# Protocols for Rydberg entangling gates featuring robustness against quasi-
static errors

Charles Fromonteil  Institute for Theoretical Physics, University of
Innsbruck, 6020 Innsbruck, Austria  Institute for Quantum Optics and Quantum
Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
Dolev Bluvstein  Department of Physics, Harvard University, Cambridge,
MA02138, USA  Hannes Pichler  Institute for Theoretical Physics, University of
Innsbruck, 6020 Innsbruck, Austria  Institute for Quantum Optics and Quantum
Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria

###### Abstract

We introduce a novel family of protocols for entangling gates for neutral atom
qubits based on the Rydberg blockade mechanism. These protocols realize
controlled-phase gates through a series of global laser pulses that are on
resonance with the Rydberg excitation frequency. We analyze these protocols
with respect to their robustness against calibration errors of the Rabi
frequency or shot-to-shot laser intensity fluctuations, and show that they
display robustness in various fidelity measures. In addition, we discuss
adaptations of these protocols in order to make them robust to atomic-motion-
induced Doppler shifts as well.

##  I Introduction

Arrays of neutral atoms trapped by optical tweezers provide a powerful
platform for implementing quantum information processing protocols [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 14, 20, 21]. A
promising approach is to encode quantum information in long-lived hyperfine
states (or other long-lived electronic states), with each atom representing a
single qubit. Single qubits can be initialized with high fidelity via optical
pumping, read out via tweezer-resolved imaging techniques, and coherently
manipulated with high fidelity via optical control. Moreover, high-fidelity
multi-qubit gates enabled by state-selective coherent excitation to strongly
interacting Rydberg states have been realized [5, 6, 7, 8, 9, 10, 11, 12, 13,
14]. In combination with coherent rearrangement techniques, this results in a
versatile, scalable quantum processing architecture [14].

In this work, we introduce novel protocols for realizing multi-qubit
entangling gates in this platform, with several appealing features: similar to
recently introduced protocols [9, 22, 23], our new protocols consist of a
sequence of global laser pulses, avoiding the requirement for local addressing
with Rydberg lasers. In contrast to previous proposals, all these pulses are
resonant with the Rydberg transition and require pulse areas that can be
simply calibrated. In addition, the protocols introduced exhibit a native
robustness against calibration errors of the Rabi frequency and low-frequency
or shot-to-shot fluctuations of the laser intensity. Moreover, we also discuss
methods to suppress errors due to Doppler shifts from thermal motion of the
atoms. Although in recent experiments the primary limitations can often be
incoherent errors such as Rydberg decay and laser scattering [14], with
improvements in technology and laser power these quasi-static errors will
become more relevant, and ultimately be important to ensure robust quantum
circuit operation.

##  II Model

We consider a pair of atoms, where two non-interacting stable internal states
of each atom represent the two qubit states $|0\rangle$ and $|1\rangle$. We
are interested in entangling gates between these qubits that are mediated via
state-selective coherent excitation from the state $|1\rangle$ to an
interacting Rydberg state $|r\rangle$. The Hamiltonian governing the dynamics
of a pair of atoms driven by a laser that induces such a laser an excitation
process is given by [9]

$\displaystyle H=$
$\displaystyle\sum_{i=1,2}\frac{\Omega}{2}\left(e^{i\varphi}|1\rangle_{i}\langle
r|_{i}+\textrm{h.c.}\right)+V|r,r\rangle\langle r,r|.$ (1)

Here, $\Omega$ and $\varphi$ are the (real) Rabi frequency and the phase of
the laser, respectively. We stress that we consider the situation of a
homogeneous, i.e., global laser field, thus eliminating the requirement for
local control. Moreover, the laser frequency is resonant with the transition
between the state $|1\rangle$ and the Rydberg state $|r\rangle$. In the
following we are interested in the situation where the atoms are placed at
distances such that the interaction energy in the state where both atoms are
in the Rydberg state, $V$, is much larger than the Rabi frequency
($V\gg|\Omega|)$. This results in a dynamical constraint that suppresses
simultaneous excitation of both atoms to the Rydberg state [5, 6, 24]. To
simplify the discussion in the following we enforce this Rydberg blockade
constraint exactly, which is formally equivalent to considering the limit
$V\rightarrow\infty$.

Owing to the global drive and the blockade constraint, the dynamics of the
four computational basis states decomposes into simple blocks. The state
$|0,0\rangle$ does not couple to the light field and is trivially invariant.
The state $|0,1\rangle$ is resonantly coupled to the state $|0,r\rangle$, with
a coupling matrix element $\Omega e^{i\varphi}/2$, forming an effective two
level system. The dynamics of the state $|1,0\rangle$ exactly mirrors this, as
$|1,0\rangle$ resonantly couples to the state $|r,0\rangle$, with the same
coupling strength $\Omega e^{i\varphi}/2$. Finally, the state $|1,1\rangle$ is
resonantly coupled to the state
$|W\rangle=\tfrac{1}{\sqrt{2}}(|1,r\rangle+|r,1\rangle)$, forming again a
closed, effective two-level system. However, in this last case the coupling
matrix element is larger and given by $\sqrt{2}\Omega e^{i\varphi}/2$. The
entire dynamics can thus be understood by considering two inequivalent
effective two level systems, which can be conveniently represented by two
Bloch spheres where the south pole is identified with a computational basis
state and the north pole with the corresponding coupled state containing a
Rydberg excitation [9]. These two Bloch spheres are depicted in Fig. 1, where
the trajectory of the Bloch vector on the left sphere depicts the dynamics of
the states $|0,1\rangle$ and equivalently $|1,0\rangle$, whereas the right
sphere shows the dynamics of the state $|1,1\rangle$ for a given pulse
sequence specified below. The Hamiltonian (1) induces rotations on both Bloch
spheres. While the rotation axis (determined by $\varphi$) is the same for
both spheres, the rotation frequency differs by a factor $\sqrt{2}$. For
future reference we define the coordinate system such that $\varphi=0$
corresponds to a rotation around the $x$ axis, while $\phi=\pm\pi/2$
corresponds to a rotation around the $\pm y$ axis (see Fig. 1). Note that the
state $|0,0\rangle$ does not evolve at all under $\eqref{eq:Hamiltonian}$.

Figure 1: Bloch sphere trajectories followed by the effective two-level system
$|0,1\rangle\leftrightarrow|0,r\rangle$ and equivalently
$|1,0\rangle\leftrightarrow|r,0\rangle$ (left) and of the effective two-level
system $|1,1\rangle\leftrightarrow|W\rangle$ (right) during the sequence (2).
The states $|0,1\rangle$ and $|1,0\rangle$ are respectively coupled to
$|0,r\rangle$ and $|r,0\rangle$ with a coupling strength $\Omega$, while
$|1,1\rangle$ is coupled to
$|W\rangle=\frac{|1,r\rangle+|r,1\rangle}{\sqrt{2}}$ with a coupling strength
$\sqrt{2}\Omega$. Both trajectories are closed, with an enclosed area of
$2\pi$, such that the three computational basis states $|1,0\rangle$,
$|0,1\rangle$ and $|1,1\rangle$ pick up a minus sign. Since the state
$|0,0\rangle$ does not evolve, this realizes a CZ gate.

##  III Controlled phase gate Protocol

We proceed by introducing two closely related pulse sequences that result in a
controlled-$Z$ gate, denoted by CZ between the two qubits. Its action on the
qubit states is given by
$\textrm{CZ}|z_{1},z_{2}\rangle=(-1)^{z_{1}z_{2}-z_{1}-z_{2}}|z_{1},z_{2}\rangle$
with ($z_{i}\in\\{0,1\\}$). We note that this definition of the CZ gate
differs from the standard one by a trivial rotation of both qubits. To realize
this gate, we use a sequence of resonant laser pulses (with fixed Rabi
frequency $\Omega$), each specified by the pulse duration $t$ and the phase of
the laser $\varphi$, or equivalently, the rotation axis. We denote the
unitaries generated by Hamiltonian (1) from a pulse of duration
$t=\alpha/\Omega$ with laser phase $\varphi=0$ by $U_{x}(\alpha)$. Analogously
we denote the operations corresponding to pulses with $\varphi=\pi$,
$\varphi=\pi/2$ and $\varphi=-\pi/2$, by $U_{-x}(\alpha)$, $U_{y}(\alpha)$ and
$U_{-y}(\alpha)$ respectively. We note that we consider square pulses here for
simplicity, however, due to the resonant nature of the pulses only the pulse
area matters for all aspects discussed in this section (experimentally,
intensity-dependent light shifts of the transition need to be appropriately
calibrated. Both our primary protocols are constructed by a sequence of 5 such
pulses, each of them with a pulse area $\Omega
t\in\\{\pi,\frac{\pi}{2},\tfrac{\pi}{\sqrt{2}},\tfrac{\pi}{2\sqrt{2}}\\}$: the
pulse sequences are given by

$\displaystyle\textrm{CZ}=U_{x}\left(\tfrac{\pi}{2\sqrt{2}}\right)U_{y}(\pi)U_{x}\left(\tfrac{\pi}{\sqrt{2}}\right)U_{y}(\pi)U_{x}\left(\tfrac{\pi}{2\sqrt{2}}\right),$
(2)

and

$\displaystyle\textrm{CZ}=U_{x}\left(\tfrac{\pi}{2}\right)U_{y}\left(\tfrac{\pi}{\sqrt{2}}\right)U_{x}(\pi)U_{y}\left(\tfrac{\pi}{\sqrt{2}}\right)U_{x}\left(\tfrac{\pi}{2}\right).$
(3)

