# Robust control and optimal Rydberg states for neutral atom two-qubit gates

Madhav Mohan1 Robert de Keijzer1 Servaas Kokkelmans1 1Department of Applied
Physics and Eindhoven Hendrik Casimir Institute, Eindhoven University of
Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands

###### Abstract

We investigate the robustness of two-qubit gates to deviations of experimental
controls, on a neutral atom platform utilizing Rydberg states. We construct
robust CZ gates – employing techniques from quantum optimal control – that
retain high Bell state fidelity $F>0.999$ in the presence of significant
deviations of the coupling strength to the Rydberg state. Such deviations can
arise from laser intensity noise and atomic motion in an inhomogeneous
coupling field. We also discuss methods to mitigate errors due to deviations
of the laser detuning. The designed pulses operate on timescales that are
short compared to the fundamental decay timescale set by spontaneous emission
and blackbody radiation. We account for the finite lifetime of the Rydberg
state in both the optimisation and fidelity calculations – this makes the
gates conducive to noisy intermediate-scale quantum experiments, meaning that
our protocols can reduce infidelity on near-term quantum computing devices. We
calculate physical properties associated with infidelity for strontium-88
atoms – including lifetimes, polarisabilities and blockade strengths – and use
these calculations to identify optimal Rydberg states for our protocols, which
allows for further minimisation of infidelity.

††preprint: APS/123-QED

##  I Introduction

Neutral atoms trapped in optical tweezers are attractive candidates for
applications in quantum information. These systems have proven scalable –
arrays of up to 324 atoms have been created [1], and large scale quantum
simulations have been carried out [2]. Quantum algorithms have also been
demonstrated on neutral atom quantum computers [3], and highly-entangled
multiqubit states have been recently generated [4, 3].

The problem of generating high-fidelity entanglement – the fundamental
resource towards constructing quantum circuits – and quantum gates is a
crucial one to the realisation of full-fledged quantum computers. In
particular, a two-qubit CZ gate combined with a full set of single-qubit
operations is sufficient to perform universal quantum computation [5].
Generating the necessary entanglement between neutral atoms involves
excitation of the atoms using laser pulses, to electronic states of high
principal quantum number $n$ – termed _Rydberg_ states. Using such a scheme,
high-fidelity entangling operations between two qubits (with Bell state
fidelity $>0.991$) have been recently demonstrated by Madjarov _et al_ [6].
Theoretical predictions of two-qubit gate fidelities $F>0.9999$ [7, 8] show
that high-fidelity gates are possible on this platform.

A high fidelity of logic gates is of importance as the fidelity limits the
depth of quantum circuits that can be constructed, and thus constrains the
complexity of the algorithms that can be implemented. Furthermore, strict
requirements are placed on gate fidelities and qubit coherence times by
quantum error correction [9], and current experimental demonstrations – while
highly promising – fall short of these requirements. This era of quantum
computing is termed the noisy intermediate scale quantum (NISQ) era [10],
where multiple sources of decoherence and infidelity are present, that must be
carefully accounted for and mitigated. Following the seminal proposal for
realising CZ gates with Rydberg atoms [11], various two-qubit gate protocols
have been proposed, that promise high fidelities. Gates that are robust to
deviations of controls have been proposed, using analytical schemes based on
adiabaticity [12, 13]. Time-optimal gates – gates that minimise the duration
of operation, thus mitigating error sources that propagate in time – have
recently been identified [14, 15]. These protocols were devised using quantum
optimal control (QOC) – a class of methods that aim to find optimal ways to
steer a quantum system from an initial state to a desired final state using
time-dependent controls [16]. Using QOC methods, quantum gates have been
engineered and experimentally demonstrated on superconducting platforms [17,
18, 19, 20], and non-classical states have been generated on the Rydberg
platform [21, 4].

Although such Rydberg experiments often employ alkali atoms, recent quantum
information architectures have been identified for alkaline-earth atoms [22,
23, 24]. Notably, the high-fidelity two qubit entanglement study of Madjarov
_et al._ employs the metastable clock state of strontium-88 (Sr) atoms along
with the electronic ground state, resulting in a long-lived qubit
configuration [25]. Moreover, the rich electronic structure of these atoms
allows for novel applications, and is expected to pave the path towards higher
gate fidelities – and in the longer term, fault-tolerant quantum computing
[26].

