# Shortcuts to adiabaticity in superconducting circuits for fast multi-partite
state generation

Francisco Andrés Cárdenas-López<EMAIL_ADDRESS>International
Center of Quantum Artificial Intelligence for Science and Technology
(QuArtist)
and Physics Department, Shanghai University, 200444 Shanghai, China
Forschungszentrum Jülich GmbH, Peter Grünberg Institute, Quantum Control
(PGI-8), 52425 Jülich, Germany  Juan Carlos Retamal  Departamento de Física,
Universidad de Santiago de Chile (USACH), Avenida Víctor Jara 3493, 9170124,
Santiago, Chile  Center for the Development of Nanoscience and Nanotechnology,
CEDENNA, Estación Central, 9170124, Santiago, Chile  Xi Chen<EMAIL_ADDRESS>Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, E-48080 Bilbao, Spain  EHU Quantum Center, University of the
Basque Country UPV/EHU, 48940 Leioa, Spain

## abstract

Shortcuts to adiabaticity provides a flexible method to accelerate and improve
a quantum control task beyond adiabatic criteria. However, their application
to fast generation of multi-partite quantum gates is still not optimized. Here
we propose the reverse-engineering approach to design the longitudinal
coupling between a set of qubits coupled to several field modes, for achieving
a fast generation of multi-partite quantum gates in photonic or qubit-based
architecture. We show that the enhancing generation time is at the nanosecond
scale that does not scale with the number of system components. In addition,
our protocol does not suffer noticeable detrimental effects due to the
dissipative dynamics. Finally, the possible implementation is discussed with
the state-of-the-art circuit quantum electrodynamics architecture.

## Introduction

It has been scrutinized that entanglement [1, 2] in multi-partite quantum
systems, plays a crucial role in quantum technologies applications such as
quantum information processing [3, 4, 5], quantum computation [6, 5], and
quantum simulation [7], respectively. Entanglement as a quantum resource can
lead to a speed-up in the running time of quantum algorithms [8, 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
methods [12, 13]. Likewise, entanglement discontinuity is an excellent
indicator to characterize phase transitions on quantum systems [14, 15, 16,
17]. Besides, in the context of quantum metrology and sensing, using entangled
states has achieved quantum-enhanced precision measurements near to the shoot-
noise limit [18, 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 platform [21] gives a step forward in
this direction with the implementation of the so-called Sørensen-Mølmer
quantum gate [22, 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
$\sigma_{\ell}^{x}\sigma^{x}_{\ell^{\prime}}$ [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 atoms [42, 43, 107, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55],
whereas quantized field modes corresponds to either LC resonators, coplanar
waveguide or stripline resonator [56, 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 interaction [61, 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 environment [65, 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 regime [69, 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 environment [74]. STA has
received renewed interest in the context of cQED since it has been generalized
to open quantum system [75, 76] permitting to design a counter-diabatic and
optimal pulses to speed up a dissipative evolution [77, 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 $\Omega_{n}$ coupled to $M$ quantized field modes of frequency
$\omega_{m}$ through time-dependent longitudinal coupling $g_{n}^{m}(t)$
governed by the following Hamiltonian ($\hbar\equiv 1$)

$\displaystyle\mathcal{H}_{n}^{m}=\sum_{n}\frac{\Omega_{n}}{2}\sigma^{x}_{n}$
$\displaystyle+$ $\displaystyle\sum_{m}\omega_{m}a_{m}^{{\dagger}}a_{m}$ (1)
$\displaystyle+$
$\displaystyle\sum_{n,m}g^{m}_{n}(t)\sigma^{x}_{n}(a_{m}^{{\dagger}}+a_{m}),~{}~{}~{}~{}$

where $\sigma^{x}_{n}$ corresponds to the $x$-component Pauli matrix
characterizing the $n$-th two-level system, and $a_{m}$ ($a_{m}^{{\dagger}}$)
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
flux [80, 81], or couple both subsystems through a superconducting quantum
interference device (SQUID) [82, 83].

