# Atomic interactions for qubit-error compensations

Michele Delvecchio<EMAIL_ADDRESS>Department of Mathematical,
Physical and Computer Sciences, University of Parma, Parco Area delle Scienze
7/A, 43124, Parma, Italy  National Institute for Nuclear Physics (INFN),
Milano Bicocca Section, Parma Group, Parco Area delle Scienze 7/A, 43124,
Parma, Italy  Francesco Petiziol<EMAIL_ADDRESS>Department of
Mathematical, Physical and Computer Sciences, University of Parma, Parco Area
delle Scienze 7/A, 43124, Parma, Italy  Institut für Theoretische Physik,
Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany  Ennio
Arimondo<EMAIL_ADDRESS>Dipartimento di Fisica E. Fermi, Università
di Pisa, Largo. B. Pontecorvo 3, 56127 Pisa, Italy  INO-CNR, Via G. Moruzzi 1,
56124 Pisa, Italy  Sandro Wimberger<EMAIL_ADDRESS>Department of Mathematical, Physical and Computer Sciences, University of
Parma, Parco Area delle Scienze 7/A, 43124, Parma, Italy  National Institute
for Nuclear Physics (INFN), Milano Bicocca Section, Parma Group, Parco Area
delle Scienze 7/A, 43124, Parma, Italy<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

Experimental imperfections induce phase and population errors in quantum
systems. We present a method to compensate unitary errors affecting also the
population of the qubit states. This is achieved through the interaction of
the target qubit with an additional control qubit. We show that our approach
works well for single-photon and two-photon excitation schemes. In the first
case, we study two reduced models (i) a two-level system in which the
interaction corresponds to an effective level shift and (ii) a three-level one
describing two qubits in the Bell triplet subspace. In the second case,
instead, a double-STIRAP process is presented with comparable compensation
efficiency with respect to the single-photon case.

##  I Introduction

Quantum computation is based on a series of unitary transformations applied to
computational qubits. Quantum states are intrinsically delicate with
decoherence being the main limitation. In addition, the unitary
transformations associated to quantum gates cannot be implemented with perfect
accuracy and their small imperfections will accumulate, leading to a
computational failure. Correction schemes must thus protect against errors.
This can be done, for instance, by error correction codes Nielsen and Chuang
(2000); Terhal (2015); Chiesa  _et al._ (2020), composite pulse sequences
Levitt (1986); Jones (2011) or other robust quantum control protocols Guéry-
Odelin  _et al._ (2019); Glaser  _et al._ (2015). Additional resources are
required in all cases: more qubits in the first case, longer times for the
qubit preparation and interrogation in the second and additional control
fields or parameter optimization loops in the latter. Dynamical phase errors
can eventually induce also errors of the qubit-level populations. Our ultimate
target is to determine and compensate the unwanted phase accumulated in the
qubit wavefunction in order to correct coherent (unitary) computational
errors.

b

Figure 1:  In (a) scheme of two interacting qubits, for instance, two-level
Rydberg atoms trapped in an optical potential and experiencing a nearest-
neighbour coupling $V$. They are driven by a common laser field, periodically
switched on for the duration $T_{1}$ and off for the time $T_{2}$, in a
sequence with temporal periodicity $T=2T_{1}+2T_{2}$ as schematized in (b).
For a perfect excitation, as with $\pi$ pulses for one-photon absorption, the
$P_{g}$ ground state occupation experiences the temporal evolution shown in
(c).

The control of the wavefunction phase accumulation has received much attention
in different contexts of quantum optics, such as the collapse and subsequent
revival of atomic coherence for Bose-Einstein condensates in Mahmud and
Tiesinga (2013); Meinert  _et al._ (2014). Phase correlation destructions and
revivals in the time evolution of dipole-blockaded Rydberg states have been
investigated under a detuned and continuous excitation in Fan  _et al._
(2019), and for periodic excitation by Fan  _et al._ (2020) within the context
of discrete time crystals. The quantum control of the phase accumulated by
laser driving for interacting Rydberg atoms was studied in Bhaktavatsala Rao
and Mølmer (2014); Beterov  _et al._ (2020). An analysis of laser
imperfections in the coherent excitation to atomic Rydberg states was
experimentally investigated in de Léséleuc  _et al._ (2018).

We introduce here the approach shown in Fig. 1(a), based on the interaction
between the "computation" qubit and an additional "correction" qubit, widely
used for the implementation of high fidelity quantum-nondemolition
measurements Walls and Milburn (1995); Nielsen and Chuang (2000). The idea of
using additional qubits for correction is taken from quantum-error-correction
protocols and fault-tolerant quantum computing Nielsen and Chuang (2000), see,
e.g., Egan  _et al._ (2021) for a recent experimental realisation using ion
qubits.

In the following, we use the additional phase created by the interaction to
compensate for the above unwanted phase in the evolution of the computational
qubit. By a proper choice of the interaction strength, we realize compensation
for a long sequence of unitary operations of the computational qubit. The
compensation efficiency is measured by the wavefunction fidelity reached at
the end of the sequence. We obtain a high efficiency for sequence numbers up
to fifty. For the long sequence of applied unitary transformations, we derive
conditions for the wavefunction phase to maintain the targeted value. In most
of the explored qubit level schemes, our approach leads to a magic condition
for the required interaction, magic because it creates compensation for a
large range of unitary transformation errors. Our approach is similar to the
method of composite pulses Levitt (1986); Jones (2011). In both cases, the
phase of the qubit wavefunction accumulated by the laser-pulse sequence
produces a more robust qubit excitation. As main difference, the composite-
pulse sequence targets a very large and stable fidelity for a single
excitation. We target instead a stable and large fidelity in a long sequence
of qubit operations. Our qubit interaction is linked to the excited state
occupation, for instance, in experimental implementations based on atomic
Rydberg excitations Saffman  _et al._ (2010a); Labuhn  _et al._ (2016); Adams
_et al._ (2019); Zhang  _et al._ (2020) or on Rydberg-dressed atomic gases
Borish  _et al._ (2020). The interaction can, in principle, be tuned
experimentally. Its amplitude depends, e.g., on the atomic quantum number, and
in addition, in the presence of the Förster resonances, it may be controlled
by an applied electric field Ravets  _et al._ (2014); Higgins  _et al._
(2017); Huang  _et al._ (2018); Beterov  _et al._ (2020). Similar tunable
interactions are present in other realisations as well, e.g., in
semiconductors Schwartz  _et al._ (2021) or in artificial atoms, as in the
ground state interactions in double quantum dots in a nanowire Taherkhani  _et
al._ (2019).

The compensation scheme is applied here to a computational qubit that, under
proper laser handling, experiences a Bloch sphere rotation and
reinitialization within a given interrogation time interval. We repeat the
same sequence on a regular basis, and introduce small errors on the laser
parameters. Therefore, the qubit accumulates an unwanted phase limiting the
computation utility. The compensation recovers its utility through the
controlled interaction with the correction qubit. In the following, we
specifically consider three situations: two models with a one-photon
excitation and one with a two-photon excitation.

