# Two-qubit atomic gates: Spatio-temporal control of Rydberg interaction

Ignacio R. Sola  corresponding author<EMAIL_ADDRESS>Departamento de Quimica
Fisica, Universidad Complutense, 28040 Madrid, Spain  Vladimir S. Malinovsky
DEVCOM Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland
20783, USA  Jaewook Ahn  Department of Physics, Korea Advanced Institute of
Science and Technology (KAIST), Daejeon 34141, Republic of Korea  Seokmin Shin
School of Chemistry, Seoul National University, 08826 Seoul, Republic of Korea
Bo Y. Chang  corresponding author<EMAIL_ADDRESS>School of Chemistry,
Seoul National University, 08826 Seoul, Republic of Korea  Research Institute
of Basic Sciences, Seoul National University, 08826 Seoul, Republic of Korea

###### Abstract

By controlling the temporal and spatial features of light, we propose a novel
protocol to prepare two-qubit entangling gates on atoms trapped at close
distance, which could potentially speed up the operation of the gate from the
sub-micro to the nanosecond scale. The protocol is robust to variations in the
pulse areas and the position of the atoms, by virtue of the coherent
properties of a dark state, which is used to drive the population through
Rydberg states. From the time-domain perspective, the protocol generalizes the
one proposed by Jaksch and coworkers [Jaksch et al., Phys. Rev. Lett., 2000,
85, 2208], with three pulses that operate symmetrically in time, but with
different pulse areas. From the spatial-domain perspective, it uses structured
light. We analyze the map of the gate fidelity, which forms rotated and
distorted lattices in the solution space. Finally, we study the effect of an
additional qubit to the gate performance and propose generalizations that
operate with multi-pulse sequences.

##  I Introduction

Due to their excellent optical addressability, [1, 2, 3, 4, 5] rich many-body
physics,[6, 7, 8, 9, 10, 11, 12] and long decoherence times,[13] neutral atoms
excited in Rydberg states have been used as vectors for different quantum
technologies, including entanglement preparation, [14, 15, 16, 17, 18, 19, 20,
21] creation of photonic entanglement, [22] quantum simulators, [23, 24, 25,
26] and quantum computation.[27, 13, 28, 19, 29, 30, 31, 32, 33, 34]

A key step involves trapping the atoms at low temperature in magneto-optical
traps (MOT). Homogeneous magnetic fields working on the MOT split the
degeneracy of the hyperfine ground states of the atoms, allowing to encode and
address separately the qubit states by optical fields through intermediate
states or using microwave fields.[35, 36, 37] Currently, it is possible to
create splittings $\Delta$ of the order of $10$ GHz[14] that in principle,
allow to drive the population from the $|0\rangle$ state independently of the
$|1\rangle$ state in the nanosecond regime.

On the other hand, due to the strong dipole-dipole interaction of Rydberg
states, [38, 39] the energy of a double excitation of Rydberg states
$|rr\rangle$ is larger than twice the energy of each atom in a Rydberg state.
For weak enough pulses (or close enough atoms) this extra energy is also
larger than the Rabi frequency driving the transition, and the $|rr\rangle$
state cannot be populated. This defines a maximum distance, called the Rydberg
blockade radius, $R_{\cal B}$, within which the well-known C-PHASE gate
protocol was proposed by Jaksch et al. [27]. The three-pulse sequence protocol
consists of a $\pi$-pulse acting on the first qubit, followed by a
$2\pi$-pulse acting on the second qubit, and a $\pi$-pulse again acting on the
first qubit. The pulse frequencies are tuned to excite the chosen Rydberg
state, $|r\rangle$ from the qubit state $|0\rangle$ (alternatively, from the
$|1\rangle$ state) so the other qubit state is decoupled. Then if the system
is initially in the $|00\rangle$ state, the first pulse moves the amplitude to
$i|r0\rangle$, the second does nothing and the third moves the amplitude to
$-|00\rangle$. If the system starts in $|01\rangle$, the first pulse acts as
before, driving the amplitude to $i|r1\rangle$, the second does nothing and
the third induces the transition to $-|01\rangle$. When the system is in
$|10\rangle$ the first pulse does nothing, the second drives the amplitude to
$-|10\rangle$ and the third does nothing. Finally, the lasers cannot induce
any transition from the $|11\rangle$ state. In the following, we will refer to
this set of operations as the Jaksch protocol (JP).

There have been several proposals to extend the JP mechanism using more robust
adiabatic excitation schemes[40, 41, 42] adding alternative processes to the
dipole blockade through dark states,[43] or addressing multi-qubit gate
generalizations.[33, 34, 44, 45] Some of these ideas extend well known optical
control adiabatic strategies [46, 47, 48, 49] for dynamics with target states
conditioned on the initial state. [50, 51, 52, 53] One disadvantage of these
schemes is the need to work with long pulses, in the microsecond regime. In
the JP scheme, this is needed to operate with independent qubits, forcing
interatomic distances of the order of $\sim 5\mu$m, with Rydberg-Rydberg
interactions, $d_{\cal B}$, of a few MHz. The time-scale gap in $\Delta^{-1}$
and $d^{-1}_{\cal B}$ offers an opportunity to speed the gates typically by
two orders of magnitude, by using denser arrays of atoms, therefore boosting
the dipole blockade such that $d_{\cal B}\sim\Delta$. Although the physics of
Rydberg states is rich and several unwanted physical processes may be involved
in working in denser media, [54, 17, 55, 56, 57, 58, 59, 60] it is important
to estimate if the JP can operate under these conditions or if one can design
other robust protocols for the gates with non-independent qubits.