In this work, we improve upon the current state of the art in two ways,
relating to two-qubit gate design and the choice of Rydberg state.

First, we use QOC methods to investigate the design of two-qubit gates
operating on a timescale comparable to the time-optimal gates, that are also
robust to deviations of _Rabi frequency_ , which quantifies the coupling
strength from the computational basis states to the Rydberg state. These
deviations arise due to laser intensity noise [6] and motion of the atom in a
spatially inhomogeneous coupling laser field [12]. We achieve excellent
robustness for gates that are near time-optimal and thus also mitigate errors
due to deviations of the detuning, such as those arising from thermal Doppler
shifts [12] and stray electric fields in the experiment. The corresponding
pulses are smooth, and adhere to realistic experimental constraints on Rabi
frequency and detuning, achievable in current experimental setups.

The second way in which we improve upon the state of the art, is with the
identification of optimal Rydberg states in Sr atoms, in order to minimise the
infidelity of our gate protocol. We achieve this by accounting for various
sources of error – including spontaneous emission from the Rydberg state,
blackbody radiation (BBR) and sensitivity to stray electric fields in the
experiment – and investigating the scaling of these errors with $n$.

The methods we develop for these results can be adapted to other Rydberg atom
experiments, and thus provide a closer link between theoretical advances and
experimental performance of quantum computers.

This work is structured as follows. In Sec. II we motivate the computation of
atomic properties and formulate both the two-qubit gate dynamics, and the
variational framework. In Sec. III, we discuss the results of our
optimisation, present two new CZ gate protocols and compare them to the time-
optimal gates. Section IV identifies the regime of optimal Rydberg states to
minimise the infidelity of our protocols.

##  II Theory and problem formulation

Figure 1: Energy level diagram of the Sr states relevant for our protocol. The
$\ket{1}$ state is coupled to the $\ket{r}$ state with the laser field. The
detuning of the laser $\Delta$ is the difference of the frequency of this
transition $\omega_{1,r}$ with the laser frequency $\omega_{L}$.

###  II.1 Modelling of atomic properties

In Rydberg platforms, one encodes quantum information in the electronic or
nuclear states of an atom. The discussion presented here considers
strontium-88 atoms, but protocols for atoms such as rubidium-87 [27],
ytterbium-171 [24] and caesium-133 [12] have been studied.

Our protocol employs the electronic ground state $5\mathrm{s}^{2}$
${}^{1}\mathrm{S}_{0}$ and the long-lived metastable clock state
$5\mathrm{s}5\mathrm{p}\,^{3}\mathrm{P}_{0}$ as qubit states $\ket{0}$ and
$\ket{1}$ respectively. Entanglement is facilitated with the Rydberg state
$\ket{r}$, $5\mathrm{s}n\mathrm{s}\,^{3}\mathrm{S}_{1}$. Figure 1 presents the
main transitions of interest in our system. Single-photon transitions are
achievable from the $\ket{1}$ state to the $\ket{r}$ state [6], which
eliminates decay from intermediate states – a source of infidelity common to
many Rydberg experiments [13].

The physical properties of atoms relevant to this discussion can be captured
with dipole matrix elements. The dipole matrix element $d_{a,b}$ between two
states $a$ and $b$ is $\bra{b}e\mathbf{x}\ket{a}$, where $e$ is the electronic
charge and $\mathbf{x}$ the position vector of the active electron. Computing
these elements is crucial to our work, as it allows us to calculate lifetimes
of the Rydberg states, interaction strengths and polarisabilities, and their
scaling with the principal quantum number of the Rydberg state, $n$. How these
quantities translate to infidelities is discussed in Sec. IV.

One can capture the physics of interest for Rydberg states with modified
eigenstates of Hydrogen atoms. The energy levels for a series of Rydberg
states is given as [28]

$E_{n}=-\frac{1}{2(n-\delta_{lj}(n))^{2}}=-\frac{1}{2(n^{*})^{2}},$ (1)

where the quantum defect $\delta_{lj}$ captures the deviation from the
hydrogenic states, $n^{*}$ is the _effective principal quantum number_ , $l,j$
are the orbital angular momentum and the total angular momentum numbers of the
valence electron, and atomic units are used.