First we consider the one-photon excitation case. Starting from the two qubits
sketched in Fig. 1, under the condition for which the laser drive does not
influence the correction qubit, the impact of the interaction between the
excited levels may be modelled for the computational qubit as an effective
shift of the excited level. This gives our system (i): a two-level system
exposed to laser pulses driving transitions between the ground and excited
state. In the second configuration, the laser drive couples only to the Bell
triplet states and thus the four two-qubit levels can be reduced to three
simply by neglecting the Bell singlet state. This three-level system is our
model (ii). There is a fundamental difference between these two models: errors
in the laser drive can occur only on the computational qubit in (i), while in
(ii) both qubits fully participate, with errors that can be modelled on both.
The compensation works well in both cases but the magic value is more stable,
producing a larger compensated area with fidelity error of the order of
$10^{-6}$, for system (i). The compensation is applied to errors in the
dynamics of the computational qubit on a Bloch sphere: rotation angle errors
and rotation axis ones, produced by imperfections in the laser intensity and
in the laser detuning, respectively. We analyze the one-photon excitation by a
$\pi$ laser pulse, allowing a transition at the quantum-speed limit, but with
a large sensitivity to the laser excitation parameters Giannelli and Arimondo
(2014). The $\pi$ pulses are employed for both the qubit excitation and its
reinitialization.

In addition to the one-photon population transfer between ground and upper
states, we analyze also a two-photon excitation scheme in which the two levels
of each qubit cannot directly be coupled by the laser, but are coupled through
a third intermediate level. The two-photon excitation analyzed in this case is
implemented via a STIRAP in cascade configuration Vitanov  _et al._ (2017)
with temporal shifted Pump/Stokes laser pulse, providing a large and stable
excitation. The application of a double-STIRAP, with inverted Pump/Stokes
pulses in the second one, produces the qubit excitation and reinitialization.
Similar double-STIRAP pulses are analyzed in Beterov  _et al._ (2013) for the
implementation of quantum logic gates with Rydberg atomic ensembles. In
several configurations, the three-level STIRAP system may be reduced to a two-
level one, where our one-photon compensation scheme may be applied. Instead,
we show that the compensation even works in the non-adiabatic regime, where
the two level simplification cannot be used Laine and Stenholm (1996); Vitanov
and Stenholm (1996); Torosov and Vitanov (2013).

The paper is organized as follows:  Sec. II describes the temporal sequence of
the computation and reinitialization operations based on the two qubits, the
computation and correction one. Such a sequence is repeated for a longer time
period allowing the execution of repeated operations. Sec. III describes the
laser driving of the qubits, the corresponding one-photon excitation models
and, finally, the double-STIRAP setup for a two-photon excitation. Moreover,
Sec. III introduces the fidelity as a measure of the compensation efficiency.
Section IV describes the results of the compensation approach, depending on
the form of the excitation, based on either single or two-photon processes.
Sec. IV also presents an alternative and practically useful compensation
search tool. Sec. V concludes our work.

##  II Qubit correction

We deal with two $j={1,2}$ qubits, the first denoted as "computational" qubit
and the second one as "correction" qubit. The computational one performs a
quantum computation operation and repeats its work $n$ times, with $n$ up to
50 in our simulations. Under laser driving, this qubit starts from the initial
ground state $|g_{1}\rangle$ at time $t=0$, is excited to the state
$|e_{1}\rangle$ to perform its operation, is reinitialized and is ready for
the next round at time $t=T$. A not perfect laser driving leads, at time $T$,
to computation/reinitialization errors that propagate in the operation
sequence. We target to compensate for these errors.

Our correction process, schematized in Fig. 1(a), is based on the interaction
of the first qubit with the correction qubit. These two qubits, supposed
either equal or different, experience in their excited states $|e_{j}\rangle$
an interaction with controllable amplitude $V$ described by the following
Hamiltonian in $\hbar$ units:

$H_{V}=V|e_{1}\rangle|e_{2}\rangle\langle e_{1}|\langle e_{2}|\,.$ (1)

By a proper choice of the interaction amplitude $V$, the phase introduced into
the computational qubit will compensate for laser driving errors.
In our description, the computational process is represented by an elementary
step, for instance a rotation of the Bloch vector by a $\pi$ angle Nielsen and
Chuang (2000). In this simple approach also the reinitialization is based on a
$\pi$ pulse. The computation/correction sequence is based on the following
steps, as in Fig. 1(b). At time $t=0$, the computational qubit, initially in
its ground state, is transferred to the state $|e_{1}\rangle$ by a laser pulse
of time $T_{1}$. In the following time interval $T_{2}$, the interaction $V$
only determines the qubit evolution. Then an additional laser pulse of
duration $T_{1}$ transfers the occupation of state $|e_{1}\rangle$ to the
ground state. An interaction of duration $T_{2}$ completes the sequence with
total time $T=2T_{1}+2T_{2}$. Under perfect laser driving, the occupation
probability of the ground state of the computational qubit follows the time
dependence in Fig. 1(c). More precisely, owing to the parity of the
ground/excited states, the laser driving takes place through either a one-
photon transition or a two-photon one, with a resonant or non-resonant
intermediate level.

In the following, we will consider two configurations for the correction qubit
in the single-photon excitation case. In the first configuration (i), denoted
as two-level q-system in the following, the control or correction qubit is not
exposed to the drive. Under this assumption, the effect of the second static
qubit on the computational one is modelled by an effective level shift of its
excited state. While in the second configuration (ii), denoted as three-level
q-system, the imperfect laser excitation drives both qubits simultaneously,
producing an equivalence to the Rydberg blockade. The last example we propose
is the error compensation in a two interacting three-level setups both driven
by the same two-photon STIRAP excitation process.

##  III Qubit laser handles

###  III.1 Single-photon excitation

For both the two- and three-level cases, the qubits interact with a laser with
detuning $\delta=\omega_{L}-\omega_{0}$ from the $\omega_{0}$ ground/excited
state transition and Rabi frequency $\Omega$. The $H_{0j}$ Hamiltonian of each
atom $(j=1,2)$ in the rotating wave approximation (RWA) in the frame rotating
with the drive, is written in units of $\hbar$ as

$H_{0j}=-\delta|e_{j}\rangle\langle
e_{j}|+\frac{\Omega}{2}\left(|g_{j}\rangle\langle e_{j}|+|e_{j}\rangle\langle
g_{j}|\right).$ (2)

The RWA, in this case, remains valid in the regime where
$\Omega\ll\omega_{L}$, namely when the operation frequencies we are interested
in are much smaller than the carrier frequency of the laser Shore (2011).
Having in mind typical quantum-optical realizations based on Rydberg atoms
this condition is usually satisfied.
The transfer to the excited states and back is produced by $\pi$-pulses with
$\delta=0$, and $\Omega T_{1}=\pi$. The ground state occupation reported in
Fig. 1(c) is obtained under these conditions.

####  III.1.1 Two-level q-system (i)

Here we suppose that the computational and the correction qubit can be
addressed independently. Then we may assume that the laser excitation does not
influence the correction one, e.g., due to a large difference in transition
frequency of two different atoms. For instance, for Rydberg atoms in the
presence of Förster resonance Ravets  _et al._ (2014), we may suppose that the
correction qubit remains in a long-lived $|e_{2}\rangle$ state, the
interaction being controlled by switching on/off an electric field. The
analysis of the two-level q-system (i) is a very useful step, because it is
simpler since the interaction reduces here just to an effective energy shift
of the computation qubit. Nevertheless this case leads to the same general
compensation scheme valid also for our more complex model (ii)

####  III.1.2 Three-level q-system (ii)

As mentioned in the previous section, in this case the laser excitation drives
both qubits simultaneously and equally. Here the Dicke states represent a
convenient basis Ficek and Tanaś (2002); Comparat and Pillet (2010); Almutairi
_et al._ (2011) for the description. Starting from the $|g_{1},g_{2}\rangle$
state at $t=0$, only the $|e_{1},e_{2}\rangle$ doubly excited state and the
$|s\rangle=(|g_{1},e_{2}\rangle+|e_{1},g_{2}\rangle)/\sqrt{2}$ symmetric
states (equivalent to the Bell triplet states) are occupied by the laser
excitation with no occupation of the antisymmetric Bell/Dicke state
(equivelent to the Bell singlet state). Therefore the wavefunction is written
as

$|\psi_{tot}\rangle=c_{gg}|g_{1},g_{2}\rangle+c_{s}|s\rangle+c_{ee}|e_{1},e_{2}\rangle\,,$
(3)

and we effectively deal with a three-state system. The Hamiltonian for the two
coupled two-level systems (qubits) and its reduction to the symmetric Dicke
states is given in app. A.