One way to extend the JP to more compact arrangements of atoms is to implement
control not only in time (encoded in the pulse sequence), but also in space
(encoded as geometrical parameters) taking into account the strength of the
light-matter interaction at the exact qubit locations. In the simplest
arrangements, the spatial control might just involve specific focusing of each
laser light at different points of the lattice formed by the atoms, not just
necessarily at the site that the atom occupies, using various TEM modes of
light.[61]

Here we examine the potential advantages of using so-called structured light
for the entangling gate implementation. Recent advances in the control of
spatial properties of light have been used to create topological
electromagnetic effects, as well as almost arbitrary complex geometrical
patterns that can be complemented with pulse shaping techniques.[62, 63, 5]
For instance, a superposition of TEM00 and TEM01 or TEM10 can be used to
create nodes and phase relations between the peak values of the fields at
nearby locations. In addition to the space-dependent intensities, such hybrid
modes (see the sketch in Fig.1) may have different phases at different
positions of the atoms. The idea is to use a symmetrical protocol where the
first pulse and its copy, focusing on the first qubit, are applied before and
after a second pulse acts mostly on the second qubit. We prove that a
particular relation in the spatial features of the pulses confers special
robustness to the gate implementation, giving raise to the Symmetric
Orthogonal Protocol or SOP that is proposed in this work. By incorporating the
basic features of spatio-temporally controlled pulses as parameters in the
Hamiltonian, we design simple models to estimate the fidelities for two- and
three-qubit systems, implementing the C-PHASE type gate. The complexity of the
system increases with the number of parameters, but what can be seen as a
drawback, may be seen as an opportunity to find novel ways to control the
system using optimization techniques.

Our study is a first demonstration of quantum control application[64, 65, 66]
to design quantum gates addressing both the spatial and temporal features of
the laser fields. Using simplified models at zero temperature and without
noise, we show that the spatio-temporal control of the fields acting locally
on each qubit can be used to achieve robust and efficient implementations of
fast entangling gates. In fact, this theoretical setup offers many more
practical implementations to control gates and prepare entanglement, which
will be addressed in the future. The speed of the designed gates justifies
some our assumptions, like neglecting decoherence, dephasing, and most
mechanisms that might lead to fidelity losses. However, more detailed studies
considering the effect of other states in the system, the presence of Stark
shifts, noise in the pulses, and the motion of the atoms, are required to
assess the practical implementation of the SOP.

##  II Gate performance for non-separated qubits: Analysis

The computational basis of a two-qubit in our system is composed of
$|00\rangle$, $|01\rangle$, $|10\rangle$ and $|11\rangle$ states. Together
with the Rydberg ancillary states: $|r0\rangle$, $|0r\rangle$, $|r1\rangle$,
$|1r\rangle$, form the basis where we will follow the evolution of the system,
as we assume that the $|rr\rangle$ state cannot be accessed by strong Rydberg
blockade. We assume that the distance between atoms is shorter than the waist
of the laser beams. As a first approximation to model the local effect of the
field on each of the qubits, we define geometrical factors, $a_{k}$ and
$b_{k}$, so the spatially and temporally dependent interaction of the laser
$k$ at qubit $\alpha$ ($\alpha=a,b$) is determined by the Rabi frequencies
$\Omega_{k}(\vec{r}_{\alpha},t)=\alpha_{k}\mu_{0r}E_{k}(t)/\hbar$. The
geometrical factors can be partially incorporated in the Franck-Condon factor
$\mu_{0r}$ so we can assume, without loss of generality, that $a_{k}$ and
$b_{k}$ are normalized to unity ($\sqrt{a_{k}^{2}+b_{k}^{2}}=1$). Using hybrid
modes of light (structured light) one can control $a_{k}$ and $b_{k}$ in a
wide range of values, including negative factors.

Figure 1: Proposed implementation of the SOP for two-qubit gates. The qubits
are located at the $x$ positions. They are subject to the pulses,
$\Omega_{1}(\vec{r},t)$, $\Omega_{2}(\vec{r},t)$, $\Omega_{3}(\vec{r},t)$
whose spatial profile is shown in the horizontal “space” axis, with local
amplitudes at the qubits designated by the geometrical factors $a_{k}$ and
$b_{k}$. Different colors represent a $\pi$ phase change in the amplitude of
the field (red - positive; blue - negative). The sequence of operations that
governs the temporal evolution of the state vectors depends on the pulse
sequence, shown in the vertical “time” axis. Conditional on the initial state
of the qubit, $\\{|00\rangle,|01\rangle,|10\rangle\\}$, the energy diagram of
the subsystem where the dynamics takes place is shown under “energy”. The
$|11\rangle$, not shown, is completely decoupled.