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 readout [78] or
implementing faster two-qubit gates [79], 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

$\displaystyle\mathcal{H}_{1}^{m}=\frac{\Omega}{2}\sigma^{x}+\sum_{m}\omega_{m}a_{m}^{{\dagger}}a_{m}+\sum_{m}g_{1}^{m}(t)\sigma^{x}(a_{m}^{{\dagger}}+a_{m}).~{}~{}~{}$
(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

$\displaystyle
U^{m}_{1}(t)=\prod_{m}\hat{\mathcal{D}}_{m}(\alpha_{m}(t)\sigma^{x}),$ (3)

where
$\hat{\mathcal{D}}_{m}(\sigma^{x}\alpha_{m}(t))=\exp\big{(}\sigma^{x}(\alpha_{m}(t)a_{m}^{{\dagger}}+\alpha_{m}^{*}(t)a_{m}))$
is the displacement operator of the $m$th field mode with
$\alpha_{m}(t)=-i\int_{0}^{t}g_{m}(s)e^{i\omega_{m}s}ds$ as the cavity
displacement. For a time-independent coupling strength, we obtain that at time
$T_{k}=(2k+1)\pi/\omega_{m}$ the field mode reaches its maximum displacement
$\alpha_{{\rm{max}}}=\pm 2g_{1}^{m}/\omega_{m}$ that depends on the qubit
state. Thus, for the system prepared in the initial state
$|\Psi(0)\rangle=|g\rangle\bigotimes_{m=1}^{M}|0\rangle_{m}$ ($|g\rangle$ is
the ground state of the two-level system, and $|0\rangle_{m}$ is the vacuum
state of the $m$th field mode) the system evolves to the so-called
Greenberger–Horne–Zeilinger cat state [84, 85, 86, 87]

$\displaystyle|\Psi(T)\rangle=\frac{1}{\sqrt{2}}\bigg{[}|e\rangle\bigotimes_{m=1}^{M}|\alpha_{{\rm{max}}}\rangle_{m}+|g\rangle\bigotimes_{m=1}^{M}|-\alpha_{{\rm{max}}}\rangle_{m}\bigg{]},~{}~{}~{}$
(4)

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

$\displaystyle|\Psi\rangle=\frac{1}{\sqrt{2}}\bigg{[}|e\rangle\bigotimes_{m=1}^{M}|\alpha_{+}\rangle_{m}+|g\rangle\bigotimes_{m=1}^{M}|\alpha_{-}\rangle_{m}\bigg{]},$
(5)

where
$|\alpha_{\pm}\rangle_{m}=(|\alpha_{{\rm{max}}}\rangle_{m}\pm|-\alpha_{{\rm{max}}}\rangle_{m})/\sqrt{2}$
is the even/odd coherent state superposition [88]. 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

$\displaystyle\mathcal{H}_{n}^{1}=\omega
a^{{\dagger}}a+\sum_{n}\frac{\Omega_{n}}{2}\sigma^{x}_{n}+\sum_{n}g^{1}_{n}(t)\sigma^{x}_{n}(a^{{\dagger}}+a).$
(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) formula [90]

$\displaystyle
U_{n}^{1}(t)=\prod_{n}e^{-iA_{n}\sigma^{x}_{x}a}\prod_{n}e^{-iA_{n}^{*}\sigma^{x}_{x}a^{{\dagger}}}\prod_{n,n^{\prime}}e^{-iB_{nn^{\prime}}\sigma^{x}_{n}\sigma^{x}_{n^{\prime}}}.$
(7)

The coefficients $A_{n}(t)$ ($A_{n}^{*}(t)$) and $B_{nn^{\prime}}(t)$ 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

$\displaystyle\frac{dA_{n}(t)}{dt}=g^{1}_{n}(t)e^{-i\omega
t},\quad\frac{dB_{nn^{\prime}}(t)}{dt}=iA_{n^{\prime}}(t)\frac{dA_{n}^{*}(t)}{dt}.$
(8)

For time-independent coupling strength $g_{n}^{1}$ with the initial conditions
$A_{n}(0)=B_{nn^{\prime}}=0$, we obtain [62]

$\displaystyle A_{n}(t)$ $\displaystyle=$
$\displaystyle\frac{ig^{1}_{n}}{\omega}(e^{-i\omega t}-1),$ (9a)
$\displaystyle B_{nn^{\prime}}(t)$ $\displaystyle=$
$\displaystyle\frac{g^{1}_{n}g^{1}_{n^{\prime}}}{\omega}[-i(e^{i\omega
t}-1)-t].$ (9b)