Note that these pulse sequences are reminiscent of the refocusing (spin echo)
techniques used in NMR [25, 26]. While it is a matter of simple algebra to
confirm that they both realize the target controlled-$Z$ gate, we find it
instructive to consider a geometric argument obtained by analyzing the path
traced out by the computational basis states on the Bloch spheres introduced
above. We consider here the case of the sequence (2), but analogous arguments
apply to (3). For this analysis it is convenient to formally split the third
pulse in the sequence into two identical pulses, each with half the pulse
area, i.e.,
$U_{x}\left(\pi/\sqrt{2}\right)=U_{x}\left(\tfrac{\pi}{2\sqrt{2}}\right)U_{x}\left(\tfrac{\pi}{2\sqrt{2}}\right)$.
With this, the pulse sequence (2) can be interpreted as a sequence consisting
of 6 pulses. Note that these 6 pulses are obtained by applying the same
3-pulse sequence twice: we can write the evolution operator generated by the 6
pulses as a repeated application of the evolution operator describing the
first three pulses, $S$, i.e., ${\rm{CZ}}=S^{2}$ with
$S=U_{x}\left(\tfrac{\pi}{2\sqrt{2}}\right)U_{y}(\pi)U_{x}\left(\tfrac{\pi}{2/\sqrt{2}}\right)$.
The 3-pulse sequence described by $S$ has a simple effect on the four
computational basis states of the two atoms: While the state $|0,0\rangle$ is
trivially invariant, $S|0,0\rangle=|0,0\rangle$, all other computational basis
states are simultaneously mapped from the south pole of the corresponding
Bloch sphere to the north pole, i.e., $S|0,1\rangle=i|0,r\rangle$,
$S|1,0\rangle=i|r,0\rangle$ and $S|1,1\rangle=i|W\rangle$ (see Fig. 1). This
can be seen by analyzing the trajectories of each computational basis state in
Fig. 1: On the left, we show the trajectories of an atom pair initially on the
state $|0,1\rangle$, which is represented by the south pole of the left
sphere. The first pulse in the sequence $S$ rotates this Bloch vector around
the $x$-axis to a point in the $yz$-plane. The second pulse implements a
$\pi$-rotation around the $y$-axis, flipping the $z$-component of the Bloch
vector. Finally, the third pulse in $S$ performs the same rotation as the
first pulse, thus aligning the Bloch vector with the north pole of the Bloch
sphere. In Fig. 1 on the right, we show the trajectories of an atom pair
initially in the state $|1,1\rangle$, which is represented by the south pole
of the right sphere. The first pulse in the sequence $S$ realizes a $\pi/2$
rotation around the $x$-axis on the Bloch sphere, aligning the Bloch vector
with the $y$-axis. The second pulse corresponds to a rotation around the
$y$-axis and thus leaves this state invariant. Finally, the third pulse in $S$
performs again a $\pi/2$ rotation around the $x$-axis, and thus rotates the
Bloch vector to the north pole of the right Bloch sphere. By the same
arguments, the second application of $S$ results in a return of the Bloch
vector from the north to the south pole in all cases, i.e.,
$S|0,r\rangle=i|0,1\rangle$, $S|r,0\rangle=i|1,0\rangle$ and
$S|W\rangle=i|1,1\rangle$. In summary, at the end of the protocol (2) all
computational basis states are mapped to themselves. In this process each of
the computational basis states picks up a phase, which determines the gate
performed in the qubit subspace. These phases can be directly inferred from
the Bloch sphere picture, since they are given by half of the solid angle
enclosed by the trajectories. For both trajectories in Fig. 1, this solid
angle is simply half of the area of the unit sphere, corresponding to a phase
of $\pi$. Thus, all computational basis states except $|0,0\rangle$ acquire a
phase of $\pi$ in the above process. This corresponds to the gate
$|z_{1},z_{2}\rangle\rightarrow(-1)^{z_{1}z_{2}-z_{1}-z_{2}}|z_{1},z_{2}\rangle$,
which is indeed the desired controlled-phase gate, CZ.

Figure 2: Bloch sphere trajectories for the controlled-($\tfrac{\pi}{2}$) gate
sequence. The phase jump between the third and fourth pulses ensures that the
area enclosed in the trajectories now represents one quarter of the sphere, so
at the end of the sequence, the states $|0,1\rangle$, $|1,0\rangle$ and
$|1,1\rangle$ have acquired a phase of $-\frac{\pi}{2}$. Again, $|0,0\rangle$
does not evolve, resulting in the desired gate.

This 6-pulse sequence can easily be modified to realize controlled-phase gates
with an arbitrary phase $\phi$, defined here as,

$\displaystyle\textrm{C}_{\phi}|z_{1},z_{2}\rangle=(e^{i\phi})^{z_{1}z_{2}-z_{1}-z_{2}}|z_{1},z_{2}\rangle.$
(4)

For this, one simply introduces a change of the laser phase by an amount $\xi$
between the first 3 and the last 3 pulses, i.e. between the first and second
application of $S$. Formally this corresponds to shifting the value
$\varphi\rightarrow\varphi+\xi$ in the last three pulses. In the Bloch sphere
picture this modifies the solid angle enclosed by each trajectory, and thus
the phase acquired by the states $|0,1\rangle$, $|1,0\rangle$ and
$|1,1\rangle$ (see Fig. 2). This allows us to build any controlled-phase gate
(4), with the phase of the gate being determined by the phase jump via
$\phi=\pi-\xi$ (see Appendix A for details). A particularly useful case is the
$\textrm{C}_{\pi/2}$ gate, where the corresponding pulse sequence is

$\displaystyle\textrm{C}_{\pi/2}=U_{y}\left(\alpha_{1}\right)U_{-x}\left(\alpha_{2}\right)U_{y}\left(\alpha_{1}\right)U_{x}\left(\alpha_{1}\right)U_{y}\left(\alpha_{2}\right)U_{x}\left(\alpha_{1}\right).$
(5)

Analogous to the CZ case, two variants of the sequence exist, corresponding to
the choices $\alpha_{1}=\tfrac{\pi}{2\sqrt{2}}$ and $\alpha_{2}=\pi$, or
$\alpha_{1}=\tfrac{\pi}{2}$ and $\alpha_{2}=\tfrac{\pi}{\sqrt{2}}$,
respectively. The Bloch sphere trajectories for the former choice are shown in
Fig. 2.

Below we consider properties of the above protocols in the presence of noise
in some control parameters. Since all the protocols defined in this section
have the same properties in this regard, we refer to them indistinctly as
Protocol I in the following.

##  IV Robustness Analysis and robust gates

CZ gate protocol | Global drive | $\mathcal{F}$ | $\mathcal{P}$ | $\mathcal{C}$ | Execution time
---|---|---|---|---|---
Jaksch et al. [5] | No | $1-4.935\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-4.935\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-4.870\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $12.57/\Omega$
Levine et al. [9] | Yes | $1-2.963\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-2.547\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-0.416\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $8.59/\Omega$
Protocol I | Yes | $1-1.878\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-1.878\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-0.329\epsilon^{4}+O\left(\epsilon^{5}\right)$ | $10.73/\Omega$
Protocol II | Yes | $1-0.329\epsilon^{4}+O\left(\epsilon^{5}\right)$ | $1-1.944\epsilon^{6}+O\left(\epsilon^{7}\right)$ | $1-0.329\epsilon^{4}+O\left(\epsilon^{5}\right)$ | $21.45/\Omega$
Table 1: Leading-order expansions for $\mathcal{F}$, $\mathcal{P}$ and
$\mathcal{C}$ as a function of the laser intensity error parameter
$\epsilon=\frac{\delta\Omega}{\Omega}$, for the protocols of [5] and [9] and
those presented here. We highlighted a vanishing susceptibility in green.
Analytical expressions for all susceptibilities can be found in Appendix H.

We note that Protocol I results in a total gate time
$T_{\textrm{I}}=\frac{\pi}{\Omega}(2+\sqrt{2})$. While this is shorter than
the protocol introduced in the seminal work by Jaksch et al. [5], it is
slightly longer than the protocol recently implemented in Ref. [9]. However,
it has several advantages. All pulses are resonant, and they require only
simple pulse times and phase jumps, potentially simplifying calibration in
experiments. More importantly, all variants possess a native robustness
against shot-to-shot fluctuations of the laser intensity, as we detail below.