Using a pulse sequence of non-overlapping pulses $\Omega_{k}(\vec{r},t)$, in
resonance between the $|0\rangle$ state of the qubit and the chosen Rydberg
state $|r\rangle$, the Hamiltonian is block-diagonal, ${\sf
H}_{k}^{V}\oplus{\sf H}_{k}^{a}\oplus{\sf H}_{k}^{b}\oplus{\sf H}^{d}$, where
${\sf H}_{k}^{V}=-\frac{1}{2}\Omega_{k}(t)\left(a|00\rangle\langle
r0|+b|00\rangle\langle 0r|+\mathrm{h.c.}\right)$ is the Hamiltonian of a
$3$-level system in $V$ configuration, acting in the subspace of
$\\{|00\rangle,|r0\rangle,|0r\rangle\\}$ states, ${\sf
H}_{k}^{a}=-\frac{a}{2}\Omega_{k}(t)\left(|01\rangle\langle
r1|+\mathrm{h.c.}\right)$ and ${\sf
H}_{k}^{b}=-\frac{b}{2}\Omega_{k}(t)\left(|10\rangle\langle
1r|+\mathrm{h.c.}\right)$ are two-level Hamiltonians acting in the subspace of
$\\{|01\rangle,|r1\rangle\\}$ and $\\{|10\rangle,|1r\rangle\\}$ respectively.
Finally, ${\sf H}^{d}=0$ is a zero Hamiltonian acting on the state
$|11\rangle$, decoupled from any field. The energy diagrams of the subsystems
are shown in Fig1.

The time-evolution operator of any of these Hamiltonians can be solved
analytically through their time-independent dressed states, that have zero
non-adiabatic couplings.[67, 68, 69] For reference, we include here the
analytical form of the time-evolution operators at the end of the pulses. For
${\sf H}_{k}^{V}$,

$U^{V}_{k}=\left(\begin{array}[]{ccc}\cos\theta_{k}&ia_{k}\sin\theta_{k}&ib_{k}\sin\theta_{k}\\\
ia_{k}\sin\theta_{k}&a_{k}^{2}\cos\theta_{k}+b_{k}^{2}&a_{k}b_{k}\left[\cos\theta_{k}-1\right]\\\
ib_{k}\sin\theta_{k}&a_{k}b_{k}\left[\cos\theta_{k}-1\right]&b_{k}^{2}\cos\theta_{k}+a_{k}^{2}\end{array}\right)$
(1)

where the mixing angle

$\theta_{k}=\frac{1}{2}\int_{-\infty}^{\infty}\Omega_{k}(t)dt=\frac{1}{2}A_{k}$

is half the pulse area. Interestingly, ${\sf H}_{k}^{V}$ supports a dark
state, $|\Phi^{0}_{k}\rangle=-b_{k}|r0\rangle+a_{k}|0r\rangle$. This is a
dressed state of zero energy, which is uncoupled to the ground state,
$|00\rangle$, that is, $|\Phi^{0}_{k}\rangle$ cannot decay nor be excited by
the field. On the other hand, the time-evolution operator that affects the
initial states $|01\rangle$ and $|10\rangle$, is

$U^{\alpha}_{k}=\left(\begin{array}[]{cc}\cos\vartheta_{k}&i\sin\vartheta_{k}\\\
i\sin\vartheta_{k}&\cos\vartheta_{k}\end{array}\right)$ (2)

where $\vartheta_{k}=\alpha_{k}\theta_{k}$ $(\alpha=a,b)$.

### Effect on $|00\rangle$

Let us predict the effect of the JP on the partially distinguishable qubits.
Excitation from $|00\rangle$ with a $\pi$-pulse prepares the entangled state
$|\Psi_{1}\rangle=ia_{1}|r0\rangle+ib_{1}|0r\rangle$. Because there is
population in both qubits and $\Omega_{2}(\vec{r},t)$ acts on both, the second
pulse can induce transitions from $|\Psi_{1}\rangle$, breaking the mechanism
of the JP. If we want $|\Psi_{1}\rangle$ to remain undisturbed, we need
$|\Psi_{1}\rangle$ to be the dark state of ${\sf H}_{2}^{V}$,
$|\Phi_{2}^{0}\rangle$, for which $a_{1}=-b_{2}$ and $b_{1}=a_{2}$. Then the
third pulse can be another copy of the first pulse, transforming
$|\Psi_{1}\rangle$ back into $-|00\rangle$.

To generalize the result we define structural vectors, which are normalized
two-component vectors formed by the geometrical factors,
$\vec{e}_{k}=\left(a_{k},b_{k}\right)$ that characterize the spatial
properties of the protocol. Success in the gate performance (starting in
$|00\rangle$) implies that the first diagonal element of the full propagator,
$U_{11}^{V}=\left(U^{V}_{3}U^{V}_{2}U^{V}_{1}\right)_{11}=-1$. Working out the
matrix multiplication we obtain