Let us notice important features playing a key role in the compensation
process. Both laser detunings and inter-qubit interaction are diagonal terms,
see Eq. (15) in app. A. From the mathematical point of view, the evolution
depends only on the sign conformity of those parameters. This feature will
apply also to the compensation results. From a physical point of view the
diagonal terms may be used to balance each other. This balance occurs in the
Rydberg blockade Saffman  _et al._ (2010a), where the interaction is large
enough to block the laser excitation to the state $|e_{1},e_{2}\rangle$.
Viceversa the tuning of those parameters has been used to enhance that
excitation as in the ultracold Rydberg atom antiblockade Ates  _et al._
(2007); Amthor  _et al._ (2010) or in the Rydberg enhancement of a room
temperature vapor Kara  _et al._ (2018). We operate with Rabi frequencies of
the laser excitation, much larger than the interaction $V$, where the above
processes should _not_ occur.

###  III.2 Two-photon excitation

In the previous sections, we have described the single-photon excitation in
the two configurations (i) and (ii). In this section, instead, we show the
laser handling for a two-photon excitation scheme. In this case, the
$|g_{j}\rangle$ ground and $|e_{j}\rangle$ $(j=1,2)$ excited states, that form
the qubits, in a cascade configuration are linked by two laser fields (denoted
as Pump and Stokes) through the intermediate state $|i_{j}\rangle$. The
Hamiltonian $H_{0j}$ representing one of the two qubits is in a frame doubly
rotating at the driving frequencies and in the rotating wave approximation

$H_{j}(t)=\ \begin{pmatrix}0&\frac{\Omega_{P}(t)}{2}&0\\\
\frac{\Omega_{P}(t)}{2}&-\Delta&\frac{\Omega_{S}(t)}{2}\\\
0&\frac{\Omega_{S}(t)}{2}&-\Delta_{2}\end{pmatrix},$ (4)

with $\Delta$ the intermediate level detuning, $\Delta_{2}$ the two-photon
detuning, $\Omega_{P}$ and $\Omega_{S}$ the Pump and Stokes Rabi frequencies,
respectively. For the resonant STIRAP excitation ($\Delta_{2}=0$) with
temporally shifted Pump/Stokes laser pulses Vitanov  _et al._ (2017), the Rabi
frequencies have the following Gaussian shaped temporal dependencies:

$\Omega_{P}(t)=\Omega_{0}e^{-\left(\frac{t-T_{1}/2}{T_{G}}\right)^{2}},\quad\Omega_{S}(t)=\Omega_{0}e^{-\left(\frac{t+T_{1}/2}{T_{G}}\right)^{2}},$
(5)

with $\Omega_{0}$ peak value and width $T_{G}/\sqrt{2}$. They are parametrized
such that the pulse crossing takes place at $t=0$, and the separation between
two subsequent maxima is $T_{1}$. For the transfer back to the ground state
with an inverted Pump/Stokes temporal dependencies, the pulse crossing occurs
at time $t=T_{2}$. The time sequence of the laser driving is represented with
dashed lines in Fig. 2, please see the right vertical scale for the Rabi
frequencies. In order to maintain generality with respect to the experimental
implementations, we work with STIRAP standard dimensionless quantities $\Omega
T_{G}$ and $\Delta T_{G}$. This double-STIRAP sequence has been investigated
previously, e.g., in Higgins  _et al._ (2017); Beterov  _et al._ (2020).

An efficient and stable STIRAP transfer is realized by imposing an adiabatic
evolution of the dark state $|D_{j}\rangle$ Arimondo (1996); Fleischhauer  _et
al._ (2005) defined as

$|D_{j}(t)\rangle=\cos\theta(t)|g_{j}\rangle-\sin\theta(t)|e_{j}\rangle,$ (6)

with $\theta(t)$ given by $\tan\theta(t)=\Omega_{P}(t)/\Omega_{S}(t)$. If the
STIRAP parameters are slowly varied, the qubit initially prepared in the
ground state $|g_{j}\rangle$ follows adiabatically the instantaneous dark
state, ending up in the target state $|e_{j}\rangle$ with very large fidelity.
Nonadiabatic coupling between the eigenstates is negligible when the
$\theta(t)$ mixing angle rate is smaller than the
$\sqrt{\Delta^{2}+\Omega_{P}^{2}+\Omega_{S}^{2}}$ separation of the
Hamiltonian eigenvalues Shore (2017). Robust and efficient STIRAP transfers
occur at large values for both Rabi frequencies and intermediate state
detuning. There, an adiabatic elimination of the intermediate state leads to
an effective two-level system where our one-photon compensation may be
applied. Instead, we investigate a nonadiabatic regime where the stability
with respect to imperfections in the laser parameter is very limited. This
occurs for low $\Delta$ values, where the fidelity presents large oscillations
as a function of the laser detuning, see, e.g., Vitanov and Stenholm (1997).
This is seen in Fig. 2, where the fidelity $\mathcal{F}=0.98$ at the end of
the first double-STIRAP is not good enough for quantum computation purposes.
The complex dependence on $\Delta$ will appear below in Fig. 8(a).

While the ground $\ket{g_{1}}$ and the excited $\ket{e_{1}}$ states of the
three-level STIRAP system are described by Eq. (4) for the computation qubit,
an identical copy forms the control qubit (j=2) that interacts with the first
one via an interaction $V$ only between the two excited states, see Eq. (1).

Figure 2: Evolution of the $P_{g1}$ population (blue solid line), and of the
$\Omega_{p},\Omega_{S}$ Rabi frequencies (dashed lines) for a total time $T$,
based on a double-STIRAP transfer sequence to the excited state and back to
the ground, as for an atomic Rydberg state. Parameters $T_{G}=1$, $T_{1}=1.2$,
$\Delta=\Delta_{2}=0$, $\Delta T_{G}=1.4$, $\Omega_{0}T_{G}=12$, and
$T_{2}=10$. The $P_{g}$ occupation scale is on the left, and the
$\Omega(t)T_{G}$ scale is on the right. The $P_{g1}$ temporal evolution over a
single $T$ sequence presents the coherent oscillations discussed in Laine and
Stenholm (1996).