To quantify such robustness, we consider several gate fidelity measures that
are relevant in different scenarios below. For a given gate protocol, we
define a continuous family of unitary operators $U(\epsilon)$, parametrized by
$\epsilon$, which measures the deviation of a control parameter (such as the
Rabi frequency) from its target value. In particular, $U(0)$ is the target
unitary operation realizing a controlled-phase gate in the qubit subspace. The
fidelity of $U(\epsilon)$ with the target gate is given by [27]

$\displaystyle\mathcal{F}=\frac{\mathrm{tr}\left(PU(\epsilon)PU^{\dagger}(\epsilon)\right)+|\mathrm{tr}\left(U(\epsilon)PU^{\dagger}({0})P\right)|^{2}}{d(d+1)},$
(6)

where $d=4$ is the dimension of the qubit subspace, and
$P=\sum_{z_{1},z_{2}\in\\{0,1\\}}|z_{1},z_{2}\rangle\langle z_{1},z_{2}|$ is
the projector onto it. The first derivative of the fidelity vanishes at
$\epsilon=0$ since $\mathcal{F}$ assumes its maximum there. Thus the fidelity
susceptibility

$\displaystyle\chi=\frac{d^{2}}{d\epsilon^{2}}\mathcal{F}\big{|}_{\epsilon=0}$
(7)

is a natural measure of the sensitivity of the protocol against variations of
the corresponding control parameter [28]. Accordingly, we call a protocol
fully robust if $\chi=0$. Another important quantity when analyzing imperfect
implementations of gate protocols is the average probability for the system to
return to the qubit subspace at the end of the protocol,

$\displaystyle\mathcal{P}=\frac{\mathrm{tr}\left(PU(\epsilon)PU^{\dagger}(\epsilon)\right)}{d}.$
(8)

Clearly, at $\epsilon=0$ the leakage out of the qubit manifold vanishes and
$\mathcal{P}=1$. Accordingly, we define a susceptibility

$\displaystyle\chi_{\mathcal{P}}=\frac{d^{2}}{d\epsilon^{2}}\mathcal{P}\big{|}_{\epsilon=0}.$
(9)

We call a protocol leakage-robust if $\chi_{\mathcal{P}}=0$. Finally, we
consider the conditional gate fidelity $\mathcal{C}$, defined as the fidelity
of the gate, conditioned on the observation of no leakage out of the qubit
subspace. It is defined as

$\displaystyle\mathcal{C}=\frac{1}{d+1}\left(1+\frac{|\mathrm{tr}\left(U(\epsilon)PU^{\dagger}({0})P\right)|^{2}}{\mathrm{tr}\left(PU(\epsilon)PU^{\dagger}(\epsilon)\right)}\right)=\mathcal{F}/\mathcal{P}.$
(10)

Leakage errors could be in principle detected and potentially converted to
erasure errors, which can be corrected with a remarkably high threshold [29].
The conditional fidelity $\mathcal{C}$ is thus particularly important, as it
quantifies the effect of the remaining errors that can not be converted to
erasure errors. Again, we define a corresponding susceptibility via

$\displaystyle\chi_{\mathcal{C}}=\frac{d^{2}}{d\epsilon^{2}}\mathcal{C}\big{|}_{\epsilon=0}.$
(11)

We call a gate conditionally robust if $\chi_{\mathcal{C}}=0$. We have the
relation $\chi=\chi_{\mathcal{P}}+\chi_{\mathcal{C}}$.

###  IV.1 Laser intensity errors

####  IV.1.1 Conditional robustness

In this section we are interested in the effect of variations of the laser
intensity on gate fidelity. Experimentally these may originate from
calibration errors or from drifts of the laser system that are slow on the
timescale of the gate execution, i.e. $T_{\textrm{I}}$. To analyze these
situations we set $\epsilon\equiv\delta\Omega/\Omega$, where $\delta\Omega$
quantifies the deviation of the Rabi frequency from its target value. The
interesting feature of Protocol I is that it is natively conditionally robust
against such errors. It is straightforward to compute the leading order
expansion of the conditional fidelity $\mathcal{C}$ for this type of error
(see Appendix C), which reads

$\mathcal{C}=1-\frac{\pi^{4}}{640}\left(13+4{\rm
cos}\left(\tfrac{\pi}{\sqrt{2}}\right)+8{\rm
cos}\left(\tfrac{2\pi}{\sqrt{2}}\right)-4{\rm
cos}\left(\tfrac{3\pi}{\sqrt{2}}\right)\right.\\\ \left.+3{\rm
cos}\left(\tfrac{4\pi}{\sqrt{2}}\right)\right)\epsilon^{4}+O\left(\epsilon^{5}\right).$
(12)

Indeed the second-order term vanishes identically, that is,
$\chi_{\mathcal{C}}=0$ and the protocol is conditionally robust.

In Table 1 we give the leading-order expansions for $\mathcal{F}$,
$\mathcal{P}$ and $\mathcal{C}$ for Protocol I, as well as for the CZ gate
protocols given in Refs. [5] and [9]. Protocol I as well as the protocol of
Ref. [5] are both conditionally robust, while the protocol of Ref. [9] is not.
Note that the leading order contribution to the conditional fidelity is an
order of magnitude smaller for Protocol I, compared to the protocol of Ref.
[5]. Moreover, we find that none of the three protocols are leakage-robust
against laser intensity errors.

####  IV.1.2 Fully robust protocol

To achieve full robustness ($\chi=0$), we now consider a variation of Protocol
I. For this, the CZ gate is realized by applying two
controlled-$(\frac{\pi}{2})$ gates in sequence, each of them realized
according to Protocol I, with the pulse sequence (5). Applying this sequence
twice obviously realizes a CZ gate, with an execution time
$T_{\textrm{II}}=2T_{\textrm{I}}$. The resulting protocol, which we call
Protocol II from now on, inherits the conditional robustness of the individual
$\textrm{C}_{\pi/2}$ sequences. Moreover, it is also leakage-robust against
laser intensity errors. This is a result of the destructive interference
between the two leakage amplitudes originating form imperfect implementations
of each $\textrm{C}_{\pi/2}$ sequence. In fact it is straightforward to
calculate the series expansion for $\mathcal{P}$,

$\mathcal{P}=1-\frac{\pi^{6}}{1024}\left(30+30{\rm
cos}\left(\tfrac{\pi}{\sqrt{2}}\right)+27{\rm
cos}\left(\tfrac{2\pi}{\sqrt{2}}\right)\right.\\\ \left.+2{\rm
cos}\left(\tfrac{3\pi}{\sqrt{2}}\right)+6{\rm
cos}\left(\tfrac{4\pi}{\sqrt{2}}\right)+{\rm
cos}\left(\tfrac{6\pi}{\sqrt{2}}\right)\right)\epsilon^{6}+O\left(\epsilon^{7}\right),$
(13)

which confirms leakage robustness, $\chi_{\mathcal{P}}=0$, but in addition
shows that the contribution in next order also vanishes. Since
$\chi_{\mathcal{P}}=\chi_{\mathcal{C}}=0$, this Protocol II is fully robust to
laser intensity errors. The leading-order expansions for ${\mathcal{F}}$,
${\mathcal{P}}$ and ${\mathcal{C}}$ in that case are presented in the fourth
row of Table 1. We note that any controlled-phase gate $\textrm{C}_{\phi}$ (4)
can similarly be realized in a fully robust way, by successively applying two
$\textrm{C}_{\phi/2}$ sequences of the Protocol I type, separated by a well-
chosen phase jump to ensure destructive interference of the leakage amplitudes
(see Appendix D).

###  IV.2 Motional Doppler shift

Another source of gate errors in experiments with Rydberg atom arrays is
thermal motion of the atoms in the traps [30]. Specifically, the corresponding
Doppler shift leads to a shot-to-shot fluctuation of the effective laser
detuning. Since the velocities of the two atoms are uncorrelated, the
corresponding Doppler shifts for the first atom, $\delta\Delta_{1}$, and for
the second atom, $\delta\Delta_{2}$, break the permutation symmetry of the
model by adding a term $-\sum_{i=1,2}\delta\Delta_{i}|r\rangle_{i}\langle
r|_{i}$ to the Hamiltonian (1). To analyze the robustness properties of the
above gate protocols we find it convenient to work with the two independent
uncorrelated parameters: $\frac{\delta\Delta_{1}+\delta\Delta_{2}}{2\Omega}$,
which we call symmetric detuning error, and
$\frac{\delta\Delta_{1}-\delta\Delta_{2}}{2\Omega}$, which we call
antisymmetric detuning error. We note that robustness against errors of
multiple parameters is equivalent to robustness against errors in each
parameter individually (see Appendix E). Note that this holds also for
leakage-robustness and conditional robustness. As a consequence, for a
protocol to be robust against Doppler errors arising from the atoms’ thermal
motion, it must be both robust to symmetric and anti-symmetric detuning errors
individually. We therefore analyze these two cases separately below.

Figure 3: Left: Pulse sequences corresponding to the CZ gate protocols
introduced here. Each global pulse is represented by one box, with the
corresponding pulse area $\alpha_{1}$ or $\alpha_{2}$. The values of
($\alpha_{1}$, $\alpha_{2}$) can be chosen from two options:
$(\tfrac{\pi}{2\sqrt{2}},\pi)$, as in (2), or
$(\tfrac{\pi}{2},\tfrac{\pi}{\sqrt{2}})$, as in (3). The laser phase for each
pulse is represented by the shape and color of the corresponding box’s border.
A filled box means the detuning error’s sign is flipped for this pulse. The
protocol duration is proportional to the number of boxes, and is indicated at
the bottom of the figure. Right: Values of the leakage ($\chi_{\mathcal{P}}$)
and conditional ($\chi_{\mathcal{C}}$) susceptibilities, for the error
parameters associated to intensity or Doppler error, for all the protocols
introduced above as well as the protocols of [5] and [9]. The values given
here correspond to a CZ gate with the choice
$(\alpha_{1},\alpha_{2})=(\tfrac{\pi}{2\sqrt{2}},\pi)$; for an arbitrary phase
and/or with $(\alpha_{1},\alpha_{2})=(\tfrac{\pi}{2},\tfrac{\pi}{\sqrt{2}})$,
the specific values can be different but the robustness properties are the
same (except for one case, denoted by a star in the table, where leakage-
robustness is only achieved in the CZ case).