$\begin{array}[]{ll}U_{11}^{V}&=c_{3}c_{2}c_{1}-(\vec{e}_{2}\vec{e}_{1})c_{3}s_{2}s_{1}-(\vec{e}_{3}\vec{e}_{2})s_{3}s_{2}c_{1}\\\
&-(\vec{e}_{3}\vec{e}_{2})(\vec{e}_{2}\vec{e}_{1})s_{3}c_{2}s_{1}-\left[\vec{e}_{3}\vec{e}_{1}-(\vec{e}_{3}\vec{e}_{2})(\vec{e}_{2}\vec{e}_{1})\right]s_{3}s_{1}\end{array}$
(3)

where we have used the compact notation $c_{k}=\cos\theta_{k}$,
$s_{k}=\sin\theta_{k}$. This formula is valid in general for a $3$-pulse
sequence on arrays of $N$ qubits, where $\vec{e}_{k}$ becomes a
$N$-dimensional vector. The condition that
$|\Psi_{1}\rangle=|\Phi_{2}^{0}\rangle$ (the dark state of ${\sf H}^{V}_{2}$)
is that $\vec{e}_{2}$ is orthogonal to both $\vec{e}_{1}$ and $\vec{e}_{3}$.
In symmetric protocols, with $\vec{e}_{3}=\vec{e}_{1}$ and
$\theta_{1}=\theta_{3}$, one obtains the equation that defines the behavior
under the SOP,

$U_{11}^{V}=\cos^{2}\theta_{1}\cos\theta_{2}-\sin^{2}\theta_{1}\ .$ (4)

For $2$-qubit systems, the normalization and orthogonality of the structural
vectors implies that there is only one free parameter, $b\equiv b_{1}=-a_{2}$,
that measures the overlap of the field on both qubits and as such, indirectly
measures the proximity of the atoms in the trap.

If $\theta_{1}=\pi/2$, as in the JP, $U_{11}^{V}=-1$ regardless of
$\theta_{2}$ and for any value of $b$. In addition, Eq.(4) guarantees a
remarkable robustness to variations in the pulse areas. Making
$\delta\theta^{\prime}_{1}=\pi/2+\delta_{1}=\theta^{\prime}_{3}$, and
$\theta^{\prime}_{2}=\pi+\delta_{2}$, one can easily obtain

$U_{11}^{\prime^{V}}=\delta_{1}^{2}\left(-1+\frac{\delta_{2}}{2}\right)-\left(1-\frac{\delta_{1}^{2}}{2}\right)^{2}=-1+(2\delta_{2}^{2}-\delta_{1}^{2})\frac{\delta_{1}^{2}}{4}\
.$ (5)

Any error in the pulse areas will only add a quartic error in $U_{11}^{V}$
($\sim\\!\delta^{4}$).

### Effect on $|01\rangle$ and $|10\rangle$

To evaluate the gate performance when the two-qubit system is initiated in
$|01\rangle$ or $|10\rangle$ we use Eq.(2). In a symmetrical sequence, the
final state will be ($\alpha=a,b$)

$U_{11}^{\alpha}=\left(U^{\alpha}_{3}U^{\alpha}_{2}U^{\alpha}_{1}\right)_{11}=\cos\left(2\alpha_{1}\theta_{1}+\alpha_{2}\theta_{2}\right)\
.$ (6)

In Jaksch protocol, ($\theta_{1}=\pi/2$, $\theta_{2}=\pi$),
$U_{11}^{\alpha}=\cos\left([\alpha_{1}+\alpha_{2}]\pi\right)=-1$ for
independent qubits, as only one component, $\alpha_{1}$ or $\alpha_{2}$,
exist. But for orthogonal geometrical factors ($a_{1}=b_{2}\equiv
a,b_{1}=-a_{2}\equiv b$),

$U_{11}^{a}+U_{11}^{b}=2\cos\left(a\pi\right)\cos\left(b\pi\right)\approx-2+b^{2}\pi^{2}$
(7)

inducing quadratic deviations in $b$ that lower the fidelity of the gate.

A compromise must be made in the choice of the pulse areas. Because
$U_{11}^{V}$ does not depend on the geometrical factors (nor on $A_{2}$), the
pulse parameters should be adjusted mainly due to their effect on $U_{11}^{a}$
and $U_{11}^{b}$. which require “rotating” the pulse areas. From Eq.(6), for
any pulse area in the SOP, we obtain

$\displaystyle
U_{11}^{a}=\cos\left(aA_{1}-\frac{1}{2}bA_{2}\right)=\cos\left(\frac{1}{2}\left[aA_{\mathrm{odd}}-bA_{\mathrm{even}}\right]\right)$
(8) $\displaystyle
U_{11}^{b}=\cos\left(bA_{1}+\frac{1}{2}aA_{2}\right)=\cos\left(\frac{1}{2}\left[bA_{\mathrm{odd}}+aA_{\mathrm{even}}\right]\right)$
(9)

where we use even and odd pulse areas. We can write Eqs.(8,9) in terms of new
mixed pulse areas $A^{\prime}_{\mathrm{odd}}$, $A^{\prime}_{\mathrm{even}}$,
where

$\left(\begin{array}[]{c}A^{\prime}_{\mathrm{odd}}\\\
A^{\prime}_{\mathrm{even}}\end{array}\right)=\left(\begin{array}[]{cc}a&-b\\\
b&a\end{array}\right)\left(\begin{array}[]{c}A_{\mathrm{odd}}\\\
A_{\mathrm{even}}\end{array}\right)$ (10)

such that $A^{\prime}_{\mathrm{odd}}=2\pi(1+2m)$,
$A^{\prime}_{\mathrm{odd}}=2\pi(1+2n)$, as in the JP, but for the new mixed
pulse areas. Although $U_{11}^{V}$ will affect the overall fidelity, we
observe that the effect of using orthogonal geometrical vectors is to rotate
clockwise by an angle $\beta=\arctan(b/a)$, the pulse areas that will give
maximal but not perfect fidelities.