The scaling of the relevant physical properties with $n$ is well understood
for alkali atoms. For instance, the lifetime due to spontaneous emission,
$\tau_{sp}$ of Rydberg states in alkali atoms scales as $(n^{*})^{3}$ [29],
whereas the BBR lifetime $\tau_{BBR}$, as well as the total lifetime
$(1/\tau_{sp}+1/\tau_{BBR})^{-1}$ (see Eq. 11) both scale as $(n^{*})^{2}$[30,
31].

For alkaline-earth atoms, the presence of two valence electrons can complicate
the physics [32], although most of the physical quantities relevant to our
study are straightforward to compute. The corresponding calculations are
carried out and detailed in Appendix A.

###  II.2 Dynamics and two-qubit gates

For a single qubit – with the $\ket{1}$ state coupled to the Rydberg state
$\ket{r}$ by a laser field – the Hamiltonian is given by

$H_{1q}=\frac{\Omega(t)}{2}(\ket{1}\bra{r}+\ket{r}\bra{1})-\Delta(t)\ket{r}\bra{r},$
(2)

where $\Omega$ is the Rabi frequency and $\Delta=\omega_{1,r}-\omega_{L}$ the
detuning of the laser respectively (with $\omega_{1,r}$ the frequency of the
transition and $\omega_{L}$ the laser frequency)— see Fig. 1. The Rabi
frequency $\Omega\propto\sqrt{P}\bra{r}ex\ket{1}$, where $P$ is the total
power of the driving laser and $\bra{r}ex\ket{1}$ is the radial dipole matrix
element (RDME) (see Eq. 8) between the clock state and the Rydberg state. In
this context, a pulse is the profile of $\Omega(t)$ and $\Delta(t)$ in time.

The Hamiltonian for a two-qubit system with both atoms coupled to the same
laser field is

$H_{2q}=H_{1q}\otimes I+I\otimes H_{1q}+V_{\mathrm{int}}\ket{rr}\bra{rr},$ (3)

where $V_{\mathrm{int}}$ is the interaction strength between two atoms excited
to Rydberg state $\ket{r}$. In our protocol we utilise the van der Waals
interaction, $V_{\mathrm{int}}=-C_{6}/R^{6}$, where $R$ is the interatomic
distance and $C_{6}$ a state-dependent coefficient.

The dynamics of the Hamiltonian in Eq. 3 can be employed to generate two-qubit
quantum gates. We work in the regime of the _Rydberg blockade_ – where the
interaction strength between the two atoms is much larger than the Rabi
frequency, such that an atom excited to state $\ket{r}$ suppresses the
excitation of adjacent atoms to $\ket{r}$ [11]. While several protocols assume
$V_{\mathrm{int}}=\infty$, this can lead to errors due to the finite blockade
strength of Rydberg states – hence, we include the finite blockade strengths
in our optimisation. For the Rydberg states considered in Sec. IV, we observe
no infidelity arising from the finite strengths, and consistently find gates
with $1-F\sim 10^{-5}$.

The general class of protocols enabled by Eq. 3 in the blockade regime has
been well studied, and is captured by the dynamics of the computational
states. The $\ket{0}$ state – and consequently, the $\ket{00}$ state – is dark
to the laser field and is not affected in our protocol. The dynamics of the
$\ket{01}$ and $\ket{10}$ states can thus be evaluated by considering the
atoms individually, and reducing them to two-level systems $\ket{1},\ket{r}$
with Rabi frequency $\Omega(t)$. Further, the dynamics of $\ket{01}$ and
$\ket{10}$ are identical if the pulse is _symmetric_ i.e., the same laser
field is applied to both the atoms, which is the case in our protocol. In the
limit of infinite blockade, the dynamics of the $\ket{11}$ state can also be
described by a two-level system of $\ket{11},\ket{b}$ for the bright state
$\ket{b}=(\ket{1r}+\ket{r1})/\sqrt{2}$ with Rabi frequency $\sqrt{2}\Omega(t)$
[12]. This description is valid as the large $V_{\mathrm{int}}$ suppresses
excitations to the $\ket{rr}$ state, and is valuable to understand how
entanglement is generated. However, in order to account for finite
$V_{\mathrm{int}}$, we will consider the three-level system
$\ket{11},\ket{b},\ket{rr}$.