####  IV.2.1 Symmetric detuning errors

A symmetric detuning error corresponds to
$\delta\Delta_{1}=\delta\Delta_{2}\equiv\delta\Delta$, so we define the error
parameter $\epsilon=\frac{\delta\Delta}{\Omega}$. For symmetric detuning
errors the permutaion symmetry is not broken and the dynamics can still be
understood in terms of the effective two-level systems with corresponding
Bloch spheres introduced above. A symmetric detuning error simply modifies the
Bloch sphere trajectories: it tilts the rotation axes and changes the rotation
angles for each of the pulses, resulting in general in trajectories that do
not close and thus in a reduced gate fidelity. Nevertheless, one can show
analytically that the implementations of the CZ gate according to Protocol I
as well as realizations of arbitrary Cϕ gates according to Protocol II are
natively leakage-robust against these symmetric Doppler errors,
$\chi_{\mathcal{P}}=0$. This is in contrast to the protocols of Refs. [5] and
[9], which do not enjoy this feature. Unfortunately, neither Protocol I nor II
are conditionally robust against symmetric detuning errors.

To address this issue, we introduce an experimental method to invert the
atomic velocities and thus echo out the Doppler shift. The central idea is to
recapture the atoms in their harmonic optical tweezer potentials (which are
otherwise turned off during the Rydberg pulses). After half a trap period, the
velocity of each atom is reversed, and the tweezers turned off again. This
effectively inverts the Doppler shift in the subsequent Rydberg pulses. We
note that in this process not only the atomic velocity, but also the atomic
position changes sign (when measured with respect to the trap minimum). An
alternative to the above method to invert the Doppler shift is to invert the
direction of the Rydberg laser (experimentally requiring calibration and
stability of the relative path lengths of the two different paths for the
laser). We note that in both these methods, fluctuations in the atomic
position introduce an uncontrolled shift of the Rydberg laser phase between
the pulses before and after this Doppler inversion. To avoid additional errors
stemming from this effect, we only consider inverting the Doppler shifts at
instances during a gate protocol, where all atomic populations are in qubit
states, i.e., after the application of full gate sequences 111This works as
long as the uncontrolled phase shifts acquired in the Doppler inversion are
small, as in typical experimental setups (Appendix G). If they are large, then
the Doppler inversion should only be applied after the application of leakage
robust sequences..

This possibility to invert the detuning error can be used to create protocols
that are fully robust against this symmetric detuning error. To do so, we
construct the CZ gate out of two successive $\textrm{C}_{\pi/2}$ sequences,
both realized according to Protocol I. We flip the sign of the detuning error
between the first and the second Cπ/2 gates, in order to echo out the relative
phase error up to leading order and thus achieve conditional robustness.
Additionally, we introduce a phase jump of $\pi$ between the two sequences, in
order to ensure destructive interference of the Rydberg amplitudes, and thus
leakage-robustness. The resulting protocol is thus fully robust. It is similar
to Protocol II up to the inversion of the error’s sign and the phase jump of
$\pi$ between the two $\textrm{C}_{\pi/2}$ gates: we hence refer to it as
Protocol II.a (see Fig. 3). Once again, this protocol can easily be adapted to
arbitrary controlled-phase gates. A fully robust $\textrm{C}_{\phi}$ gate is
realized by applying two successive $\textrm{C}_{\phi/2}$ gates (each
following Protocol I): with an appropriate choice of the laser phase as well
as an inverted detuning error for the second gate, the resulting gate sequence
is fully robust against symmetric detuning errors.

The values of the leakage and conditional susceptibilities for symmetric
detuning errors for the various protocols are given in Fig. 3. The full
leading-order expansions of $\mathcal{F}$, $\mathcal{P}$ and $\mathcal{C}$ are
given in Appendix H.

####  IV.2.2 Antisymmetric detuning errors

Now we consider antisymmetric variations of the frequency, which corresponds
to opposite detuning errors for the two atoms, i.e.,
$\frac{\delta\Delta_{1}}{\Omega}=-\frac{\delta\Delta_{2}}{\Omega}\equiv\epsilon$.
We first note that conditional robustness of a protocol against symmetric
detuning errors implies that the same protocol also is conditionally robust
against antisymmetric detuning errors. This is trivially true for the state
$|0,0\rangle$, and can also be easily seen for the states $|0,1\rangle$ and
$|1,0\rangle$: if the atoms are prepared in either of these two states, the
effect of a symmetric and an anti-symmetric detuning error are the same, since
only one of the atoms couples to the laser. Thus, for these states, the
robustness of the dynamics against symmetric detuning errors directly implies
robustness against antisymmetric errors as well. The same is true also for the
remaining state $|1,1\rangle$, but the analysis requires more care. In the
presence of anti-symmetric detuning errors, the dynamics can no longer be
reduced to an effective two-level system, but instead three states must be
considered: the perturbation leads to an effective coupling of strength
$\Omega\epsilon$ between $|W\rangle$ and the singlet state
$|A\rangle=\frac{|r,1\rangle-|1,r\rangle}{2}$. Specifically, the dynamics of
this three-level system is described by the Hamiltonian

$\displaystyle
H_{3}=\Omega\left(\frac{\sqrt{2}}{2}e^{i\varphi}|1,1\rangle\langle
W|-\epsilon|W\rangle\langle A|\right)+\textrm{h.c.}$ (14)

The form of this Hamiltonian directly implies that the diagonal matrix element
of the corresponding evolution operator, $\langle 1,1|U(\epsilon)|1,1\rangle$,
has no contribution in first order in $\epsilon$, which in turn results in a
conditional robustness of the gate protocol against anti-symmetric detuning
errors.

We now discuss leakage-robustness against anti-symmetric detuning errors.
Again, for $|0,0\rangle$, $|1,0\rangle$ and $|0,1\rangle$, protocols that are
leakage-robust against symmetric errors imply the same for anti-symmetric
errors. For $|1,1\rangle$ the dominant leakage error stems form population in
the $|A\rangle$ state, since population in the $|W\rangle$ state is of higher
order in $\epsilon$. This can be seen from a penetrative analysis of the
Hamiltonian (14). To obtain a leakage-robust pulse sequence the final
population in $|A\rangle$ must thus be cancelled up to second-order. This
requirement can be expressed as $\langle
A|\partial_{\epsilon}U(0)|1,1\rangle=0$.

This can be satisfied by a new type of sequence, which we call Protocol III
(see Fig. 3). In the same spirit as Protocols II and II.a, this variant
consists in splitting the CZ gate into two $\textrm{C}_{\pi/2}$ gates.
However, each of these gates is now realized according to Protocol II, that
is, using a sequence of 12 pulses. We invert the detuning error (see the
previous subsection) of the second $\textrm{C}_{\pi/2}$ sequence, which
ensures conditional robustness of the CZ gate against symmetric and
antisymmetric Doppler errors. Furthermore, we introduce a phase difference of
$\tfrac{\pi}{2}$ between the first and second $\textrm{C}_{\pi/2}$ gates,
which results in a destructive interference of all error terms, ensuring
$\langle A|\partial_{\epsilon}U(0)|1,1\rangle=0$. Finally, since both
$\textrm{C}_{\pi/2}$ gates are realized with Protocol II, they are leakage-
robust against symmetric frequency fluctuations, so the states $|1,0\rangle$
and $|0,1\rangle$ undergo robust dynamics as well. Thus, this new protocol,
whose total duration is now $T_{\textrm{III}}=4T_{\textrm{I}}=42.90/\Omega$,
is fully robust to symmetric and anti-symmetric Doppler frequency errors. The
pulse sequence and associated susceptibilities are shown in Fig. 3.

The protocols II.a and III have the added advantage of having simultaneous
(conditional or full) robustness against intensity and Doppler errors. Indeed,
so far we have discussed robustness of gate protocols against intensity and
Doppler errors separately, but in practice it would be desirable to identify
protocols that are simultaneously robust against both. The table in Fig. 3
gives us the corresponding information: we see that Protocols II.a and III are
the most interesting in this context. Indeed, the former is conditionally
robust against variations of all three error parameters corresponding to
intensity or Doppler errors, and the latter is fully robust against them all
(at the cost of a longer execution time).

Finally, we note that the required pulse time can be compressed if we relax
the requirement that Doppler inversion be performed between full gates. This
can be interesting in the limit where the atoms are tightly trapped, such that
fluctuations of the atoms’ positions are very small relative to the optical
wavelength, and thus the error due to the positional phase shift is negligible
compared to other error sources. We introduce two additional protocols in
Appendix F where the Doppler shift inversion is realized while the qubits are
in the Rydberg state, that result in conditionally- or fully-robust dynamics
against intensity and Doppler fluctuations with shorter exectution times.

##  V Conclusion

We have introduced novel protocols to realize arbitrary controlled-phase gates
between two neutral atoms based on resonant global excitation to Rydberg
states and the Rydberg blockade mechanism. A main feature of these protocols
is that they exhibit robustness against certain quasi-static coherent errors.
Our analysis focused on effects due to errors in calibration of the Rabi
frequency or errors arising from slow fluctuations of the laser intensity, as
well as errors from Doppler shifts. We note that in current experimental
setups, other error sources, such as spontaneous emission from Rydberg states
or from intermediate states that are virtually populated during the Rydberg
excitation process, often dominate the error budget and a detailed analysis of
each setup is necessary to assess whether or not these protocols can
contribute to improved gate fidelity in practice. Finally we note that the
ideas introduced here for two-qubit gates also generalize to a multi-qubit
setting (see Appendix B).