##  III Gate performance for non-separated qubits: Numerics

Figure 2: Dependence of the final amplitude of the starting state
($|00\rangle$, $|01\rangle$ and $|10\rangle$ on deviations from $\pi$ in the
pulse areas of odd pulses ($\delta A_{1}=\delta A_{3}=\delta A$), and of
$2\pi$ in the second pulse ($\delta A_{2}=2\delta A$), for different
geometrical factors.

As discussed in the previous section, the orthogonality of the geometrical
factors assures that population passage in the $V$ subsystem comprising the
$|00\rangle$, $|r0\rangle$ and $|0r\rangle$ states, goes through a dark state.
It is so robust that errors in the pulse areas only imply quartic deviations
in $U^{V}_{11}$ regardless of the value of $b$, that is, of the proximity of
the qubits. This is shown In Fig.2 (solid line), where $U_{11}$ are computed
for the different subsystems as a function of variations in pulses areas from
the JP, assuming $\delta\\!A=\delta\\!A_{1}=\delta\\!A_{2}/2$. On the other
hand, $U^{\alpha}_{11}$ increases quadratically around the minima (dashed
black lines in Fig.2) and become disaligned for $|01\rangle$ and $|10\rangle$
as $b$ increases. Thus, the desired value of $U_{11}^{\alpha}=-1$ may fall
outside of the flat band that characterizes the behavior of $U_{11}^{V}$
around its minima, disrupting the gate’s efficiency.

Figure 3: Gate fidelity of the (a) JP and (b) SOP for $b^{2}=0.1$, as a
function of the pulse areas (in units of $\pi$) for the solutions with minimal
pulse area.

To optimize the gate one needs to change the pulse areas in specific
directions. The smallest optimal pulse areas that maximize the fidelity are
shown in Fig.3, for $b^{2}=0.1$. To avoid a distortion in the map of solutions
(henceforth fidelity map) we represent the map as a function on the area
acting on each qubit, $A_{\mathrm{odd}}=A_{1}+A_{3}$ and
$A_{\mathrm{even}}=A_{2}$, rather than $A_{1}$ and $A_{2}$. We obtain a
maximum fidelity of $0.96$ displaced to larger areas in $A_{\mathrm{odd}}$ and
smaller areas in $A_{\mathrm{even}}$. The overall pulse area,
$A_{T}=|A_{\mathrm{odd}}|+|A_{\mathrm{even}}|=3.7\pi$, is however smaller than
in the JP ($4\pi$). On the other hand, the overall robustness of the gate,
measured as the area occupied by the maximum in the space of solutions, is
practically the same.

But this is only the minimal pulse area implementation. In the JP, other
solutions are possible modulo area $2\pi$ in the areas of the first and third
pulses, $A_{1}$ and $A_{3}$, and modulo $4\pi$ in $A_{2}$. Hence the family of
protocols with symmetric pulses ($A_{3}=A_{1}$) satisfying
$A_{\mathrm{odd}}=2\pi(1+2m)$ and $A_{\mathrm{even}}=2\pi(1+2n)$, where $m,n$
are integers, give perfect fidelity in the absence of noise or perturbations.
While in the JP all the different protocols give the same fidelity, this is
not the case for the SOP. Exploring solutions for larger pulse areas, better
fidelities ($F\geq 0.98$) are found, as shown in Fig.4.

The fidelity map for the JP is a regular lattice with spacings $\Delta
A_{\mathrm{odd}}=\Delta A_{\mathrm{even}}=4\pi$. In the SOP, the lattice is
rotated with respect to the JP, with a rotation angle of $\beta=\arctan(b/a)$,
in agreement with Eq.(10). There are some distortions as $b^{2}$ increases,
regaining a perfect, but different, symmetrical pattern as one reaches
$b^{2}=0.5$ ($\beta=\pi/4$). All maps for different $b$ have approximately the
same number of maxima, separated a minimum distance of $4\pi$, so the density
of solutions is conserved. However, except for the $b=0$ case, not all the
fidelities at the maxima reach unity. Typically, larger areas are needed to
find better solutions, e.g. $A_{\mathrm{odd}}=-6.1\pi$,
$A_{\mathrm{even}}=0.9\pi$ for a total area of $A_{T}=7\pi$ and a peak
fidelity of $F=0.99$ for $b^{2}=0.2$. On the other hand, the minimal pulse
area at which the first maxima appears decreases with $b$, and for $b^{2}=0.5$
one observes $F=0.8$ using $\Omega_{2}(\vec{r},t)$ only, or
$\Omega_{1}(\vec{r},t)$ and $\Omega_{3}(\vec{r},t)$, for a total area of only
$2.42\pi$.