###  III.3 Fidelity/Infidelity

The efficiency of the correction qubit approach will be measured at the times
$t=nT$ by the fidelity $\mathcal{F}$, or the infidelity
$\mathcal{I}=1-\mathcal{F}$, between the final and the initial (ground) state
$|g_{1}\rangle$ of the first qubit. For the two-state q-system (i), presented
in Sec. III.1.1, the straightforward fidelity is, given an arbitrary state of
the first qubit $\ket{\psi_{1}}$,

$\mathcal{F}_{1}(t=nT)=\left|\langle g_{1}|\psi_{1}(nT)\rangle\right|^{2}.$
(7)

For the three-level setup of Sec. III.1.2, instead, the required fidelity is
obtained by performing a reduction (partial trace Nielsen and Chuang (2000))
to the total density matrix. Using the $\psi_{\rm tot}$ wavefunction in the
Dicke state basis of Eq. (3), the fidelity is given by

$\displaystyle\mathcal{F}_{2}$
$\displaystyle=\text{Tr}[(\ket{g_{1}}\bra{g_{1}}\otimes\mathbb{1}_{2})(\ket{\psi_{\rm
tot}}\bra{\psi_{\rm tot}})]$
$\displaystyle=|c_{gg}(nT)|^{2}+\frac{1}{2}|c_{s}(nT)|^{2}\,.$ (8)

This fidelity contains the predicted occupation of the computational ground
state and an additional term associated to the identity of the two qubits.

Figure 3: 2D contour plots of the logarithmic scale $\mathcal{I}$ infidelity
in the ($\epsilon$, $VT_{2}$) plane for a rotation error. The colour scale
below (c) applies to the plots (a-c). The black/white regions indicate
infidelity values larger/smaller than the color scale limits. Plot of
$\mathcal{I}_{1}(t=T)$ in (a), $\mathcal{I}_{1}(t=50T)$ in (b), and
$\mathcal{I}_{2}(t=50T)$ in (c, d). In (a), and (b) the error compensation
corresponds to the configuration (i) of Sec. III.1.1, in (c, d) the model (ii)
of Sec. III.1.2. In (d) the interaction is continuously active.

##  IV Fidelity compensation

In typical applications as considered here, the Rabi frequency $\Omega$ can be
much larger than the interaction $V$ between control and computational qubit.
In this case, the duration $T_{1}$ of a $\pi$-pulse is much shorter than the
time $T_{2}$ needed for the interaction to have a sizeable effect. In the
limit $T_{1}\ll T_{2}$, it is then justified to treat $V$ as being non-zero
only during the interval $T_{2}$. This assumption enables a simpler analytical
treatment adopted in the following expressions, while the general
configuration is addressed numerically.

We now address typical errors that lead to imperfect population transfer: in
Sec. IV.1.1 an error in the rotation angle of the Bloch vector of the form
$\sigma_{x}$, and in Sec. IV.1.3 of the rotational axis of the form
$\sigma_{z}$.

Given, e.g., the well known long lifetime of the excited Rydberg states
Saffman  _et al._ (2010b), other effects of decoherence can be safely
neglected.

###  IV.1 Single-photon excitation

####  IV.1.1 Rotation angle error

For the Rabi $\pi$ pulse applied over the $T_{1}$ time, we introduce a
relative error $\epsilon$ given by

$\Omega T_{1}=\pi(1+\epsilon)$ (9)

associated to the Bloch vector rotation angle Nielsen and Chuang (2000).
At the time $t=T=2T_{1}+2T_{2}$, after the first sequence with two $\pi$
pulses and an acquired phase from the excited state evolution, the ground
state fidelity $\mathcal{F}_{1}(t=T)$ for the two-state setup is

$\displaystyle\mathcal{F}_{1}(t=T)=$ $\displaystyle
1-\frac{1}{2}\sin^{2}\left(2\pi\epsilon\right)\left[1+\cos\left(VT_{2}\right)\right]$
$\displaystyle\approx$ $\displaystyle
1-2(\pi\epsilon)^{2}\left[1+\cos\left(VT_{2}\right)\right],$ (10)

in the second line at small $\epsilon$ values. The above expressions report an
important feature associated to all the compensation schemes here examined.
The $VT_{2}=(2m+1)\pi$ value, with $m$ integer, is "magic" because it produces
a strong recovery of the fidelity for the driving error. The excited state
phase shift leads to a positive interference in the wavefunction evolution at
all the times $t=nT$. At the magic values, the interaction steps preceding and
following the second laser pulse produce the exact reverse of the first pulse
rotation angle, as shown in Appendix B. The complete qubit sequence reduces to
the identity, and the initial state is exactly restored for any value of
$\epsilon$. While a full recovery is obtained for all $n$ values at the magic
value, Eq. (10) reports a fidelity very close to the one for interaction
strengths near the magic value. This phase compensation mechanism has a strong
analogy with the spin echo and dynamical decoupling techniques Viola  _et al._
(1999); Lidar and Brun (2013), where an arbitrary phase shift is compensated
by a double laser excitation. It is also analog to the anti-resonances of the
quantum kicked-rotor model where each pulse exactly counteracts the preceding
one due to a proper choice of free phase evolution in-between the two pulses
Izrailev (1990); Sadgrove and Wimberger (2011).

With the laser acting on the computational qubit only, namely in our model
(i), the plots in the $(\epsilon,VT_{2})$ plane report the resulting
infidelity $\mathcal{I}_{1}(t=T)$ in Fig. 3(a) and $\mathcal{I}_{1}(t=50T)$ in
(b). These plots, as most of the following ones, are limited on the horizontal
to the $|\epsilon|\leq 0.015$ value, easily reached in experiments, and to
$|VT_{2}|\leq 5$ owing to the vertical periodicity (phase) of the
compensation. The infidelity range reported in the plots corresponds to
standard quantum computation experiments. The plots highlight that the
compensation scheme works very well. However, increasing the qubit cycling
number, the compensation range around the magic value becomes narrow, and
requires a more precise choice of the interaction parameter. The compensation
efficiency appears clearly from the data of Fig. 4 where the compensated
fidelity $\mathcal{F}_{1}$ vs. $n$ is compared to the fidelity reached in
multiple operations in the absence of the compensation. For highlighting the
robustness of the compensation, notice that the chosen parameter $V$ is only
close to the magic value. Therefore, a small imperfection in the product
$VT_{2}$ does not affect the efficiency of the compensation.

In the three-level q-model (ii), instead, the simultaneous driving of both the
computation and compensation qubit leads to slightly different features in the
time evolution, but the compensation is still produced by the combination of
the in-phase laser excitation and the phase acquired here by the state
$|e_{1},e_{2}\rangle$ with the laser excitation off. For $V$ off while the
laser is on, an analytical solution of the Dicke state time evolution,
followed by a projection on the single qubit space, produces the fidelity at
time $t=T$ reported in Appendix A with a complex dependence on sine and cosine
functions. At small $\epsilon$ values the fidelity, from Eqs. (16) and (17),
becomes

$\mathcal{F}_{2}(t=T)\approx
1-\frac{1}{2}(\pi\epsilon)^{2}\left[1+\cos\left(VT_{2}\right)\right].$ (11)

Only for one rotation at time $t=T$, the phase shift $\pi$ remains magic.
However, as the number $n$ of interrogations increases, the behavior of the
setup (ii) is different from the case (i). The fidelity of the three-level
model remains overall higher than the one of the two-level one, as can be
appreciated from a comparison between the plots (b) and (c) of Fig. 3. It
should be mentioned here that from the time $t=2T$ onwards the contribution of
the Dicke/Bell symmetric state remains below the $10^{-5}$ level, therefore
with an operating compensation, the computational qubit is in perfect shape
for continuing its job.

Figure 4: Fidelity $\mathcal{F}_{1}(t=nT)$ vs. the sequence number $n$ at a
given error $\epsilon=0.007$: in the absence of compensation (solid blue line)
and for $VT_{2}=2.5$ (red dashed line) close to the optimal compensation for a
laser driving of the setup (i).

