_{g}^{(n)}\) is the charge bias, φ0 = ℏ/2e is the reduced quantum flux. Following the standard procedure of circuit quantization, and assuming that the junction forming the flux-qubit is identical and the embedded ones are different by a factor α(n) we arrive the following Hamiltonian

$${{{{{{{\mathcal{H}}}}}}}}= \mathop{\sum }\limits_{m}^{M}{\omega }_{m}{a}_{m}^{{{{\dagger}}} }{a}_{m}+\mathop{\sum }\limits_{n=1}^{N}\frac{{\Omega }_{n}}{2}{\sigma }_{n}^{z}\\ +\mathop{\sum}\limits_{n,m}{g}_{n}^{m}({\varphi }_{x}^{n}){\sigma }_{n}^{z}({a}_{m}^{{{{\dagger}}} }+{a}_{m}).$$
(23)

The detailed derivation of the above Hamiltonian can be found in Supplementary Note 2. Here am\(({a}_{m}^{{{{\dagger}}} })\) correspond to the annihilation (creation) bosonic operator describing the mth mode of the CPWR with frequency ωm. Moreover, \({\sigma }_{n}^{z}\) is the z-component Pauli matrix describes the flux-qubit in the two-level approximation with transition frequency Ωm. Finally, \({g}_{n}^{m}({\varphi }_{x}^{n})\) is the flux-dependent coupling strength between the CPWR with the N two-level systems. Notice that we have neglected all the capacitive interactions since we are assuming the Josephson energy as the leading energy contribution in the artificial atom. Likewise, due to the presence of the embedded junction in the CPWR, we obtain an antisymmetric spatial profile in the current/voltage of the resonator. This eliminates the coupling between the artificial atom and the node \(\psi ({x}_{n}^{+},t)\) of the CPWR, leading to a longitudinal interaction between both subsystems.

Fig. 6: circuit quantum electrodynamic implementation.
figure 6

Schematic illustration of our experimental proposal consisting of a λ/2 coplanar waveguide resonator (CPWR) of length d galvanically coupled to b N artificial atoms C(n) formed by three Josephson junctions \({E}_{{J}_{\ell }}^{(n)}\) threaded by an external magnetic flux \({\phi }_{x}^{(n)}\). Moreover, we biased each artificial atom through a voltage source \({V}_{g}^{(n)}\) coupled through a capacitor \({C}_{g}^{(n)}\). We model the coplanar waveguide resonator as a series of LC circuits characterized by the capacitance and inductance per unit of length c and l, respectively.

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Conclusion

In this article, we have proposed the reverse-engineering method to design a longitudinal interaction to generate multi-partite entangled states in photonic and qubit-based systems. Such approach suggests a modulation that satisfies the desired criteria under suitable conditions. In our case, we constrain the modulation to achieve a fixed cavity displacement to generate photonic multi-partite entangled states. In contrast, we impose the vanishing of some unwanted interaction to implement a Sørensen-Mølmer quantum gate to generate multi-partite qubit states, respectively. As a result, we obtain a fast generation time (less than tens of nanoseconds) with the current state-of-the-art superconducting quantum circuits architecture. Notably, the gating time does not scale with the number of either field modes or qubits, allowing us to generate entangled quantum states containing many parties. Since the generation time is shorter than the coherence time of the subsystem, the unavoidable effect of the environment does not induce a significantly detrimental impact on the final fidelity generation. The renewed attention to the protocols provided by STA in the context of cQED may pave the way to accelerate and improve several quantum tasks by engineering the adequate pulses sequences and modulation that are fundamentally constrained in the quantum platform.