The dynamics mentioned above lead to accumulation of phases on the states –
these are state-dependent, due to the different Rabi frequencies. By setting a
constraint on the final phases, and applying an additional single-qubit
rotation, one can realise CPHASE gates [15] characterised by the
transformation
$\ket{00}\to\ket{00},\ket{01}\to\ket{01},\ket{10}\to\ket{10},\ket{11}\to\mathrm{e}^{i\phi}\ket{11}$.
At $\phi=\pi$, the CZ gate is obtained.

To account for various sources of error, corrections can be made to Eqns. 2
and 3. The finite lifetime of the Rydberg state is accounted for by including
the term $H_{\mathrm{decay}}=-i\frac{\Gamma}{2}\ket{r}\bra{r}$ in Eq. 2, with
the total decay width $\Gamma$ defined in Eq. 11. This assumes that the decay
occurs only to states outside of the computational subspace, and thus slightly
overestimates the error [33]. For the
$5\mathrm{s}n\mathrm{s}^{3}\mathrm{S}_{1}$ Rydberg state in Sr, branching
ratios of the decay processes have been calculated, that verify the validity
of this assumption [25]. Deviations of the Rabi frequency and detuning,
$\delta\Omega,\delta\Delta$ are included by making the substitutions
$\Omega(t)\to\Omega(t)+\delta\Omega$ and $\Delta(t)\to\Delta(t)+\delta\Delta$
in Eq. 2.

Another question of interest for experiments is the choice of Rydberg state
that minimises the infidelity of a gate. This question has been considered in
some detail. For higher-lying states, we expect higher infidelity arising from
the sensitivity of $\ket{r}$ to stray electric fields [13], but a decrease in
the infidelity arising from decay of $\ket{r}$. The scaling of intrinsic gate
error with $n$, for the gate proposed by Jaksch _et al_ [11] was presented for
Rubidium [34]. However, how the physical quantities translate into errors is
protocol dependent and it is difficult to interpret the results in context of
the robust gates we will obtain, or of other recent gates proposed.

Further, as we consistently find high-fidelity pulses in the presence of
finite blockade strengths for the Sr Rydberg states considered, the discussion
by Saffman _et al._ [34] of mitigating this error by going to a higher $n$ –
or varying interatomic distances – does not apply here.

###  II.3 Variational optimisation

We now discuss the dynamics formulation of Sec. II.2 in the context of quantum
optimal control (QOC), towards variational optimisation.

As mentioned in Sec.I, time-optimal CZ and $\mathrm{C_{2}Z}$ gates – aiming to
find the fastest possible gate protocol – have been obtained using QOC for a
Rydberg atom platform [14]. In the rest of this work, we will compare and
contrast our robust gates to these specific time-optimal gates. Analytic
protocols which are robust to deviations of experimental controls have also
been proposed [35, 12] – it is thus a worthwhile question whether robustness
can be included in a variational optimisation framework.

Numerical robustness has been studied for two-qubit gates on the Rydberg
platform before [36, 37] – and while these studies demonstrate excellent
robustness is possible, we note certain differences. For instance, robust
gates were obtained with single-site addressability, for atoms with two-photon
transitions to $\ket{r}$ [36] – the control parameters of the obtained pulses
oscillate on a time scale that is difficult to implement experimentally. In
comparison to these pulses, the pulses we find are smooth, and do not require
single-site addressability, significantly easing experimental implementation
[35]. As these are optimised for one-photon transitions in Sr, the fidelities
$F$ are inherently higher, but also cannot be directly compared to fidelities
of the two-photon gate as the experimental setups considered differ.