## Acknowledgements

We thank Manuel Endres, Mikhail Lukin, and Nishad Maskara for helpful
discussions. We acknowledge financial support from the ERC Starting grant QARA
(grant no. 101041435), and an ESQ Discovery Grant, the Army Research Office
(grant no. W911NF-21-1-0367) and the DARPA ONISQ program (grant no.
W911NF2010021). D.B. acknowledges support from the NSF Graduate Research
Fellowship Program (grant DGE1745303) and The Fannie and John Hertz
Foundation.

Note added — While finalizing this work we became aware of Ref. [32].

## Appendix

##  Appendix A Arbitrary controlled-phase gate

Here we detail how to use Protocol I to realize a controlled-phase gate with
arbitrary phase $\phi$ (see Eq. (4)). The idea is to use the same pusle
sequence as in eq. (3) (interpreted as a 6-pulse sequence as discussed in the
main text), but change the phase of the laser in the last three pulses by an
amount $\xi$. The full evolution unitary is thus (we use here the sequence (3)
as our basis, but the analysis is the same with (2))

$\displaystyle
U=U_{x,\xi}(\tfrac{\pi}{2})U_{y,\xi}(\tfrac{\pi}{\sqrt{2}})U_{x,\xi}(\tfrac{\pi}{2})U_{x}(\tfrac{\pi}{2})U_{y}(\tfrac{\pi}{\sqrt{2}})U_{x}(\tfrac{\pi}{2}),$
(15)

where the subscript $\xi$ indicates that the phase is shifted by $\xi$.
Defining

$P_{\xi}=\prod_{i=1,2}e^{i\xi}|r\rangle_{i}\langle r|_{i}+|1\rangle_{i}\langle
1|_{i}+|0\rangle_{i}\langle 0|_{i},$

one can write $U_{x,\xi}(\theta)=P_{\xi}^{\dagger}U_{x}(\theta)P_{\xi}$. Using
the fact that $P_{\xi}^{\dagger}P_{\xi}=\mathbb{1}$, this leads to

$\displaystyle U=P_{\xi}^{\dagger}S\,P_{\xi}S,$ (16)

where
$S=U_{x}(\tfrac{\pi}{2})U_{y}(\tfrac{\pi}{\sqrt{2}})U_{x}(\tfrac{\pi}{2})$.

As seen in the main text, $S$ is designed so that
$S|1,0\rangle\propto|r,0\rangle$, $S|0,1\rangle\propto|0,r\rangle$ and
$S|1,1\rangle\propto\frac{|r,1\rangle+|1,r\rangle}{\sqrt{2}}$. All these
states have exactly one atom in the Rydberg state, so for
$|z_{1},z_{2}\rangle\neq|0,0\rangle$, we have

$\displaystyle P_{\xi}S|z_{1},z_{2}\rangle=e^{i\xi}S|z_{1},z_{2}\rangle.$ (17)

Therefore,

$\displaystyle
U|z_{1},z_{2}\rangle=P_{\xi}^{\dagger}S\,P_{\xi}S|z_{1},z_{2}\rangle=e^{-i\left(\pi-\xi\right)}|z_{1},z_{2}\rangle.$
(18)

Finally, since $U|0,0\rangle=|0,0\rangle$, it follows that for any
$|z_{1},z_{2}\rangle$,

$\displaystyle
U|z_{1},z_{2}\rangle=\left(e^{i(\pi-\xi)}\right)^{z_{1}z_{2}-z_{1}-z_{2}}|z_{1},z_{2}\rangle,$
(19)

which is a controlled-$\phi$ gate with $\phi=(\pi-\xi)$, up to a single-qubit
rotation. In summary, to realize a controlled-phase gate of phase $\phi$, one
needs to separate both halves of the 6-pulse sequence by a phase jump
$\xi=\pi-\phi$.

##  Appendix B Extension to more than two qubits

For practical applications and the implementation of quantum circuits, it is
useful to be able to realize entangling gates for more than two qubits.
Indeed, although all multiqubit gates can be decomposed as a product of one-
and two-qubit gates, the number of gates required scales quickly with the
number of qubits, and can make such decompositions impractical. In this
section, we therefore discuss the extension of the protocols presented above
to more than two atoms. We now consider three atoms that are all in the same
blockade radius: as a result, no more than one of them may be excited to the
Rydberg state at a time. Same as above, the dynamics thus decomposes as a
block-diagonal evolution, but there is now a third effective two-level system
to consider, corresponding to the state $|1,1,1\rangle$ and with effective
Rabi frequency $\sqrt{3}\Omega$.

Our main target is now the Controlled-Controlled-Z gate (CCZ), which is
defined as
${\rm{CCZ}}|z_{1},z_{2},z_{3}\rangle=(-1)^{z_{1}z_{2}z_{3}}|z_{1},z_{2},z_{3}\rangle$.
We numerically find a sequence $S_{3}$ of five resonant global pulses, which
brings all three two-level systems from their ground state to their excited
state: writing a pulse with pulse area $\alpha=\Omega t$ and phase $\xi$ as
$U_{\xi}(\alpha)$, the five-pulse sequence $S_{3}$ reads

$\displaystyle
S_{3}=U_{0}(\alpha_{1})U_{\xi_{2}}(\alpha_{2})U_{\xi_{3}}(\alpha_{3})U_{\xi_{2}}(\alpha_{2})U_{0}(\alpha_{1})$
(20)

where $\alpha_{1}=1.088$, $\alpha_{2}=1.955$, $\alpha_{3}=5.373$,
$\xi_{2}=1.552$, and $\xi_{3}=1.593$. Just like in Sec. III, applying this
sequence twice brings all computational basis states back to themselves, and
gives them a minus sign (except $|0,0,0\rangle$). This is equivalent to the
above definition of the CCZ gate up to exchanging the states $|0\rangle$ and
$|1\rangle$. The total execution time for this protocol is $T=22.84/\Omega$,
which is only about twice as long as the two-qubit protocol introduced in the
main text. Moreover, this protocol can also be adapted to realize arbitrary
Controlled-Controlled-Phase gates ($\textrm{CC}_{\phi}$), by introducing a
phase jump between the two successive applications of $S_{3}$.

Figure 4: Bloch sphere representation of the 3-qubit CCZ gate sequence. The
coupled superposition states $|W_{2}\rangle$ and $|W_{3}\rangle$ are defined
respectively as $|W_{2}\rangle=\frac{|0,1,r\rangle+|0,r,1\rangle}{\sqrt{2}}$
and
$|W_{3}\rangle=\frac{|1,1,r\rangle+|1,r,1\rangle+|r,1,1\rangle}{\sqrt{3}}$;
the associated coupling strengths are respectively $\sqrt{2}\Omega$ and
$\sqrt{3}\Omega$. For all three Bloch spheres, the solid angle enclosed in the
trajectory represents half the sphere, so all computational basis states
except $|0,0,0\rangle$ acquire a phase of $\pi$ during this sequence.

Finally, the protocol introduced here retains the robustness properties of its
two-qubit counterpart against laser intensity fluctuations. Indeed, the
division of the sequence in two applications of $S_{3}$, which maps the south
poles of all three Bloch spheres to their north poles, ensures conditional
robustness for the same reasons as those given in Appendix C. Moreover, in the
same way as in Sec. IV, dividing a $\textrm{CC}_{\phi}$ gate into two
successive $\textrm{CC}_{\phi/2}$ gates, separated by a well-chosen phase jump
(which is zero in the CCZ case), gives rise to a destructive interference
which ensures leakage-robustness of the gate at the cost of a doubled
execution time.

Extension of these principles to more than three qubits is similarly possible,
but the numerical cost of finding a sequence that brings all effective two-
level systems from their ground to excited state increases with the number of
atoms considered. Still, once such a sequence is found, the robustness ideas
developed in Sec. IV apply to the resulting gate protocol as well.