For the case when the interaction $V$ is applied also within the laser
excitation period $T_{1}$, the full analytical solution is not available.
Therefore we rely on numerical simulations, such as shown in Fig. 3(d), for
the three-level model driving as in all the following figures. The response to
the compensation is greatly modified with an optimal compensation that is
almost independent of the error $\epsilon$ (for the explored range) at new
magic values $VT_{2}=\pm 2.8$ and $VT_{2}=\pm 3.9$. These values converge to
the previous magic ones at larger values of the ratio $T_{2}/T_{1}$.

Figure 5: In (a) $|\tilde{P}(\nu)|$ Fourier spectrum of the population
difference for the two-level setup (i) at $\epsilon=0.1$ and for $VT_{2}=1$.
The sidebands and the splitting are well distinguishable only at these large
$\epsilon$ values. They coalesce into $\nu_{0}$ for the magic compensation
value, represented schematically by the red curve. In (b) we show
$1-\tilde{P}(\nu_{0})$ on a logarithmic scale vs. the parameter plane
$(\epsilon,VT_{2})$ for the rotation error. The white regions indicate
infidelity values smaller than the minimum of the color scale.

####  IV.1.2 Compensation search tool

Although the study of the fidelity allows us to identify parameter values
giving the desired compensation at fixed interrogation times $t=nT$, these
values may not guarantee the same fidelity for different $n$. Therefore, we
present now a simple approach for determining the compensation parameters
guaranteeing high fidelity for all periods $nT$. It is based on the Fourier
components of the population difference $P(t)=P_{g}(t)-P_{e}(t)$. At
$\epsilon=0$ and at perfect compensation, $P=P_{g}-P_{e}$ is a periodic
function whose Fourier spectrum contains a single component at $\nu_{0}=1/T$.
At the $\epsilon\neq 0$ rotation error, a splitting into two sidebands arises
in the population difference of the Fourier spectrum $\tilde{P}(\nu)$ at the
frequencies $\nu_{0}(1\pm\epsilon T_{1})$, as from the two-level Rabi
evolution for $\pi$ pulses. The interaction $V$ modifies the sideband
positions appearing in Fig. 5(a), with a sinusoidal dependence on $VT_{2}$. It
also introduces additional weaker sidebands at even larger $\epsilon$ values.
At the interaction values of best compensation, the main sideband ideally
coalesce into the $\nu_{0}$ value. The $1-\tilde{P}(\nu_{0})$ amplitude of the
Fourier spectrum peak vs. the $(\epsilon,VT_{2})$ parameters allows a simple
determination of the compensation range, as in Fig. 5(b). The analogy between
this plot and that of Fig. 3(c) highligths the utility of this search tool in
the full parameter space for the compensation.

Figure 6: Two-level q-system (i): 2D plots of the $\mathcal{I}_{2}(t=50T)$
infidelity plot vs. the $(\delta T_{1},VT_{2})$ parameters for the laser
detuning error. Panel (a) is obtained for the case of $V$ off in the periods
when the laser is on. In (b) the interaction is on at all times. Here, the
tilted black line corresponds to the Rydberg enhancement condition
$V=2\delta$, see main text.

####  IV.1.3 Rotation axis error

We examine here the compensation of an error associated to the Bloch vector
rotation axis Nielsen and Chuang (2000). We suppose that the laser detuning
$\delta$ error from resonance is given by

$\delta T_{1}=\pi\epsilon\,,$ (12)

where $\delta T_{1}\equiv\delta\cdot T_{1}$.  In a $\pi$ pulse the Bloch
vector rotates by the following angle

$\Omega_{\text{eff}}T_{1}=\left[\pi^{2}+(\delta
T_{1})^{2}\right]^{1/2}\approx\pi(1+\frac{1}{2}\epsilon^{2})\,,$ (13)

to be compared to Eq. (9) of the previous case. For small $\epsilon$ values,
the fidelity at the time $T$ for the model (i) configuration is

$\begin{split}\mathcal{F}_{1}(t=T)\approx&1-2\epsilon^{2}\left[1-\cos(VT_{2})\right]\\\
&-\pi\epsilon^{3}(2T_{2}+1)\sin(VT_{2})\,.\end{split}$ (14)

Both cosine and sine dependencies on $VT_{2}$ appear in the present fidelity.
These features lead to the compensation shown in Fig. 6(a). This figure may be
compared to Fig. 3(c), also for the horizontal scale owing to the detuning
error definition of Eq. (12). Within the central region of low $\epsilon$
error, the magic $VT_{2}=\pi$ value does not appear clearly. The sine term of
Eq. (14) leads to an asymmetric response of the $(\delta T_{1},VT_{2})$ plot,
breaking the analogy with the Rabi error where a $\delta$ symmetry is valid
for all the $\epsilon$ values.

The results for an interaction $V$ acting at all times are shown in Fig. 6(b).
The diagonal line dependence of the top figure is greatly enhanced. For a
given value of $V$, the compensation is effective within a more limited range
of the detuning error. In order to obtain a wider range of error compensation,
the role of $V$ within the laser interaction period is reduced by decreasing
the ratio $T_{1}/T_{2}$.

The diagonal response in Fig. 6(a) for very small values of $\delta
T_{1}(\approx 0.05$) and in (b) globally, is evidence of the Rydberg
antiblockade/enhancement. The doubly excited state for the two atoms is
reached for $\delta=V/2$, i.e., $\omega_{L}=\omega_{0}+V/2$, taking into
account our definition of the interaction energy $V$ provided to the two
atoms. This condition corresponds to the black line shown in Fig. 6(b). The
excitation of both qubits associated to the enhancement allows for the
computational qubit to acquire the phase shift required for the compensation.
Notice that, owing to our choice of the laser driving parameters, $\Omega\gg
V$, the Rydberg enhancement, and also the blockade, are supposed to be
negligible Saffman  _et al._ (2010a). However, even a weak enhancement may
contribute essentially to the more sensitive phase-compensation.

###  IV.2 Double-STIRAP compensation

Figure 7: The $P_{g1}$ population (thick blue solid line) and $P_{i1}$ (thin
orange line) vs. $t$ time up to $5T$ for a double-STIRAP operation with
$\Delta T_{1}=1.4$ and parameters as in Fig. 2. (a) without compensation and
(b) with an applied compensation $VT_{2}=2$. Notice the large increase in
fidelity produced by the compensation, from $\mathcal{F}_{2}(t=5T,V=0)\simeq
0.43$ to $\mathcal{F}_{2}(t=5T,VT_{2}=2)\simeq 0.96$.

We now turn to the two interacting STIRAP configurations introduced in Sec.
III.2. In the case of the two-photon resonant excitation $\Delta_{2}=0$, we
examine the compensation for the non-adiabatic regime in which the fidelity is
very sensitive to the intermediate-level detuning $\Delta$. As a consequence
of the phase accumulation, such a poorly controlled response is enhanced in a
long sequence of double-STIRAP pulses. The rapid decrease of the fidelity with
the $n$ number, is shown in Fig. 7(a) where the $P_{g1}$ population of the
initial state is plotted vs. time up to $t=5T$. Driving parameters are as in
Fig. 2. An important new feature is the large $P_{i1}$ population occupation
of the intermediate state, contributing to less efficient transfer processes
with increasing sequence number $n$. The complex dependence of the infidelity
on $\Delta$ is presented in Fig. 8(a) evidencing its drastic increase around
the value $\Delta T_{1}=1.4$.