By choosing $\tau=2\pi/\omega$, the coefficient $A_{n}(\tau)$ vanishes and
$B_{nn^{\prime}}(\tau)\equiv\theta_{nn^{\prime}}=2\pi
g^{1}_{n}g^{1}_{n^{\prime}}/\omega^{2}$. Consequently, the time evolution
operator reduces to

$\displaystyle U_{n}^{1}(\tau)$ $\displaystyle=$
$\displaystyle\prod_{n,n^{\prime}}\exp\bigg{[}-i\theta_{nn^{\prime}}\sigma^{x}_{n}\sigma^{x}_{n^{\prime}}\bigg{]}$
(10) $\displaystyle=$ $\displaystyle\prod_{n\neq
n^{\prime}}\bigg{[}\cos(\theta_{nn^{\prime}})\mathbb{I}+i\sin(\theta_{nn^{\prime}})\sigma^{x}_{n}\sigma^{x}_{n^{\prime}}\bigg{]},$

corresponding to the Sørensen-Mølmer quantum gate [22, 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 system
[91, 92]. If we initialize the system in the state
$|\Phi(0)\rangle=\bigotimes_{n=1}^{N}|g\rangle_{n}\otimes|0\rangle$, and
choose $\theta_{nn^{\prime}}=\pi/4$, the state evolves to a GHZ state [93]

$\displaystyle|\Phi(\tau)\rangle=\frac{1}{\sqrt{2}}\bigg{[}\bigotimes_{n=1}^{N}|g\rangle_{n}+e^{i\pi(N+1)/2}\bigotimes_{n=1}^{N}|e\rangle_{n}\bigg{]}|0\rangle,$
(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 $T=47.61~{}\rm{(ns)}$, whereas for the multi-partite operation
with qubits, the gate time is $T=19~{}\rm{(ns)}$ [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 regime
[69, 70, 71, 72]. A way to accelerate the generation time without demanding
larger coupling strength relies upon the technique of STA [74], 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 method
[73, 78], to design the coupling strength $g^{n}_{1}$ and $g^{1}_{n}$ such
that the gating time is shortened significantly.

Figure 1: Coupling strength modulation. Reversely engineered coupling strength
$g_{c}(t)$ (continuous line) and $g(t)$ (dotted) as a function of the
dimensionless time $t/t_{f}$ for (a) the GHZ for photonic systems, blue,
orange and green data corresponds to the pulse considering
$\alpha(t_{f})=1,3,5$, respectively. Whereas (b) stands for the GHZ in qubits.
The numerical simulations are preformed with the following system parameters
(a) $\omega/2\pi=6.6~{}{\rm{GHz}}$, $g/2\pi=21~{}{\rm{MHz}}$, and (b)
$\Omega/2\pi=10~{}{\rm{GHz}}$, $\omega/2\pi=1~{}{\rm{GHz}}$, and
$g/2\pi=144~{}{\rm{MHz}}$.

### Reverse engineering protocol

Afterwards defining the model, we will discuss how to design the coupling
strengths $g^{n}_{1}$ and $g^{1}_{n}$ in the frame of reverse-engineering
approach [73, 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
$|\Psi(x,t)\rangle=\exp{(-it\epsilon)}\mathcal{U}(t)|\varphi(x,t)\rangle$,
where $\epsilon=\omega(n+1/2)$ and
$|\varphi(x,t)\rangle=\sqrt[4]{m\omega_{r}/\pi}\exp(-m\omega_{r}x^{2}/2)$
corresponds to the eigenfunction for the uncoupled cavity with Hamiltonian
$\omega_{r}a^{{\dagger}}a$ and $|\xi\rangle\equiv|g\rangle$ describes the
ground state of the qubit. Additionally, $\mathcal{U}(t)$ stands for a unitary
transformation that eliminates the qubit-field mode coupling [94, 95, 96]