##  Appendix C Conditional robustness of Protocol I

####  C.0.1 Requirement for conditional robustness

In this section we show that Protocol I is conditionally robust against
intensity fluctuations. For this we first note that a gate is conditionally
robust if $\langle z_{1},z_{2}|\partial_{\epsilon}U(0)|z_{1},z_{2}\rangle=0$
for $(z_{1},z_{2})\in\\{(0,1),(1,0),(1,1)\\}$, where
$\partial_{\epsilon}U(0)\equiv\partial_{\epsilon}U(\epsilon)|_{\epsilon=0}$.
Indeed, in that case we have (keeping in mind that
$U(\epsilon)|0,0\rangle=|0,0\rangle$ for any $\epsilon$)

$\displaystyle|\mathrm{tr}$
$\displaystyle\left(U(\epsilon)PU^{\dagger}({0})P\right)|^{2}$
$\displaystyle\qquad=|\sum_{{\boldsymbol{z}}}\langle{\boldsymbol{z}}|U(\epsilon)|{\boldsymbol{z}}\rangle\langle{\boldsymbol{z}}|U(0)|{\boldsymbol{z}}\rangle|^{2}$
$\displaystyle\qquad=|4-\frac{\epsilon^{2}}{2}\sum_{{\boldsymbol{z}}\neq(0,0)}\langle{\boldsymbol{z}}|\partial_{\epsilon}^{2}U(0)|{\boldsymbol{z}}\rangle+O\left(\epsilon^{4}\right)|^{2}$
$\displaystyle\qquad=4\left(4-\epsilon^{2}\sum_{z_{1},z_{2}}Re\left[\langle{\boldsymbol{z}}|\partial_{\epsilon}^{2}U(0)|{\boldsymbol{z}}\rangle\right]\right)+O\left(\epsilon^{4}\right),$

and

$\displaystyle\mathrm{tr}$
$\displaystyle\left(PU(\epsilon)PU^{\dagger}({\epsilon})\right)$
$\displaystyle\qquad=1+\sum_{{\boldsymbol{z}}\neq(0,0)}|\langle{\boldsymbol{z}}|U(\epsilon)|{\boldsymbol{z}}\rangle|^{2}$
$\displaystyle\qquad=1+\sum_{{\boldsymbol{z}}\neq(0,0)}|1-\frac{\epsilon^{2}}{2}\langle{\boldsymbol{z}}|\partial_{\epsilon}^{2}U(0)|{\boldsymbol{z}}\rangle+O\left(\epsilon^{4}\right)|^{2}$
$\displaystyle\qquad=4-\epsilon^{2}\sum_{z_{1},z_{2}}Re\left[\langle{\boldsymbol{z}}|\partial_{\epsilon}^{2}U(0)|{\boldsymbol{z}}\rangle\right]+O\left(\epsilon^{4}\right),$

which, from the expressions of $\mathcal{F}$ and $\mathcal{P}$, leads to

$\mathcal{C}=\frac{\mathcal{F}}{\mathcal{P}}=1-O\left(\epsilon^{4}\right).$

This is indeed the definition of conditional robustness.

We note that the requirement $\langle
z_{1},z_{2}|\partial_{\epsilon}U(0)|z_{1},z_{2}\rangle=0$ corresponds to
saying that there is no first-order phase error in the $|z_{1},z_{2}\rangle$
component of the final state. Indeed, this matrix element gives the first-
order term of the series expansion of $\langle
z_{1},z_{2}|U(\epsilon)|z_{1},z_{2}\rangle$, which is equal to $i$ times the
first-order phase error.

####  C.0.2 Absence of phase error in Protocol I

We thus need to show that $\langle
z_{1},z_{2}|\partial_{\epsilon}U(0)|z_{1},z_{2}\rangle=0$ for all qubit
states. We write $U$ as (using the same notations as the previous section)

$\displaystyle U(\epsilon)=P_{\xi}^{\dagger}S(\epsilon)\,P_{\xi}S(\epsilon),$

where $S(\epsilon)$ is defined analogous to $S$. Suppressing the explicit
dependence on $\epsilon$ and introducting the notation
$U^{\prime}\equiv\partial_{\epsilon}U$ and
$S^{\prime}\equiv\partial_{\epsilon}S$ for simplicity, we can then write
$\partial_{\epsilon}U$ as

$\displaystyle
U^{\prime}=P_{\xi}^{\dagger}S^{\prime}\,P_{\xi}S+P_{\xi}^{\dagger}S\,P_{\xi}S^{\prime}.$
(21)

Defining $|\uparrow_{z_{1},z_{2}}\rangle=S(0)|z_{1},z_{2}\rangle$, we have

$S(0)|\uparrow_{z_{1},z_{2}}\rangle=S(0)^{2}|z_{1},z_{2}\rangle=-|z_{1},z_{2}\rangle,$

hence

$S^{\dagger}(0)|z_{1},z_{2}\rangle=-|\uparrow_{z_{1},z_{2}}\rangle,$

From this, and using the fact that (since $|\uparrow_{z_{1},z_{2}}\rangle$ has
exactly one atom in the Rydberg state)
$P_{\xi}|\uparrow_{z_{1},z_{2}}\rangle=e^{i\xi}|\uparrow_{z_{1},z_{2}}\rangle$
and
$\langle\uparrow_{z_{1},z_{2}}|P_{\xi}=e^{i\xi}\langle\uparrow_{z_{1},z_{2}}|$,
we can write

$\displaystyle\begin{split}\langle
z_{1},z_{2}|U^{\prime}|z_{1},z_{2}\rangle&=e^{i\xi}\left(\langle
z_{1},z_{2}|S^{\prime}|\uparrow_{z_{1},z_{2}}\rangle\right.\\\
&\qquad\qquad\left.-\langle\uparrow_{z_{1},z_{2}}|S^{\prime}|z_{1},z_{2}\rangle\right)\end{split}$
(22)

We now take advantage of the block-diagonal evolution of the system to
consider each computational basis state’s evolution as an independent two-
level system, with ground state $|z_{1},z_{2}\rangle$ and excited state
$|\uparrow_{z_{1},z_{2}}\rangle$.

We consider the restrictions of all 8-by-8 operators to the two-level system
$\\{|z_{1},z_{2}\rangle,|\uparrow_{z_{1},z_{2}}\rangle\\}$ from now on.

$S(\epsilon)$ is the product of three single-pulse evolution operators,
$S(\epsilon)=\prod_{p=1}^{3}U_{p}(\epsilon)$ with $U_{p}=e^{-iH_{p}t_{p}}$,
where the product goes over the three pulses of $S$ (implying an order in the
product) and the subscript $p$ specifies which pulse each operator corresponds
to. For the two-level system
$\\{|z_{1},z_{2}\rangle,|\uparrow_{z_{1},z_{2}}\rangle\\}$, each single-pulse
unitary $U_{p}$ is a Bloch sphere rotation, so their product can also be
expressed as one: the restriction of $S(\epsilon)$ to
$\\{|z_{1},z_{2}\rangle,|\uparrow_{z_{1},z_{2}}\rangle\\}$ is

$\displaystyle S(\epsilon)$
$\displaystyle=e^{i\frac{\theta(\epsilon)}{2}{\boldsymbol{n}}(\epsilon)\cdot{\boldsymbol{\sigma}}}$
(23)
$\displaystyle=e^{i\left(\frac{\theta(0)}{2}{\boldsymbol{n}}(0)\cdot{\boldsymbol{\sigma}}+\frac{\epsilon}{2}\partial_{\epsilon}\left(\theta(\epsilon){\boldsymbol{n}}(\epsilon)\right)\cdot{\boldsymbol{\sigma}}\right)+O\left(\epsilon^{2}\right)},$
(24)

where ${\boldsymbol{\sigma}}$ is the vector containing the three Pauli
matrices of the two-level system
$\\{|z_{1},z_{2}\rangle,|\uparrow_{z_{1},z_{2}}\rangle\\}$. The commutator of
two Pauli matrices is either proportional to another Pauli matrix or 0. Thus,
one can rewrite this (using the Baker-Campbell-Hausdorff formula) as

$\displaystyle S(\epsilon)$
$\displaystyle=e^{i\epsilon\frac{\alpha}{2}{\boldsymbol{m}}\cdot{\boldsymbol{\sigma}}}S(0)+O\left(\epsilon^{2}\right)$
(25)

with some real scalar $\alpha$ and vector ${\boldsymbol{m}}$. From this one
gets

$\displaystyle S(\epsilon)$
$\displaystyle=S(0)+i\frac{\alpha}{2}\left({\boldsymbol{m}}\cdot{\boldsymbol{\sigma}}\right)S(0)\epsilon+O\left(\epsilon^{2}\right).$
(26)

The derivative of $S$ is given by the second term of this expression. Finally,
this means that

$\displaystyle\langle z_{1},z_{2}|S^{\prime}|\uparrow_{z_{1},z_{2}}\rangle$
$\displaystyle=-i\frac{\alpha}{2}\langle
z_{1},z_{2}|{\boldsymbol{m}}\cdot{\boldsymbol{\sigma}}|z_{1},z_{2}\rangle$
$\displaystyle=i\frac{\alpha}{2}{\boldsymbol{m}}\cdot{\boldsymbol{e}}_{z}$

and

$\displaystyle\langle\uparrow_{z_{1},z_{2}}|S^{\prime}|z_{1},z_{2}\rangle$
$\displaystyle=i\frac{\alpha}{2}\langle\uparrow_{z_{1},z_{2}}|{\boldsymbol{m}}\cdot{\boldsymbol{\sigma}}|\uparrow_{z_{1},z_{2}}\rangle$
$\displaystyle=i\frac{\alpha}{2}{\boldsymbol{m}}\cdot{\boldsymbol{e}}_{z}.$

Plugging this into (22) finally shows that, for all $|z_{1},z_{2}\rangle$, we
have

$\langle z_{1},z_{2}|\partial_{\epsilon}U(0)|z_{1},z_{2}\rangle=0,$

which, as shown in the beginning of this section, implies that the protocol is
conditionally robust.

##  Appendix D Leakage-robustness of Protocol II

Here we show that Protocol II is leakage-robust against laser intensity
errors, for an arbitrary controlled-phase gate.

The gate is realized by applying two successive controlled-phase gates of half
the target phase, such that the resulting gate is the desired one. One also
needs to introduce a phase jump between the two successive gates. The pulse
sequence can be written as

$\displaystyle
U(\epsilon)=\left(P_{\xi}^{\dagger}G(\epsilon)P_{\xi}\right)G(\epsilon),$ (27)

where $\phi$ is the target phase, $G$ the 6-pulse sequence corresponding to a
controlled-$(\tfrac{\phi}{2})$ gate (see Sec. III and Appendix A), and $\xi$
the phase jump between the two gates. As in the previous section, we denote
$\partial_{\epsilon}U$ as $U^{\prime}$ and $\partial_{\epsilon}G$ as
$G^{\prime}$.