Figure 8:  In (a) $n=5$ infidelities vs. the intermediate level detuning
$\Delta$, blue line without compensation and symmetric with respect to the
sign of $\Delta$; orange line for an applied compensation $VT_{2}=2$ and not
symmetric with respect to the sign of $\Delta$. The black vertical line marks
the $\Delta T_{1}=1.4$ value of Figs. 2 and 7. The other STIRAP parameters are
those given in Fig. 2. (b) two-dimensional $(\Delta T_{1},VT_{2})$ plot of the
$1/T$ Fourier spectrum component for the compensation search. The compensation
is efficient in different regions of the parameter plane. The black vertical
line identifies the $\Delta T_{1}=1.4$ value, as in (a).

We analyze the Fourier spectrum to determine the compensation values when the
interaction $V$ is applied at all times. As an example, Fig. 7(b) shows the
good and stable fidelity of the computational qubit up $n=5$ when an
interaction compensation $VT_{2}=2$ is applied for the laser parameters of the
figure. Very similar results are obtained if the interaction is switched on
only for the $T_{2}$ periods. The two-dimensional plot in Fig. 8(b) for the
central Fourier component vs. the $(\Delta T_{1},VT_{2})$ parameters, shows
the presence of regions where the compensation is very efficient. The
compensation remains efficient for a reasonable large range of interaction
amplitudes. However, for a maximal compensation, a tuning of the interaction
amplitude with the detuning value is required. It should be pointed out that
the best compensation is obtained for a sign conformity between $\Delta$ and
$V$, as for the data of Fig. 6. This response suggests a hidden role of the
Rydberg enhancement, in appearance not connected to the $(\delta,V)$ relation
discussed above for the one-photon case. In Fig. 8(b) the tilted parallel
lines corresponding to constant compensation reflect the sinusoidal dependence
on the parameter $VT_{2}$ appearing in the expressions for the fidelity
reported above.

##  V Conclusions

We have introduced a model for robust quantum control, in which driving errors
affecting a computational qubit are corrected or compensated via the
interaction with an additional correction qubit. For the qubit Bloch vector,
the standard errors on rotation angle and axis have been considered.
Compensation schemes allowing a very efficient recovery of the fidelity are
determined. The fidelity remains very high for up to our choice of fifty qubit
operations for the parameters investigated here, but longer sequences could be
explored on equal footing. An optimal control approach may be used to
determine the compensation in the simultaneous presence of both rotation angle
and axis errors. For ultracold atomic qubits, the interaction is naturally
provided by the Rydberg interactions that represent a very efficient tool for
quantum simulations or quantum computation Saffman  _et al._ (2010b); Jaksch
_et al._ (2000). Using the Förster resonances, the interaction amplitude is
easily controlled also with a fast temporal response. For most of the explored
schemes we have demonstrated that the compensation is not very sensitive to
the precise value of the interaction strength or, equivalently, the
interaction time $T_{2}$. In addition the required corrections are within
range typically accessible in experimental realizations, and might help to
improve substantially the fidelities reached in novel setups such as based on
optomechanics Fedoseev  _et al._ (2021).

The quantum control of a long sequence of qubit operations is determined by
numerical calculations. However, the examination of the $t=T$ single sequence
by analytical calculation, leads to compensation requirements with a precision
good enough for exploring longer temporal sequences. Because the compensation
efficiency becomes more critical at larger sequence numbers, it should be
tuned performing numerical tests.

Our scheme has some analogy with the fidelity increase obtained in quantum
gates via Rydberg interactions by driving simultaneously the control and
target qubits as examined in Beterov  _et al._ (2013); Saffman  _et al._
(2020). It will be interesting to explore the application of our compensation
to quantum gates Bhaktavatsala Rao and Mølmer (2014); Wu  _et al._ (2021);
Zhang  _et al._ (2020). The scheme could also be extended to more elaborated
STIRAP protocols, as that recently introduced in Di Stefano  _et al._ (2016)
for artificial atoms. An additional feature to be investigated is the role of
qubit dissipation within the times of our compensation, in particular in view
of solid-state, e.g., superconducting qubit implementations Blais  _et al._
(2021).

##  VI Acknowledgments

One of the authors (E.A.) thanks G. La Rocca and D. Rossini for inspiring
discussions. Special thanks goes to G. Falci for a critical reading of the
manuscript.

## Appendices

##  Appendix A Three-level q-system Hamitonian and fidelity

The aim of this Appendix is to derive the Hamiltonian for the three-level
q-system, once again for the case where the interaction acts only within the
$T_{2}$ periods. The solution in the Dicke basis leads to the fidelity of the
computational qubit.

Within the single-qubit basis the Hamiltonian is given by the sum of
$H_{01}\otimes I+I\otimes H_{02}$, with $I$ the identity matrix and $H_{0j}$
from Eq. (2), and of the interaction term of Eq. (1). As in Sec. II, the
convenient basis to study the present evolution of two qubits is composed by
the three symmetric Dicke states of Eq. (3). Within that basis the Hamiltonian
is described by the following matrices

$\displaystyle H_{0}=\begin{pmatrix}0&\frac{\Omega}{\sqrt{2}}&0\\\
\frac{\Omega}{\sqrt{2}}&\delta&\frac{\Omega}{\sqrt{2}}\\\
0&\frac{\Omega}{\sqrt{2}}&2\delta\\\ \end{pmatrix},$ $\displaystyle
H_{V}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&0&V\\\ \end{pmatrix}\,,$ (15)

for the laser-on periods and interaction-only periods, respectively. Notice
the cooperative $\sqrt{2}$ increase of the Rabi frequency for laser excitation
to the symmetric state $|s\rangle$ with respect to Eqs. (2) and (4). For
$\delta=0$, the evolution equations are equivalent to the Bloch equations for
a spin-1 system in resonance with the driving field. The interaction produces
a $VT_{2}$ phase shift of the $|e_{1},e_{2}\rangle$ doubly excited state.

For the calculation of fidelity from Eq. (8) the following Dicke state
occupations are required:

$\displaystyle\begin{split}|c_{gg}(T)|^{2}=&\frac{1}{4}\Big{[}\left(1-\cos(2\Omega
T_{1})\right)-\frac{1}{2}\left(1-\cos(\Omega T_{1})\right)^{2}\times\\\
&\times\left(1-\cos(VT_{2})\right)\Big{]}^{2}+\\\
&+\frac{1}{16}\left[1-\cos(\Omega
T_{1})\right]^{4}\sin^{2}(VT_{2})\end{split}$ (16)
$\displaystyle\begin{split}|c_{s}(T)|^{2}=&\frac{1}{4}\sin^{2}(\Omega
T_{1})\Big{[}\cos^{2}(\Omega T_{1})\left(5+3\cos(VT_{2})\right)+\\\
&+\left(2\cos(\Omega
T_{1})+1\right)\left(1-\cos(VT_{2})\right)\Big{]}\,.\end{split}$ (17)

Using Eq. (8) the fidelity at time $(t=T)$ is obtained as

$\displaystyle\mathcal{F}_{2}(t=T)=|c_{gg}(T)|^{2}+\frac{1}{2}|c_{s}(T)|^{2}\,.$
(18)

Iterating such a procedure, one can obtain analytical expressions of the
fidelity at periods $n>1$. However, due to its growing complexity, a software
such as Mathematica is helpful for obtaining the rather lengthy algebraic
expressions with increasing $n$. In the Delvecchio (2021) we report the
Mathematica notebook for the study of the reduced three-level q-system showed
in sec III.1.2.