Figure 4: Fidelity map as a function of the pulse areas (in units of $\pi$)
for different geometrical factors: (a) $b^{2}=0$, (b) $b^{2}=0.1$, (c)
$b^{2}=0.2$, (d) $b^{2}=0.5$. Figure 5: Dependence of the fidelity of the gate
on the geometrical factor $b$ for different values of the pulse areas. Using
the minimal pulse areas of the JP, we show how the fidelity decays more slowly
along the SOP protocol conditions (black line) than when the atoms become
closer without imposing the orthogonality (black dashed line).

Instead of plotting how the fidelity map varies as a function of the pulse
areas for fixed geometrical factors, we can fix the areas and vary $b$. This
is done in Fig.5. Only a few possible choices for the pulse areas are shown.
The black line shows how the fidelity falls with the JP parameters,
$A_{1}=\pi$, $A_{2}=2\pi$, as $b$ increases. In terms of $A_{\mathrm{odd}}$
and $A_{\mathrm{even}}$ we classify this protocol with a couple of numbers
$(A_{\mathrm{odd}},A_{\mathrm{even}})=(2,2)$ in units of $\pi$. Other
implementations of the JP scheme without minimal pulse areas, as $(2,6)$,
decay faster than the $(2,2)$ because of the larger accumulated pulse area,
but surprisingly recover and work perfectly at different values of $b$.
Protocols that do not belong to the JP, as $(8,6)$, where $A_{\mathrm{odd}}$
is not of the form $A_{\mathrm{odd}}=2\pi(1+2m)$, fail at $b=0$ but also
provide high fidelities at certain values of $b$.

Although natural multiples of $\pi$ for the pulse areas usually work
relatively well, the ratios between the areas do not need to be natural
numbers. The number of solutions is dense. For instance, taking into account
the rotation of the optimal pulse areas depending on $b$, we show how the
fidelity changes for the protocol $(-6.1,0.9)$, which maximizes the fidelity
at $b^{2}=0.2$ [see Fig.4(c)]. More surprisingly, one also finds solutions
that do not require the three-pulse strategy, like $(14,0)$ in which the
second pulse does not participate, but nevertheless one achieves high
fidelities at low and large $b$. These protocols are related to approximate
solutions of Diophantine equations, where the gate mechanism does not rely on
the dark state. They will be explored elsewhere.

The curves in Fig.5 show that the fidelities decay following a quadratic
behavior when the geometrical factors depart from the optimal values, as
expected. However, the decay is often slower at larger $b$. (And the effect is
even more noticeable if $F$ is plotted against $b$, instead of $b^{2}$.)
Interestingly, the SOP guarantees a slower decay in the presence of the other
qubit. Compare the black line with the black dashed line. In the black dashed
line we start in the JP conditions and show what happens when the atoms
approach (the qubits are no longer independent) but we do not use structured
light. Then $\vec{e}_{2}=(b,a)$ instead of $(-b,a)$. In solid black line we
use the SOP. As long as the displacement of the atoms preserves the
orthogonality of the structural vectors, the decay in the fidelity of the SOP
is clearly slower.

##  IV Three-qubit systems

Figure 6: Fidelity map of the ${\cal P}_{ab}^{-}$ gate as a function of the
pulse areas (in units of $\pi$) in a 3-qubit system. (a) JP scheme with
$c_{k}^{2}=0.1$ for all pulses. (b) SOP with $b^{2}=0.1$ and $c_{k}^{2}=0.1$.
(c) SOP with optimized $c_{k}$ imposing symmetric and orthogonal conditions in
qubits $a$ and $b$. (d) Fidelity performing at optimized $a_{k}$ and $b_{k}$
for $c^{2}_{k}=0.1$, imposing symmetric conditions.

The protocol proposed in this work implies the use of denser arrays of trapped
atoms. It is then important to analyze its robustness to the presence of other
qubits not involved in the two-qubit gate. In this section, we check the
efficiency of the SOP to prepare the gate ${\cal P}^{-}_{ab}$, which is the
C-PHASE gate acting on qubits $a$ and $b$ in the presence of qubit $c$. This
gate must operate exactly as with two qubits, regardless of the state of the
third qubit, hence the diagonal of the ${\cal P}^{-}_{ab}$ matrix has the
signature $\mathrm{diag}\\{-1,-1,-1,-1,-1,-1,1,1\\}$ when the basis is ordered
as
$\\{|000\rangle,|010\rangle,|100\rangle,|001\rangle,|101\rangle,|011\rangle,|110\rangle,|111\rangle\\}$.
The theoretical treatment follows closely the analysis in Section.2 and the
main equations (4), (6) remain valid by adding a third component, $c_{k}$,
which gives the geometrical factor on qubit $c$, to the structure vector
$\vec{e}_{k}$.

For independent qubits, as in the JP, the result is the same as for two
qubits, see Fig.4(a). However, if the third qubit is close to the other two
($c_{k}^{2}=0.1$ for all pulses), even if the second and first qubits are
sufficiently apart ($b^{2}=0$), the fidelity already decreases as shown in
Fig.6(a). The lattice of solutions looks like in the two-qubit system, but
some local maxima can be rather smaller than one, although high fidelity
solutions still exist.

On the other hand, if one starts with nearby qubits $a$ and $b$ and implements
the SOP, is the presence of qubit $c$ more disrupting? The answer is in
Fig.6(b), where we assumed that $c_{k}^{2}=0.1$ for all pulses, and we applied
the SOP to the remaining qubits forcing orthogonality and symmetric
conditions.