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

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

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

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

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

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

$\displaystyle g_{c}(t_{0})=\dot{g}_{c}(t_{0})=\ddot{g}_{c}(t_{0})=0,$ (15a)
$\displaystyle g_{c}(t_{f})=\dot{g}_{c}(t_{f})=\ddot{g}_{c}(t_{f})=0,$ (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 $t_{f}$ the final cavity displacement be larger
than we can. In this scenario, together with the boundary conditions in Eq.
(15a) and Eq. (15b), we add the following constrains

$\displaystyle\alpha(t_{f})=-i\int_{0}^{t_{f}}g(s)e^{i\omega
s}ds=d_{{\rm{max}}},$ (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

$\displaystyle A_{n}(t_{f})=0,\quad B_{nn^{\prime}}(t_{f})=\frac{\pi}{4}.$
(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 $\phi_{x}(t)$.
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.
(8) 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 $c_{k}$ such $g_{c}(t)$ fulfills the boundary
condition given in Eq. (15a), Eq. (15b) together with the Eq. (16) for the
photonic case, and Eqs. (17) for the qubits, respectively.

Figure 2: Reversed enginnered modulation. Reversely engineered (a) absolute
cavity displacement $|\alpha(t)|$ as a function of the dimensionless time
$t/t_{f}$, blue, orange and green lines correspond to the maximal cavity
displacement of $d_{{\rm{max}}}=\\{1,3,5\\}$, respectively. (b) and (c) stand
for the real (blue) and imaginary (green) parts of the reversely engineered
$|A_{n}(t)|$ and $|B_{nn^{\prime}}(t)|$ coefficients, respectively. The
numerical simulations are performed with the same parameters as those in Fig.
1.

We numerically calculate the coefficients $c_{k}$ using the gradient descent
minimization package of python [97]. Fig. 1 shows the auxiliary variable
$g_{c}(t)$ and the coupling strength $g(t)$ obtained through Eq. (14) as a
function of the dimensionless time $t/t_{f}$ for three different maximal
cavity displacement $d_{{\rm{max}}}=\\{1,3,5\\}$, 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 $g_{c}(t)$ 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 $t/t_{f}$ depicted in
Fig. 2. For the coupling strength $g/2\pi=21~{}{\rm{MHz}}$ [80] for the
photonic system, we obtain a generation time $t_{f}=\pi/(40g)\equiv
3.74~{}\rm{(ns)}$ achieving $\alpha(t_{f})=d_{{\rm{max}}}$. Whereas, the
qubit-based GHZ state can be generated within the timescale
$t_{f}=\pi\omega/(80g^{2})\equiv 1.89~{}\rm{(ns)}$ for a coupling strength
$g/2\pi=114~{}{\rm{MHz}}$ [62] where the conditions $A_{n}(t_{f})=0$ and
$B_{n,n^{\prime}}(t_{f})=\pi/4$ are fulfilled, respectively.

### Multi-partite photonic state

Figure 3: Multi-partite photonic state (a) Wigner representation of the
reduced density matrix of the first field mode at final time $t_{f}$ 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 $\pm d_{{\rm{max}}}=3$. (d)
Fidelity $\mathcal{F}$ between the target Greenberger–Horne–Zeilinger state in
Eq. (5) with the state $|\Psi(t)\rangle$ for two different maximal cavity
displacement $d_{{\rm{max}}}=1$ to the blue line, and $d_{{\rm{max}}}=3$ for
the green line, respectively. The numerical simulations are carried out with
the same parameters as those in Fig. 1.

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 $g_{1}^{m}(t)\equiv g(t)$ $\forall m$. We
quantify the Wigner function $W(x,p)$ of the reduced density matrix of the
first field modes given by
$\rho_{f_{1}}(t)={\rm{Tr}}_{\\{f_{2},...,f_{M},q\\}}[|\Psi(t)\rangle\langle\Psi(t)|]$,
where the sub-indexes $f_{k}$, 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 $|\alpha\rangle$ corresponds to a Gaussian centered in
$(\mathrm{Re}(\alpha),\mathrm{Im}(\alpha))$ [88], at the final STA time
$t_{f}$, where $\rho_{f_{1}}(t_{f})=(1/2)(|d_{{\rm{max}}}\rangle\langle
d_{{\rm{max}}}|+|-d_{{\rm{max}}}\rangle\langle-d_{{\rm{max}}}|)$, the Wigner
function represents two Gaussians centered at $\pm d_{{\rm{max}}}$. Fig. 3(a)
shows the Wigner function on the phase space corresponding to two points far
apart to each other from a distance $2d_{{\rm{max}}}$. Likewise, Fig. 3(b)-(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
$p=\pm d_{{\rm{max}}}$. In contrast, in the momentum projection, we have to
see two overlapping Gaussian centered at $x=\pm d_{{\rm{max}}}$, respectively.
From Fig. 3(c) we see a wide region where $W(x,p)$ takes negative values,
showing the non-local nature of the multi-partite entangled state [88].