Towards the optimisation, we employ the augmented Lagrangian trajectory
optimiser (ALTRO) method [38], used before to study robust QOC for
superconducting qubits [39]. Notably, ALTRO can satisfy constraints up to a
specified tolerance by using an augmented Lagrangian method (ALM) to
adaptively adjust Lagrange multiplier estimates for the constraint functions.
This makes the optimiser well suited for our problem, which considers multiple
constraints. In this method, one formulates the problem to be solved as a
trajectory optimisation problem [39]. The cost function we use is

$J(\mathbf{c})=(\mathbf{x}_{N}-\mathbf{x}_{T})^{\intercal}Q_{N}(\mathbf{x}_{N}-\mathbf{x}_{T})+\sum_{k=1}^{N-1}\mathbf{c}_{k}^{\intercal}R_{k}\mathbf{c}_{k},$
(4)

where the augmented control vector $\mathbf{c}$ encapsulates the experimental
controls of our system, $\mathbf{x}$ is the augmented state vector consisting
of the variables the control can influence – such as the state populations and
phases, $\mathbf{x}_{T}$ denotes the target augmented state, $\mathbf{x}_{N}$
is the augmented state vector at the final time step and the sum runs over
time steps of the discretised problem. $R_{k}$ and $Q_{N}$ are diagonal weight
matrices on the control at each time step $k$, and on the augmented state at
the final time step $N$, respectively. Such a cost function can be efficiently
computed with ALTRO [38, 39]. The number of time steps $N=100$ for all the
results presented in this paper, and we observe negligible errors due to this
discretisation.

With one notable exception discussed later on, the augmented controls and
states used in this work are

$\mathbf{c}=\begin{pmatrix}\dot{\Omega}\\\\[7.0pt]
\dot{\Delta}\end{pmatrix},\;\;\;\hat{\mathbf{x}}=\begin{pmatrix}\partial_{\Omega}\Psi\\\\[6.5pt]
\partial_{\Delta}\Psi\\\\[6.5pt] \phi\\\\[6.5pt] P_{\mathrm{tot}}\\\\[6.5pt]
\overline{T}_{\ket{r}}\end{pmatrix},$ (5)

where $\Psi=(\Psi^{01},\Psi^{11})$ are the time-dependent wavefunctions
associated with initial computational basis states $\ket{01}$ and $\ket{11}$,
$\phi$ is the CPHASE gate angle as introduced in Sec. II.2, $P_{\mathrm{tot}}$
refers to the total population in the computational basis states and
$\overline{T}_{\ket{r}}$ is the average integrated Rydberg lifetime (see Eq.
17). We use the notation $\dot{a}=da/dt$. $\hat{\mathbf{x}}$ is the reduced
state vector and consists of the terms in $\mathbf{x}$ that are penalised in
Eq. 4 – for the remaining terms, the corresponding element in $Q_{N}$ is set
to zero. The complete augmented state vector $\mathbf{x}$ is given and
motivated in Appendix B. In Eq. 5, it is implicit that $\mathbf{c}$ has
different elements $\mathbf{c}_{k}$ at each time step $k$. The dynamics of the
system are governed by the time-dependent Schrödinger equation.

Below, we motivate our formulation of the problem and the individual terms in
Eq. 5. As smooth pulses are more feasible from an experimental point of view
[35], we set $\dot{\Omega}$ and $\dot{\Delta}$ as the augmented controls –
instead of $\Omega$ and $\Delta$ – in order to penalise pulses with
discontinuous jumps, with the weights $R_{k}$ in Eq. 4.

For a full description of the dynamics, we would have to consider the
wavefunctions associated with all basis states,
$\Psi^{00},\Psi^{01},\Psi^{10}$ and $\Psi^{11}$. We can, however, leave out
$\Psi^{00}$ as this state is not coupled to the Rydberg states in our
protocol. Furthermore, as the protocol is symmetric, $\Psi^{01}$ and
$\Psi^{10}$ adhere to the same dynamics – thus we can reduce the computational
expense by optimising for only one of the two basis states in this formalism.

There are various ways to encode robustness into the optimisation objective
[36, 37]. We do so by including $\partial_{\Omega}\Psi$ and
$\partial_{\Delta}\Psi$ as augmented states, in order to penalise the
sensitivity of the desired wavefunction to deviations of the experimental
parameters $\Omega$ and $\Delta$ respectively. This method – termed the
derivative method – is intuitive and computationally inexpensive. For single-
qubit gates on the superconducting platform, it achieves significantly higher
robustness at lower computational cost compared to other available methods
[39].