As in the $2$-qubit case, the fidelity map is rotated by approximately the
same angle as before, with a slight shift due to $c_{k}$. But the highest
fidelities are now clearly smaller than one ($F_{\mathrm{max}}\sim 0.85$),
making the protocol less useful.

What if the $c_{k}$ parameters are optimized? Can one control the position of
qubit $c$ (or the spatial profile of the laser located in this qubit) such
that the fidelity increases, for fixed values of the other geometrical
parameters? Fig.6(c) shows that the third qubit cannot be used to improve the
fidelity of the ${\cal P}_{ab}^{-}$ gate. Here we project at every value of
$(A_{\mathrm{odd}},A_{\mathrm{even}})$ the best fidelity obtained by
optimizing $c_{k}$, using symmetric pulses [$\Omega_{3}(t)=\Omega_{1}(t)$] and
forcing orthogonal conditions over the subspace of the first two qubits, as in
SOP. The parameters are found by using a simplex optimization with linear
constraints.[70] The optimization does not change drastically the fidelity
map, which shows the same angle of rotation, but the picture becomes blurry.
By controlling $c_{k}$ (with $c_{k}^{2}\geq 0.1$) the protocols that appear as
local maxima become more robust (the peaks are like plateaux) and the
fidelities at low maxima increase, but the highest fidelities are still $F\sim
0.85$.

Perfect fidelities can be obtained for most pulse areas in the SOP if the
geometrical factors of all qubits involved are controlled (forcing a minimal
value of $\alpha_{k}^{2}\geq 0.1$ and keeping $c_{k}=0.1$ fixed). This is
shown in Fig.6(d), where the minimum fidelity chosen is $0.7$ in the map
(close to the maximum fidelity in Fig.6(b)). The patterns of solutions display
a regular, non-rotated lattice, similar to that in Fig.6(a), but with plateaux
rather than maxima, that show high fidelities everywhere except for small
pulse areas. Even at minimal pulse area of $A_{T}=4\pi$ one can find protocols
with $F\geq 0.99$. Clearly, there are many protocols that implement efficient
and robust $2$-qubit gates in systems of $3$ not fully distinguishable qubits,
but they require finer control. In particular, in optimizing all the geometric
parameters we enforce the symmetry ($\vec{e}_{3}=\vec{e}_{1}$) but not the
orthogonalization, so the working protocols are not necessarily of the SOP
type.

##  V Multipulse sequences

Figure 7: Fidelity map for $M$-pulse sequence extensions of the SOP with (a)
M=2, (b) M = 3, (c) M = 4, (d) M = 5. We fixed $b^{2}=0.1$ in all cases.

The SOP protocol follows a set of rules that can be easily extended to other
multipulse sequences, with $M$ number of pulses. We here propose the Extended
Symmetric Orthogonal Protocol or ESOP, with the following features:

  1. 1.

In the $M$-pulse sequence all odd pulses are equal to each other (copies of
the same pulse) as well as all even pulses: $A_{k+2}=A_{k}$,
$\vec{e}_{k+2}=\vec{e}_{k}$.

  2. 2.

Odd and even pulses are orthogonal to each other:
$\vec{e}_{k+1}\cdot\vec{e}_{k}=0$.

Under these conditions it can be shown that the number of surviving terms in
$U_{11}^{V}$ [see Eq.(1)] is minimal. In particular, for 2-pulse sequences,
there is only one term: $U_{11}^{V}=\cos\theta_{2}\cos\theta_{1}$. For $3$ and
more pulses, there are always terms involving products of the sines of the
areas of even or odd pulses:

$\displaystyle\textnormal{For M=3:
}U_{11}^{V}=\cos^{2}\theta_{\mathrm{odd}}\cos\theta_{\mathrm{even}}-\sin^{2}\theta_{\mathrm{odd}}$
(11) $\displaystyle\textnormal{For M=4:
}U_{11}^{V}=\cos^{2}\theta_{\mathrm{odd}}\cos^{2}\theta_{\mathrm{even}}-\sin^{2}\theta_{\mathrm{odd}}-\sin^{2}\theta_{\mathrm{even}}$
(12) $\displaystyle\textnormal{For M=5:
}U_{11}^{V}=\cos^{3}\theta_{\mathrm{odd}}\cos^{2}\theta_{\mathrm{even}}-3\sin^{2}\theta_{\mathrm{odd}}-\sin^{2}\theta_{\mathrm{even}}$
(13)

where $\theta=A/2$. On the other hand, for the $|10\rangle$ and $|01\rangle$
states, the unitary evolution terms can always be written as

$U_{11}^{\alpha}=\cos\left(\sum_{k}^{M}\alpha_{k}\theta_{k}\right)\hskip
14.22636pt(\alpha=a,b).$ (14)

For $M=2$ there are no terms depending on $\sin\theta_{\mathrm{odd}}$,
constraining too much the possible solutions. In the JP and, to a lesser
extent in the SOP with three pulses, this was the main term that forced the
pulse areas of the odd pulses to be odd multiples of $\pi$, leading to optimal
fidelities. The operating mechanism for the gate performance with two pulses
is different. Still, it is possible to achieve high fidelities in a smaller
set of protocols.