The sequence $G$ is designed such that
$G(0)|z_{1},z_{2}\rangle=e^{i\tfrac{\phi}{2}f(z_{1},z_{2})}|z_{1},z_{2}\rangle$
where $f(z_{1},z_{2})=z_{1}z_{2}-z_{1}-z_{2}$, and one can check that with
$Q=|r,0\rangle\langle r,0|+|0,r\rangle\langle 0,r|+|W\rangle\langle W|$ the
projector onto the manifold of coupled Rydberg states,
$QG(0)=e^{i\tfrac{\phi}{2}}Q$, so at $\epsilon=0$,

$\displaystyle QU^{\prime}|z_{1},z_{2}\rangle$
$\displaystyle=Q\left[\left(P_{\xi}^{\dagger}G^{\prime}P_{\xi}\right)G+\left(P_{\xi}^{\dagger}GP_{\xi}\right)G^{\prime}\right]|z_{1},z_{2}\rangle$
$\displaystyle=Q\left[e^{i\tfrac{\phi}{2}f(z_{1},z_{2})}e^{-i\xi}+e^{i\tfrac{\phi}{2}}\right]G^{\prime}|z_{1},z_{2}\rangle,$
(28)

where we used the fact that $QP_{\xi}^{\dagger}=e^{-i\xi}Q$ (since all the
states in $Q$’s manifold have exactly one atom in the Rydberg state). For
$|z_{1},z_{2}\rangle=|0,0\rangle$, this vanishes since
$G^{\prime}|0,0\rangle=0$, and for $|z_{1},z_{2}\rangle\neq|0,0\rangle$, we
have $f(z_{1},z_{2})=-1$, so

$\displaystyle QU^{\prime}(0)|z_{1},z_{2}\rangle$
$\displaystyle=Q\left[e^{-i\left(\xi+\tfrac{\phi}{2}\right)}+e^{i\tfrac{\phi}{2}}\right]G^{\prime}(0)|z_{1},z_{2}\rangle$
(29)

Thus, for $\xi=\pi-\phi$, we have $QU^{\prime}(0)|z_{1},z_{2}\rangle=0$. This
leads to $QU(\epsilon)P=O\left(\epsilon^{2}\right)$.

From this, since $(P+Q)U(\epsilon)=U(\epsilon)(P+Q)$, we can write

$\displaystyle\mathcal{P}$
$\displaystyle=\frac{1}{d}\mathrm{tr}\left(PU(\epsilon)PU(\epsilon)^{\dagger}\right)$
$\displaystyle=1-\frac{1}{d}\mathrm{tr}\left(PU(\epsilon)QU(\epsilon)^{\dagger}P\right)$
$\displaystyle=1-O\left(\epsilon^{4}\right)$

which means $\chi_{\mathcal{P}}=0$.

We note that for the CZ gate case, this analysis leads to a phase jump
$\xi=\pi-\pi=0$, which explains why no phase jump is needed in the CZ case.

##  Appendix E Multivariate robustness

In practice, more than one control parameter may fluctuate. We prove here that
a gate is robust to simultaneous variations of two or more parameters if it is
robust with respect to each control parameter separately.

We replace the error parameter $\epsilon$ with a vector of control parameters
${\boldsymbol{\epsilon}}=(\epsilon_{1},\epsilon_{2},\dots)$. Since the
fidelity assumes its maximum at ${\boldsymbol{\epsilon}}={\boldsymbol{0}}$,
its Hessian matrix
$H_{ij}=\partial_{\epsilon_{i}}\partial_{\epsilon_{j}}\mathcal{F}({\boldsymbol{\epsilon}})|_{{\boldsymbol{\epsilon}}={\boldsymbol{0}}}$,
is negative semi-definite. In particular, the 2-by-2 submatrix
$\big{(}\begin{smallmatrix}H_{ii}&H_{ij}\\\
H_{ji}&H_{jj}\end{smallmatrix}\big{)}$ is also negative semi-definite, so its
determinant must be positive, such that (as $H_{ij}=H_{ji}$)

$\displaystyle\begin{split}&H_{ii}H_{jj}-H_{ij}^{2}\geq 0\\\
&(\partial_{\epsilon_{i}}\partial_{\epsilon_{j}}\mathcal{F})^{2}\leq(\partial_{\epsilon_{i}}^{2}\mathcal{F})(\partial_{\epsilon_{j}}^{2}\mathcal{F}).\end{split}$
(30)

If a protocol is robust to variations of the two parameters $\epsilon_{i}$ and
$\epsilon_{j}$, the right hand-side is zero, so the left hand-side must also
be zero, meaning that the sequence is also robust to simultaneous
fluctuations.

##  Appendix F Short robust protocols with Doppler inversion in the Rydberg
state

Figure 5: CZ pulse sequences for Protocols I.a ($T_{\textrm{I}}=10.73/\Omega$)
and II.b ($T_{\textrm{II}}=21.45/\Omega$), and susceptibilities of these
sequences against intensity and Doppler errors.

We now introduce two new protocols that take advantage of tightly trapped
atoms, i.e. we assume that the effects of positional phase shift due to the
Doppler inversion scheme can be neglected. In that case, one can create a
protocol that is conditionally robust to (symmetric and antisymmetric) Doppler
errors using only 6 pulses. This protocol (which we call Protocol I.a) simply
consists in realizing a 6-pulse Protocol I, but reversing the Doppler shift
for the last three pulses (second half) of the sequence. Computing the
expansions of the conditional susceptibilities in that case shows that the
sequence is indeed conditionally robust to symmetric and antisymmetric Doppler
errors. In addition, due to being the same as Protocol I up to the Doppler
inversion, this protocol is also conditionally robust against laser intensity
errors.

The second protocol we introduce is a fully robust variant of Protocol I.a.
Its structure follows naturally from the same ideas as in the main text: we
split the $\textrm{C}_{\phi}$ gate into two Protocol I.a-style
$\textrm{C}_{\phi/2}$ gates, and separate them by the right phase difference
(0 for the CZ gate) to ensure destructive interference of the leakage
amplitudes. Computing the expansions of the fidelities in this case shows that
this protocol (which we call Protocol II.b) is fully robust against intensity
and Doppler errors. The CZ pulse sequences and associated susceptibilities for
Protocols I.a and II.b are given in Fig. 5.

##  Appendix G Analysis of the Doppler echo scheme

####  G.0.1 The scheme

In this section, we further describe the Doppler echo scheme discussed in the
text and provide further analysis. Each individual atom resides in its own
individual optical tweezer, which acts as a harmonic oscillator potential. The
atoms are typically not cooled all the way to the ground state of this quantum
harmonic oscillator, and in particular are in a thermal state of the
oscillator [33], which can be described as the atom being in a random coherent
state with random amplitude and phase. The motional state of the atom at any
point in time can thus be understood as simply oscillating in a classical
harmonic oscillator potential with some random amplitude and phase of
oscillation. The atom’s position and velocity in the classical harmonic
oscillator as a function of time will be

$\displaystyle x(t)=x_{0}\cos(\omega_{0}t+\phi)$ $\displaystyle
v(t)=v_{0}\sin(\omega_{0}t+\phi)$

where $\omega_{0}$ is the trap frequency and $x_{0},v_{0},\phi$ are random
position, velocity, and phase. The Doppler error is proportional to the
velocity $v(t)$ of the atom at the time that the gate is performed. The time
of the gate (during which the tweezer is typically turned off [9]) is
significantly shorter than the trap frequency, and so the atom velocity and
position in the oscillator does not significantly change during the gate. Upon
recapturing the atoms in the tweezer and waiting half a trap period, i.e.
$\pi/\omega_{0}$, the atom position and velocity then reverse:

$\displaystyle x(\pi/\omega_{0})=-x(0)$ $\displaystyle
v(\pi/\omega_{0})=-v(0)$

As such, the Doppler error $\propto v$ will reverse in sign, as will the
positional fluctuation of the atom about the center of the potential. This can
be used to echo and suppress Doppler-induced dephasing. With all population
stored in the qubit states $\\{|0\rangle$, $|1\rangle\\}$ after a gate, any
Rydberg laser phase noise or dephasing between $|1\rangle$ and $|r\rangle$
during the half-trap-period oscillation will cause no effect since all phase
information is stored between $|0\rangle$ and $|1\rangle$.

However, if one leaves population in the Rydberg state $|r\rangle$ during the
half-trap-period oscillation, several complications arise. First, there can be
significant decoherence between $|1\rangle$ and $|r\rangle$ from both
inhomogeneous light shifts from the tweezer and from spin-motion entanglement
induced by the different trapping potentials for $|1\rangle$ and $|r\rangle$
(in fact, $|r\rangle$ is anti-trapped for many settings [14]). To avoid such
decoherence, one requires a magic trapping potential, i.e. such that both
$|1\rangle$ and $|r\rangle$ experience the same trapping potential, which can
be made possible in various settings [34].