Furthermore, we also calculate the Fidelity
$\mathcal{F}(\rho,\sigma)=({\rm{Tr}}[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}])^{2}$
($\rho$ and $\sigma$ are arbitrary density matrices) between the state
$|\Psi(t)\rangle$ 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 $d_{{\rm{max}}}=\\{1,3\\}$ as depicted in
Fig. 3(d). We observe that the reversed engineering protocol is insensitive to
the final displacement $d_{{\rm{max}}}$, where we appreciate fidelities close
to one for both modulations.

Figure 4: Fidelity as function of the number of subsystems. Fidelity
$\mathcal{F}$ at the shortcut to adiabaticity time $t_{f}$ 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
$d_{\rm{max}}=3$, 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.

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 ($t_{f}\equiv
3.2~{}\rm{(ns)}$), which is one order of magnitude shorter than the same
protocol without the use of the reverse engineering approach
($T=47.61~{}\rm{(ns)}$). Moreover, our reverse engineering approach permit us
fixing the final cavity displacement beyond the maximal distplacement
$\alpha_{\rm{max}}=\pm 2g_{1}^{m}/\omega_{m}$. 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 $t_{f}$ 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 $t_{f}$ for each case. Those results are depicted in
Fig. 4(a), 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 $N_{{\rm{max}}}=|d_{\rm{max}}|+1$. 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.

### Multi-partite qubit state

Figure 5: Revsersed engineering multi-qubit state generation. (a) Population
evolution of the two-qubit states $|gg\rangle$ (continuous blue), $|ge\rangle$
(continuous orange), $|eg\rangle$ (dotted green), and $|ee\rangle$ (continuous
red) computed from the Hamiltonian in Eq. (6) through the reverse- engineering
protocol as a function of the dimensionless time $t/t_{f}$, we observe that at
final time the state evolves to a GHZ state. (b) Fidelity $\mathcal{F}$ as a
function of $t/t_{f}$ between the GHZ state of two qubits with the state
$|\Phi(t)\rangle$. The numerical simulations are carried out with the same
parameters as those in Fig. 1.

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., $g_{n}^{1}(t)=g(t)$ $\forall n$. 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
$\rho_{q}(t)={\rm{Tr}}_{f}[|\Phi(t)\rangle\langle\Phi(t)|]$, where the sub-
index $f$ corresponds to the trace over the field mode, where
$|\Phi(t)\rangle$ is the solution of the Schödinger equation for the
Hamiltonian given in Eq. (6). Fig. 5 (a) shows the population evolution for a
system composing by $N=2$ qubits. We see that the occupation probabilities of
the symmetric subspace $\\{|ge\rangle,|eg\rangle\\}$ 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 $A_{n}(t)$ only vanishes at $t=0$, and $t=t_{f}$,
respectively. Likewise, the population of the state $|ee\rangle$ keeps
changing until $t_{f}$ achieves its optimal value. In Fig. 5(b) 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 $t=t_{f}$ by increasing the number of qubits from $N=\\{2-12\\}$.
Fig. 4(b) shows the maximal fidelity as a function of the number of qubits. It
is observed that the fidelities $\mathcal{F}$ are bounded between $98\%-99\%$,
thus demonstrating that the gating time does not scale with the number of
qubits for achieving a large fidelity.