##  Appendix B Two-level q-system with $VT_{2}=\pi$

The present target is to derive the magic compensation value from the temporal
evolution of the two-level q-system, with time separated actions of laser and
interaction. For the case of rotation angle error with an arbitrary Rabi
frequency $\Omega$ including the $\epsilon$ error as from Eq. (9), the
evolution operator at time $T=2T_{1}+2T_{2}$ is

$U(T)=e^{-iVT_{2}\ket{e_{1}}\bra{e_{1}}}e^{-i\frac{\Omega
T_{1}}{2}\sigma_{x1}}e^{-iVT_{2}\ket{e_{1}}\bra{e_{1}}}e^{-i\frac{\Omega
T_{1}}{2}\sigma_{x1}}$ (19)

with $\sigma_{x1}$ the Pauli matrix for the computational qubit. We
demonstrate that for $VT_{2}=\pi$ the operator $U(T)$ corresponds to the
identity matrix, for all the $\Omega$ values.

Let us consider the first three exponentials in Eq. (19) r.h.s.. Written in
matrix form, they explicitly read

$\begin{split}\begin{pmatrix}e^{-iVT_{2}}&0\\\
0&1\end{pmatrix}&\begin{pmatrix}\cos(\frac{\pi}{2}(1+\epsilon))&&-i\sin(\frac{\pi}{2}(1+\epsilon))\\\
-i\sin(\frac{\pi}{2}(1+\epsilon))&&\cos(\frac{\pi}{2}(1+\epsilon))\end{pmatrix}\times\\\
\times&\begin{pmatrix}e^{-iVT_{2}}&0\\\ 0&1\end{pmatrix}\,,\end{split}$ (20)

Setting $VT_{2}=\pi$ and performing the matrix product, we obtain

$\begin{pmatrix}\cos(\frac{\pi}{2}(1+\epsilon))&&i\sin(\frac{\pi}{2}(1+\epsilon))\\\
i\sin(\frac{\pi}{2}(1+\epsilon))&&\cos(\frac{\pi}{2}(1+\epsilon))\end{pmatrix}=e^{i\frac{\Omega
T_{1}}{2}\sigma_{x}}\,.$ (21)

This operator performs a rotation of the Bloch vectors equal and opposite to
that induced by the first pulse of the sequence in Eq. (19). Inserting the
latter result in that equation we obtain

$U(T)=e^{i\frac{\Omega T_{1}}{2}\sigma_{x}}e^{-i\frac{\Omega
T_{1}}{2}\sigma_{x}}=I\,,$ (22)

demonstrating the magic value compensation, independent of the $\epsilon$
rotation angle error.

The application of the evolution operator to the case of the rotation axis
error unfortunately does not produce a similar simple interpretation since
both errors cannot be exactly corrected by the same procedure from above. One
may apply, however, an optimal control approach to find the optimal value for
the best simultaneous compensation of both errors.