Even with a magic trapping potential, dephasing can still occur between
$|1\rangle$ and $|r\rangle$ if population is left in $|r\rangle$ for the half-
trap-period oscillation. Specifically, the reversal of the atom position along
the direction of the laser is in fact precisely a Doppler shift: the atom
moves a distance $2x(0)$ which will cause it to accumulate phase within the
wavelength of the Rydberg laser by an amount $k\times 2x(0)$, where $k$ is the
net momentum of the Rydberg laser (for a single laser this is just
$k=2\pi/\lambda$ where $\lambda$ is the laser wavelength, and for a two-photon
counter-propagating excitation scheme this is the net momentum between the two
lasers $k_{1}-k_{2}$). Since $x(0)$ is randomly drawn from the thermal
distribution of the atom position, the phase accumulation $k\times 2x(0)$ will
now lead to a random phase fluctuation between $|1\rangle$ and $|r\rangle$.
Quantitatively, the thermal spread of the atom can be attained from the
equipartition theorem: for a massive particle $m$ the root-mean-square spread
in position will be $x_{\text{RMS}}=\sqrt{k_{B}T/(m\omega_{0}^{2})}$, where
$k_{B}$ is Boltzmann’s constant and $T$ is the atom temperature. For 87Rb,
with typical trap frequencies on the scale of $\omega_{0}\sim 2\pi\times 100$
kHz and atom temperatures on the scale of $\sim$ 5 $\mu$K [14, 33], this will
give $x_{\text{RMS}}\sim$ 35 nm. For the two-photon scheme used for 87Rb [14,
9], this would lead to a characteristic phase accumulation of $k\times
2x_{\text{RMS}}\sim 2\pi\times 0.1$ rad. This is a non-negligible phase
accumulation. Thus, it is useful to determine the effect of such a fluctuation
for the protocols discussed above, in the cases where it cannot be suppressed.

####  G.0.2 Effect of the positional phase shift, by protocol

The first thing to consider is whether this phase fluctuation, by itself,
causes a gate infidelity. There are two different cases in the protocols
discussed in Sec. IV and Appendix F: either the Doppler inversion occurs
between two controlled-phase gates, while the atoms are in the qubit manifold
(Protocol II.a and Protocol III), or it happens in the middle of a 6-pulse
gate sequence, while they are in the Rydberg-excited state (Protocols I.a and
II.b). In the first case, a different laser phase does not modify the action
of the second $\textrm{C}_{\phi/2}$ gate: the two gates still add up normally
and give the desired $\textrm{C}_{\phi}$ gate. The phase shift thus has no
effect in that case. In the second case, however, the phase shift effectively
modifies the phase jump between the two parts of a 6-pulse gate. This results
in a change of the gate phase (see Appendix A), which can also be seen as a
conditional infidelity ($\mathcal{C}\neq 1$). The protocol in that case is not
robust to the phase shift.

For Protocol II.a and Protocol III, where this finite phase shift alone does
not cause errors, one can also consider how much the conditional and leakage-
robustness against other errors are affected by it. In fact, this phase
fluctuation does not compromise the conditional robustness, but this
phenomenon affects the leakage robustness since the phase jump which ensures
destructive interference of the leakage amplitudes is effectively modified.
However, this effect is a higher-order one: the population left in the
$|r\rangle$ state between the two gates is quadratic, so the small correction
added by the phase shift gives a high-order correction to the fidelity.

From this we see that, when the positional phase fluctuation is not dominated
by the other errors, Protocols II.a and III, which are robust to this error,
are more advantageous than Protocols I.a and II.b, which are not. On the other
hand, if this effect is suppressed, Protocols I.a and II.b would have the
advantage of their shorter execution time.

##  Appendix H Expressions of the power series for $\mathcal{F}$,
$\mathcal{P}$ and $\mathcal{C}$

In the following, we present the analytical expressions of the expansions
shown in Table 1, as well as the fidelity expansions of the protocols
introduced in the main text and appendix, for symmetric and antisymmetric
Doppler-induced frequency errors.

To make the analytical expressions lighter, we introduced the notations
(denoting $C_{n}\equiv{\rm cos}\left(\frac{n\pi}{\sqrt{2}}\right)$):

$\displaystyle A$ $\displaystyle\equiv C_{1}^{2}+C_{1}+1$ $\displaystyle B$
$\displaystyle\equiv 13+4C_{1}+8C_{2}-4C_{3}+3C_{4}$ $\displaystyle D$
$\displaystyle\equiv 30+30C_{1}+27C_{2}+2C_{3}+6C_{4}+C_{6}.$

The values presented in Tables 3 and 4 can similarly be expressed
analytically. We present numerical values here to ease comparison between
different protocols.

CZ gate protocol | $\mathcal{F}$ | $\mathcal{P}$ | $\mathcal{C}$ | Execution time
---|---|---|---|---
Jaksch et al. [5] | $1-\frac{\pi^{2}}{2}\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-\frac{\pi^{2}}{2}\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-\frac{\pi^{4}}{20}\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $T_{J}\equiv 4\pi/\Omega=12.57/\Omega$
Protocol I | $1-\frac{A\pi^{2}}{4}\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-\frac{A\pi^{2}}{4}\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-\frac{\pi^{4}}{640}B\epsilon^{4}+O\left(\epsilon^{5}\right)$ | $T_{\textrm{I}}\equiv(2+\sqrt{2})\pi/\Omega=10.73/\Omega$
Protocol II | $1-\frac{\pi^{4}}{640}B\epsilon^{4}+O\left(\epsilon^{5}\right)$ | $1-\frac{\pi^{6}}{1024}D\epsilon^{6}+O\left(\epsilon^{7}\right)$ | $1-\frac{\pi^{4}}{640}B\epsilon^{4}+O\left(\epsilon^{5}\right)$ | $T_{\textrm{II}}\equiv 2(2+\sqrt{2})\pi/\Omega=21.45/\Omega$
Table 2: Analytical expressions of the fidelity expansions given in Table 1 in the main text. CZ gate protocol | $\mathcal{F}$ | $\mathcal{P}$ | $\mathcal{C}$ | Execution time
---|---|---|---|---
Jaksch et al. [5] | $1-2.480\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-1.480\epsilon^{2}+O\left(\epsilon^{6}\right)$ | $12.57/\Omega$
Levine et al. [9] | $1-3.000\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-0.077\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $1-2.923\epsilon^{2}+O\left(\epsilon^{3}\right)$ | $8.59/\Omega$
Protocol I | $1-4.314\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-3.124\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-4.314\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $10.73/\Omega$
Protocol II | $1-17.256\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-6.249\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-17.256\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $21.45/\Omega$
Protocol I.a (2) | $1-2.018\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-2.018\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-0.786\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $10.73/\Omega$
Protocol I.a (3) | $1-4.091\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-4.091\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-2.011\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $10.73/\Omega$
Protocol II.a (2) | $1-7.035\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-6.249\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-0.786\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol II.a (3) | $1-8.260\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-6.249\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-2.011\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol II.b (2) | $1-0.786\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-O\left(\epsilon^{6}\right)$ | $1-0.786\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol II.b (3) | $1-2.011\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-O\left(\epsilon^{6}\right)$ | $1-2.011\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol III (2) | $1-1.570\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-O\left(\epsilon^{6}\right)$ | $1-1.570\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $42.90/\Omega$
Protocol III (3) | $1-4.021\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-O\left(\epsilon^{6}\right)$ | $1-4.021\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $42.90/\Omega$
Table 3: Leading-order expansions of $\mathcal{F}$, $\mathcal{P}$ and $\mathcal{C}$ for symmetric detuning fluctuation ($\epsilon=\delta\Delta/\Omega$). Expressions are no longer always the same if we choose sequence (2) or (3) as our basis for pulse area values. CZ gate protocol | $\mathcal{F}$ | $\mathcal{P}$ | $\mathcal{C}$ | Execution time
---|---|---|---|---
Jaksch et al. [5] | $1-6.428\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-5.428\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $12.57/\Omega$
Levine et al. [9] | $1-11.772\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-3.417\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-8.355\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $8.59/\Omega$
Protocol I (2) | $1-17.637\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-6.132\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-11.505\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $10.73/\Omega$
Protocol I (3) | $1-19.313\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-7.808\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-11.505\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $10.73/\Omega$
Protocol II (2) | $1-58.284\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-12.264\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-46.020\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $21.45/\Omega$
Protocol II (3) | $1-61.636\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-15.616\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-46.020\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $21.45/\Omega$
Protocol I.a (2) | $1-2\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-2\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-\frac{4}{5}\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $10.73/\Omega$
Protocol I.a (3) | $1-3.591\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-3.591\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-2.580\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $10.73/\Omega$
Protocol II.a (2) | $1-12.264\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-12.264\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-171.462\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol II.a (3) | $1-15.616\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-15.616\epsilon^{2}+O\left(\epsilon^{4}\right)$ | $1-272.779\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol II.b (2) | $1-\frac{4}{5}\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-O\left(\epsilon^{6}\right)$ | $1-\frac{4}{5}\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol II.b (3) | $1-2.580\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-O\left(\epsilon^{6}\right)$ | $1-2.580\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $21.45/\Omega$
Protocol III (2) | $1-676\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-513\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-163\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $42.90/\Omega$
Protocol III (3) | $1-1090\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-837\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $1-253\epsilon^{4}+O\left(\epsilon^{6}\right)$ | $42.90/\Omega$
Table 4: Leading-order expansions of $\mathcal{F}$, $\mathcal{P}$ and
$\mathcal{C}$ for antisymmetric detuning fluctuations
($\epsilon=\frac{\delta\Delta_{1}}{\Omega}=-\frac{\delta\Delta_{2}}{\Omega}$).

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