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
processes [61, 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 DRAG
[99, 100], FAQUAD [101], and GRAPE [102, 103]. 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

$\displaystyle\frac{d\rho(t)}{dt}$ $\displaystyle=$
$\displaystyle-i[\mathcal{H}_{n}^{m},\rho(t)]+\sum_{m=1}^{M}\kappa_{m}\mathcal{L}[a_{m},\rho(t)]$
(18) $\displaystyle+$
$\displaystyle\sum_{N=1}^{N}(\gamma_{n}\mathcal{L}[\sigma^{-}_{n},\rho(t)]+\gamma_{\phi,n}\mathcal{L}[\sigma^{z}_{n},\rho(t)]),$

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 a open quantum system
[98]. $\kappa_{m}$ and $\gamma_{n}$ stand for the relaxation ratio of the
$m$-th cavity and the $n$-th qubits, respectively. Finally, $\gamma_{\phi,n}$
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 noise [54, 104]. Another source of relaxation relies upon
quasiparticle decay [105, 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 atom [107]. Another source of error
is produced by the longitudinal coupling with the quasiparticle bath that
randomly changes the energy of the artificial atom [108, 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 to a flat energy spectrum in
the artificial atom for achieving reduced flux sensitivity
$\partial[\Omega_{n}]/\partial[\phi_{x}]\approx 0$, which can be achieved with
larger Josephson junctions [105].

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., $\kappa_{m}=\kappa$,
$\gamma_{n}=\gamma$, and $\gamma_{\phi,n}=\gamma_{\phi}$, respectively. We
start by preparing the systems in the following state
$\rho_{1}^{m}(0)=|g\rangle\langle g|\bigotimes_{m=1}^{M}|0_{m}\rangle\langle
0_{m}|$ for the photonic case, and
$\rho_{n}^{1}(0)=\bigotimes_{n=1}^{N}|g_{n}\rangle\langle
g_{n}|\otimes|0\rangle\langle 0|$ for the pqubit based, respectively. We let
them evolve until $t_{f}$, 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 $\kappa/2\pi=1~{}{\rm{MHz}}$ [80] for the field
modes, and $\gamma/2\pi=1/T_{1}$, with $T_{1}=40~{}{\rm{\mu s}}$ and
$\gamma_{\phi}/2\pi=1/T_{2}$ with $T_{2}=40~{}{\rm{\mu s}}$ [110] for the two-
level system, respectively.

### Physical implementation

Figure 6: Circuit quantum electrodynamic implementation. Schematic
illustration of our experimental proposal consisting of (a) a $\lambda/2$
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.

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-qubit [50, 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 it [111] (see Fig. 6(a)).
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. 6(b). The circuit
Lagrangian reads

$\displaystyle\mathcal{L}=\sum_{n=1}^{N}\big{[}\mathcal{L}_{\rm{q}}^{(n)}+\mathcal{L}_{\rm{int}}^{(n)}\big{]}+\mathcal{L}_{\rm{CPWR}},$
(19)

where

$\displaystyle\mathcal{L}_{\rm{q}}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{C_{\Sigma_{1}}^{(n)}}{2}[\dot{\phi}_{1}^{(n)}]^{2}-q_{g}^{(n)}\dot{\phi}_{1}^{(n)}+E_{J_{1}}^{(n)}\cos\bigg{(}\frac{\phi_{1}^{(n)}}{\varphi_{0}}\bigg{)}$
(20) $\displaystyle+$ $\displaystyle
E_{J_{2}}^{(n)}\cos\bigg{(}\frac{\phi_{1}^{(n)}-\phi_{x}^{(n)}}{\varphi_{0}}\bigg{)},$
$\displaystyle\mathcal{L}_{\rm{int}}^{(n)}$ $\displaystyle=$ $\displaystyle-
C_{J_{1}}^{(n)}\dot{\phi}_{1}^{(n)}\dot{\psi}(x_{n}^{-},t)-C_{J_{2}}^{(n)}\dot{\phi}_{1}^{(n)}\dot{\psi}(x_{n}^{+},t)$
(21) $\displaystyle-$ $\displaystyle
C_{J_{2}}^{(n)}\dot{\phi}_{1}^{(n)}\dot{\phi}_{x}^{(n)}-C_{J_{2}}^{(n)}\dot{\psi}(x_{n}^{+},t)\dot{\phi}_{x}^{(n)}$
$\displaystyle+$ $\displaystyle
E_{J_{1}}^{(n)}\sin\bigg{(}\frac{\phi_{1}^{(n)}}{\varphi_{0}}\bigg{)}\frac{\psi(x_{n}^{-},t)}{\varphi_{0}}$
$\displaystyle+$ $\displaystyle
E_{J_{2}}^{(n)}\sin\bigg{(}\frac{\phi_{1}^{(n)}-\phi_{x}^{(n)}}{\varphi_{0}}\bigg{)}\frac{\psi(x_{n}^{+},t)}{\varphi_{0}},$
$\displaystyle\mathcal{L}_{\rm{CPWR}}$ $\displaystyle=$
$\displaystyle\int_{0}^{d}\bigg{[}\frac{c}{2}[\partial_{t}\psi(x,t)]^{2}-\frac{1}{2l}[\partial_{x}\psi(x,t)]^{2}\bigg{]}dx$
(22) $\displaystyle+$
$\displaystyle\sum_{n=1}^{N}\bigg{[}\frac{C_{\Sigma_{2}}^{(n)}}{2}[\partial_{t}\psi(x_{n}^{-},t)]^{2}+\frac{C_{\Sigma_{3}}^{(n)}}{2}[\partial_{t}\psi(x_{n}^{+},t)]^{2}$
$\displaystyle-$ $\displaystyle
C_{J_{E}}^{(n)}\partial_{t}\psi(x_{n}^{-},t)\partial_{t}\psi(x_{n}^{+},t)\bigg{]}.$