## References

  * Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
  * Terhal (2015) B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015), arXiv:1302.3428v7.
  * Chiesa  _et al._ (2020) A. Chiesa, E. Macaluso, F. Petiziol, S. Wimberger, P. Santini, and S. Carretta, Molecular nanomagnets as qubits with embedded quantum-error correction, J. Phys. Chem. Lett. 11, 8610 (2020), pMID: 32936660.
  * Levitt (1986) M. H. Levitt, Composite pulses, Progr. Nucl. Magn. Res. Spectrosc. 18, 61 (1986).
  * Jones (2011) J. A. Jones, Quantum computing with NMR, Progr. Nucl. Magn. Res. Spectrosc. 59, 91 (2011), arXiv:1011.1382v1.
  * Guéry-Odelin  _et al._ (2019) D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019), arXiv:1904.08448v2.
  * Glaser  _et al._ (2015) S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Training Schrödinger´s cat: quantum optimal control - strategic report on current status, visions and goals for research in europe, Eur. Phys. J. D 69, 279 (2015), arXiv:1508.00442v1.
  * Mahmud and Tiesinga (2013) K. W. Mahmud and E. Tiesinga, Dynamics of spin-1 bosons in an optical lattice: Spin mixing, quantum-phase-revival spectroscopy, and effective three-body interactions, Phys. Rev. A 88, 023602 (2013), arXiv:1304.7565v1.
  * Meinert  _et al._ (2014) F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, M. Gröbner, and H.-C. Nägerl, Interaction-Induced Quantum Phase Revivals and Evidence for the Transition to the Quantum Chaotic Regime in 1D Atomic Bloch Oscillations, Phys. Rev. Lett. 112, 193003 (2014), arXiv:1309.4045v2.
  * Fan  _et al._ (2019) C.-H. Fan, H.-X. Zhang, and J.-H. Wu, In-phase and antiphase dynamics of rydberg atoms with distinguishable resonances, Phys. Rev. A 99, 033813 (2019).
  * Fan  _et al._ (2020) C.-H. Fan, D. Rossini, H.-X. Zhang, J.-H. Wu, M. Artoni, and G. C. La Rocca, Discrete time crystal in a finite chain of Rydberg atoms without disorder, Phys. Rev. A 101, 013417 (2020), arXiv:1907.03446v2.
  * Bhaktavatsala Rao and Mølmer (2014) D. D. Bhaktavatsala Rao and K. Mølmer, Robust Rydberg-interaction gates with adiabatic passage, Phys. Rev. A 89, 030301 (2014), arXiv:1311.5147v1.
  * Beterov  _et al._ (2020) I. I. Beterov, D. B. Tretyakov, V. M. Entin, E. A. Yakshina, I. I. Ryabtsev, M. Saffman, and S. Bergamini, Application of adiabatic passage in Rydberg atomic ensembles for quantum information processing, J. Phys. B: At. Mol. Opt. Phys. 53, 182001 (2020), arXiv:2001.06352v1.
  * de Léséleuc  _et al._ (2018) S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, Analysis of imperfections in the coherent optical excitation of single atoms to Rydberg states, Phys. Rev. A 97, 053803 (2018), arXiv:1802.10424v1.
  * Walls and Milburn (1995) D. F. Walls and G. J. Milburn,  _Quantum optics / D.F. Walls, G.J. Milburn_ , springer study ed. ed. (Springer-Verlag Berlin ; New York, 1995) pp. xii, 351 p. :.
  * Egan  _et al._ (2021) L. Egan, D. M. Debroy, C. Noel, A. Risinger, D. Zhu, D. Biswas, M. Newman, M. Li, K. R. Brown, M. Cetina, and C. Monroe, Fault-tolerant control of an error-corrected qubit, Nature 598, 281 (2021).
  * Saffman  _et al._ (2010a) M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with Rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010a), arXiv:0909.4777v3.
  * Labuhn  _et al._ (2016) H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum ising models, Nature 534, 10.1038/nature18274 (2016), arXiv:1509.04543v3.
  * Adams  _et al._ (2019) C. S. Adams, J. D. Pritchard, and J. P. Shaffer, Rydberg atom quantum technologies, J. Phys. B: At. Mol. Opt. Phys. 53, 012002 (2019), arXiv:1907.09231.
  * Zhang  _et al._ (2020) C. Zhang, F. Pokorny, W. Li, G. Higgins, A. Pöschl, I. Lesanovsky, and M. Hennrich, Submicrosecond entangling gate between trapped ions via Rydberg interaction, Nature 580, 345 (2020), arXiv:1908.11284v1.
  * Borish  _et al._ (2020) V. Borish, O. Marković, J. A. Hines, S. V. Rajagopal, and M. Schleier-Smith, Transverse-field Ising dynamics in a Rydberg-dressed atomic gas, Phys. Rev. Lett. 124, 063601 (2020), arXiv:1910.13687v3.
  * Ravets  _et al._ (2014) S. Ravets, H. Labuhn, D. Barredo, L. Béguin, T. Lahaye, and A. Browaeys, Coherent dipole-dipole coupling between two single Rydberg atoms at an electrically-tuned Förster resonance, Nature Physics 10, 914 (2014), arXiv:1405.7804v1.
  * Higgins  _et al._ (2017) G. Higgins, F. Pokorny, C. Zhang, Q. Bodart, and M. Hennrich, Coherent control of a single trapped rydberg ion, Phys. Rev. Lett. 119, 220501 (2017), arXiv:1708.06387.
  * Huang  _et al._ (2018) X.-R. Huang, Z.-X. Ding, C.-S. Hu, L.-T. Shen, W. Li, H. Wu, and S.-B. Zheng, Robust Rydberg gate via Landau-Zener control of Förster resonance, Phys. Rev. A 98, 052324 (2018), arXiv:1806.09775v4.
  * Schwartz  _et al._ (2021) I. Schwartz, Y. Shimazaki, C. Kuhlenkamp, K. Watanabe, T. Taniguchi, M. Kroner, and A. Imamoglu, Electrically tunable feshbach resonances in twisted bilayer semiconductors, Science 374, 336 (2021), https://www.science.org/doi/pdf/10.1126/science.abj3831 .
  * Taherkhani  _et al._ (2019) M. Taherkhani, M. Willatzen, E. V. Denning, I. E. Protsenko, and N. Gregersen, High-fidelity optical quantum gates based on type-ii double quantum dots in a nanowire, Phys. Rev. B 99, 165305 (2019).
  * Giannelli and Arimondo (2014) L. Giannelli and E. Arimondo, Three-level superadiabatic quantum driving, Phys. Rev. A 89, 033419 (2014), arXiv:1402.1299v1.
  * Vitanov  _et al._ (2017) N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, Stimulated Raman adiabatic passage in physics, chemistry, and beyond, Rev. Mod. Phys. 89, 015006 (2017), arXiv:1605.00224v2.
  * Beterov  _et al._ (2013) I. I. Beterov, M. Saffman, E. A. Yakshina, V. P. Zhukov, D. B. Tretyakov, V. M. Entin, I. I. Ryabtsev, C. W. Mansell, C. MacCormick, S. Bergamini, and M. P. Fedoruk, Quantum gates in mesoscopic atomic ensembles based on adiabatic passage and Rydberg blockade, Phys. Rev. A 88, 010303 (2013), arXiv:1212.1138v3.
  * Laine and Stenholm (1996) T. A. Laine and S. Stenholm, Adiabatic processes in three-level systems, Phys. Rev. A 53, 2501 (1996).
  * Vitanov and Stenholm (1996) N. V. Vitanov and S. Stenholm, Non-adiabatic effects in population transfer in three-level systems, Opt. Comm. 127, 215 (1996).
  * Torosov and Vitanov (2013) B. T. Torosov and N. V. Vitanov, Composite stimulated Raman adiabatic passage, Phys. Rev. A 87, 043418 (2013), arXiv:1306.0699v1.
  * Shore (2011) B. Shore,  _Manipulating quantum structures using laser pulses_ (Cambridge University Press, 2011).
  * Ficek and Tanaś (2002) Z. Ficek and R. Tanaś, Entangled states and collective nonclassical effects in two-atom systems, Phys. Rep. 372, 369 (2002), arXiv:quant-ph/0302082v1.
  * Comparat and Pillet (2010) D. Comparat and P. Pillet, Dipole blockade in a cold Rydberg atomic sample, J. Opt. Soc. Am. B 27, A208 (2010), arXiv:1006.0742v1.
  * Almutairi  _et al._ (2011) K. Almutairi, R. Tanaś, and Z. Ficek, Generating two-photon entangled states in a driven two-atom system, Phys. Rev. A 84, 013831 (2011).
  * Ates  _et al._ (2007) C. Ates, T. Pohl, T. Pattard, and J. M. Rost, Antiblockade in Rydberg excitation of an ultracold lattice gas, Phys. Rev. Lett. 98, 023002 (2007), arXiv:physics/0605111v1.
  * Amthor  _et al._ (2010) T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, Evidence of antiblockade in an ultracold Rydberg gas, Phys. Rev. Lett. 104, 013001 (2010), arXiv:0909.0837.
  * Kara  _et al._ (2018) D. Kara, A. Bhowmick, and A. K. Mohapatra, Rydberg interaction induced enhanced excitation in thermal atomic vapor, Scientific Reports 8, 3236 (2018), arXiv:1710.05573v1.
  * Arimondo (1996) E. Arimondo, Coherent population trapping in laser spectroscopy (Elsevier, 1996) pp. 257–354.
  * Fleischhauer  _et al._ (2005) M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Electromagnetically induced transparency: Optics in coherent media, Rev. Mod. Phys. 77, 633 (2005).
  * Shore (2017) B. W. Shore, Picturing stimulated Raman adiabatic passage: a STIRAP tutorial, Adv. Opt. Phot. 9, 563 (2017).
  * Vitanov and Stenholm (1997) N. V. Vitanov and S. Stenholm, Properties of stimulated Raman adiabatic passage with intermediate-level detuning, Opt. Comm. 135, 394 (1997).
  * Saffman  _et al._ (2010b) M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010b).
  * Viola  _et al._ (1999) L. Viola, E. Knill, and S. Lloyd, Dynamical decoupling of open quantum systems, Phys. Rev. Lett. 82, 2417 (1999), arXiv:quant-ph/9809071v2.
  * Lidar and Brun (2013) D. A. Lidar and T. A. Brun,  _Quantum Error Correction_ (Cambridge University Press, 2013).
  * Izrailev (1990) F. M. Izrailev, Simple models of quantum chaos: Spectrum and eigenfunctions, Phys. Rep. 196, 299 (1990).
  * Sadgrove and Wimberger (2011) M. Sadgrove and S. Wimberger, Chapter 7 - A Pseudoclassical Method for the Atom-Optics Kicked Rotor: from Theory to Experiment and Back (Academic Press, 2011) pp. 315–369.
  * Jaksch  _et al._ (2000) D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000).
  * Fedoseev  _et al._ (2021) V. Fedoseev, F. Luna, I. Hedgepeth, W. Löffler, and D. Bouwmeester, Stimulated Raman Adiabatic Passage in Optomechanics, Phys. Rev. Lett. 126, 113601 (2021), arXiv:1911.11464v2.
  * Saffman  _et al._ (2020) M. Saffman, I. I. Beterov, A. Dalal, E. J. Páez, and B. C. Sanders, Symmetric Rydberg controlled-$z$ gates with adiabatic pulses, Phys. Rev. A 101, 062309 (2020), arXiv:1912.02977v3.
  * Wu  _et al._ (2021) J.-L. Wu, Y. Wang, J.-X. Han, S.-L. Su, Y. Xia, Y. Jiang, and J. Song, Resilient quantum gates on periodically driven Rydberg atoms, Phys. Rev. A 103, 012601 (2021), arXiv:2101.02328v1.
  * Di Stefano  _et al._ (2016) P. G. Di Stefano, E. Paladino, T. J. Pope, and G. Falci, Coherent manipulation of noise-protected superconducting artificial atoms in the lambda scheme, Phys. Rev. A 93, 051801 (2016), arXiv:1509.05562v2.
  * Blais  _et al._ (2021) A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).
  * Delvecchio (2021) M. Delvecchio, Two periodically driven interacting qubits (2021), mathematica notebook, available online: https://notebookarchive.org/2021-06-bjbq4hz.