Fig.7(b) is the same result shown in Fig.4(b), repeated here to facilitate the
comparison. Surprisingly, for $M=4$ the map is similar to that with $M=2$ in
spite of $U_{11}^{V}$ having more terms. The symmetry of the cosine and sine
terms on $\theta_{\mathrm{odd}}$ and $\theta_{\mathrm{even}}$ effectively
constraints the possible solutions so the fidelity map is similar to that with
$M=2$. The same will happen in all sequences with even number of pulses.
Solutions with odd number of pulses always provide richer fidelity maps, with
higher fidelities available at more protocols. Although solutions can be
easily generalized to any number of pulses, the highly-constraining nature of
the ESOP schemes makes these protocols probably unnecessary, as they do not
improve the results of the SOP. This might not be the case, however, when the
effects of noise are taken into account. Intensity fluctuations that are
proportional to the peak intensity of the lasers will affect more strongly
those protocols that use stronger fields, that is, with larger pulse areas.
The effect will be dominated by the pulse with larger area in the sequence,
rather than by the sum of all pulse areas. It is possible to find optimal
protocols with $5$-pulses that distribute the pulse area among all the fields,
so that the peak intensities in each field are smaller than in the similar
protocol with $3$-pulses. Obviously, if the parameters are unconstrained and
full optimization is performed to find the solutions, then having more pulses
and more parameters will clearly provide more protocols, as will be shown in
subsequent studies.

##  VI Summary and Discussion

We have studied the performance of a two-qubit gate in a denser array of
trapped atoms under perfect conditions (zero temperature, no external noise)
in the regime of strong Rydberg blockade. Extending the well-known Jaksch
protocol (JP), we have proposed a novel implementation, called the Symmetric
Orthogonal Protocol (SOP), by controlling both the pulse sequence and the
spatial properties of the fields, using structured light. The Hamiltonian of
the light-atomic system was modeled using geometrical factors, that measure
the amplitudes of the fields at the location of the qubits, allowing to obtain
analytical formulas for the propagators.

From the time-domain perspective, the scheme generalizes the JP with three
pulses that operate symmetrically in time, but with different pulse areas.
From the spatial-domain perspective, the scheme uses hybrid modes of light.
The geometrical factors form orthogonal vectors in the SOP, which allow to
decouple the effect of odd and even pulses in the sequence, by using a
coherent dark state that drives the population through the Rydberg states. The
SOP protocol is as robust as the JP to variations in the pulse areas and more
robust to changes in the position of the atoms along certain directions.
Implementations with maximal fidelity form a lattice of solutions in the space
of the pulse areas, which is rotated with respect to the lattice of solutions
in the JP.

We have analyzed the effect of adding a third atom in the proximity of the
two-qubit system. The fidelity in the SOP decays more rapidly than in the JP.
High-fidelity solutions could not be found by just controlling the position of
the third atom, but rather the geometrical factors at the first two-qubits
must be optimized for every choice of pulse areas. Finally, we have proposed
natural generalizations of the SOP to multipulse sequences, showing that
sequences with odd number of pulses form richer lattices with a denser number
of solutions than sequences with even number of pulses.

The SOP shows great promise for possible implementations of fast two-qubit
gates. Working in the strong dipole blockade regime ($d_{\cal B}>\Delta$) one
can in principle accelerate the gate a factor of $200$. However, high
fidelities are achieved in the SOP using larger pulse areas than in the JP,
typically by a factor of $2$ to $10$. Hence, gates with comparable or slightly
worse fidelity and equal robustness could be in principle prepared at
durations of the order of $20$ to $100$ times shorter, moving the scale from
the microsecond to the nanosecond regime.

From the physical point of view, the SOP operates as the JP, so one can expect
a similar sensitivity to decoherence and noise. However, because the atoms are
much closer, the dipole blockade is much larger and the gate time is much
shorter, the effect of the thermal motion of the atoms, Rydberg-Rydberg
couplings or spontaneous emission, is almost negligible. Only fluctuations in
the field intensities (hence pulse areas) as well as in the position of the
atoms, leading to changes in the geometrical factors, have some impact on the
fidelities. Preliminary estimates of these effects show about $1$% reduction
of the peak fidelities working at $\sim 25\mu$K temperature. However, further
comprehensive studies are needed to properly quantify the effect of noise.
Equally important will be to assess the current limitations (time durations,
spatial resolution) in using structured light specific to the proposed design.
Hybrid modes of light may find many applications in quantum technologies,
beyond the present proposal. On the other hand, full optimization of both the
pulse areas and the geometrical factors will probably allow refinements in the
implementations, the surface of which has only been explored in the present
contribution.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This research was supported by the Quantum Computing Technology Development
Program (NRF-2020M3E4A1079793) and the National Creative Research Initiative
Grant (NRF- 2014R1A3A2030423). IRS also thanks the BK21 program (Global
Visiting Fellow) for the stay during which this project started and for
support from MINECO CTQ2015-65033-P and MINECO PID2021-122796NB-I00. SS
acknowledges support from the Center for Electron Transfer funded by the Korea
government(MSIT)(NRF-2021R1A5A1030054).

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