Here, $\phi_{1}^{(n)}$ and $\psi(x,t)$ 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, $\varphi_{0}=\hbar/2e$ is the reduced quantum flux. Following the
standard procedure of circuit quantization, and assuming that the junction
forming the flux-qubit are identical and the embedded ones is different by a
factor $\alpha^{(n)}$ we arrive a the following Hamiltonian

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{m}^{M}\omega_{m}a^{{\dagger}}_{m}a_{m}+\sum_{n=1}^{N}\frac{\Omega_{n}}{2}\sigma_{n}^{z}$
(23) $\displaystyle+$
$\displaystyle\sum_{n,m}g_{n}^{m}(\varphi_{x}^{n})\sigma_{n}^{z}(a_{m}^{{\dagger}}+a_{m}).$

The detailed derivation of the above Hamiltonian can be found in the
Supplementary Note 2. Here $a_{m}$ $(a^{{\dagger}}_{m})$ correspond to the
annihilation (creation) bosonic operator describing the $m$th mode of the CPWR
with frequency $\omega_{m}$. Moreover, $\sigma^{z}_{n}$ is the $z$-component
Pauli matrix describing the flux-qubit in the two-level approximation with
transition frequency $\Omega_{m}$. 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.

## 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.

## ACKNOWLEDGMENTS

F. A. C. L thanks to A. Parra-Rodriguez, for helpful discussion. This work is
supported by NSFC (12075145), STCSM (Grants No. 2019SHZDZX01-ZX04), EU FET
Open Grant EPIQUS (899368), HORIZON-CL4-2022-QUANTUM-01-SGA project 101113946
OpenSuperQPlus100 of the EU Flagship on Quantum Technologies, the Basque
Government through Grant No. IT1470-22, the project grant PID2021-126273NB-I00
funded by MCIN/AEI/10.13039/501100011033 and by “ERDFA way of making Europe”
and “ERDF Invest in your Future”, Nanoscale NMR and complex systems
(PID2021-126694NB-C21) and QUANTEK project (Grant No. KK-2021/00070). F. A. C.
L. thanks to the German Ministry for Education and Research, under QSolid,
Grant no. 13N16149. J. C. R acknowledges to Financiamiento Basal para Centros
Científicos y Tecnológicos de Excelencia (Grant No. AFB220001). X. C.
acknowledges ayudas para contratos Ramón y Cajal–2015-2020 (RYC-2017-22482).

## COMPETING INTERESTS

The authors declare no competing interests.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author, F. A. C. L, upon reasonable request.

## AUTHOR CONTRIBUTIONS

All authors originate the project. F. A. C. L. carried out the numerical
simulations in collaboration with JC. R. X.C. provides the methods of STA. All
authors contributed to verifying the results and writing the manuscript.

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