$\phi_{\ket{11}}$, and $\phi_{\ket{10}}=\phi_{\ket{01}}$ denote the phases
accumulated by the computational states at the end of the gate. Enforcing the
condition

$\phi=\phi_{\ket{11}}-2\phi_{\ket{01}}=(2m+1)\pi,$ (6)

for $m$ an integer, results in a CZ gate [15], up to one-qubit rotations. High
fidelity in the noiseless case is ensured by including $\phi$ in $\mathbf{x}$
and $\pi$ as the corresponding element in $\mathbf{x}_{T}$ – such that the
condition in Eq. 6 is ensured – and by including $P_{\mathrm{tot}}$ (see Eq.
15) in $\mathbf{x}$ such that the controls return the computational basis
states to themselves (up to the acquired phase), in the final augmented state
$x_{N}$.

$\overline{T}_{\ket{r}}$ captures the total time spent in the Rydberg state,
averaged across the computational basis states – and Sec. III uses this as a
metric to compare decay errors from $\ket{r}$ between different protocols. For
a given configuration, minimising $\overline{T}_{\ket{r}}$ reduces the error
due to spontaneous emission and BBR.

To ensure that the solver does not trade optimisation of $\phi$ and
$P_{\mathrm{tot}}$ with other terms in $\mathbf{x}$, we also include the
appropriate constraints for these terms (treated with the ALM method) in the
variational optimisation. Constraints on Rabi frequency and detuning are also
implemented, and will be discussed in Sec. III.

To the best of our knowledge, our study is the first to use the derivative
method [39] to study robustness both on the Rydberg platform and for two-qubit
gates – we expect use case scenarios for robust state preparation, multiqubit
gates and quantum algorithms inspired by QOC [40].

##  III Constraint-specific robust gates

(a) Protocol A : $T=10/\Omega_{\mathrm{max}},|\Delta(t)|\leq
1.25\Omega_{\mathrm{max}}$

(b) Protocol B : $T=15/\Omega_{\mathrm{max}},|\Delta(t)|\leq
3.25\Omega_{\mathrm{max}}$

Figure 2: Two protocols to implement robust CZ gates. We highlight the
smoothness of the pulses, the fast time scales (compared to decay from
$\ket{r}$, see Eq. 12), and realistic constraints on detuning, ensuring that
the gates can be implemented on current experimental setups.

We now discuss the results of the optimisation as detailed in Sec. II.3. We
will quantify the performance of our gates by computing the Bell state
fidelities $F$. As discussed, in the absence of sources of error – except
finite $V_{\mathrm{int}}$, which is always included in our optimisation –
tight constraints are implemented to ensure infidelity of the Bell state
$(1-F)\sim 10^{-5}$. While the results of this section can be understood by
expressing quantities in units of the maximum Rabi frequency
$\Omega_{\mathrm{max}}$, we set $\Omega_{\mathrm{max}}=2\pi\times 6.8$ MHz –
the Rabi frequency demonstrated for the $n=61$ state in Sr [6] – to provide a
better understanding of the physical values and time scales. We set the
lifetime of $\ket{r}$ as $\tau_{n=61}=96.5\mathrm{\mu s}$ – the calculation of
the lifetime is detailed in Appendix A. Unless specified otherwise, the
fidelities in this section account for decay from the Rydberg state and finite
blockade strengths.

We use the ARC3.0 library [41] to compute the van der Waals coefficient
$C_{6}=-2\pi\times 181$GHz $\mu\mathrm{m}^{6}$ – we find the $n=61$ state is
well-resolved above $R\approx 2\mu$m. The interatomic axis is taken
perpendicular to the quantisation axis of $\ket{r}$. We set $R=3.5\mu$m in the
simulation, both for the optimisations and fidelity calculations – an
experimentally feasible value – and find negligible errors from the finite
blockade strength. The physical parameters used in our simulation are
collected in Table 1. To provide context, we will also compare our results to
the time-optimal pulses [14, 15].

To illuminate our method, the role of constraints and relevant parameters (as
introduced in Sec. II.3), we present two gates robust to deviations of Rabi
frequency $\delta\Omega$ – Protocols A and B – in Fig. 2.

Figure 3: Fidelities $F$ of the protocols from Fig. 2 against variations in
$\delta\Omega$, plotted in the absence of other error sources. Compared to the
time-optimal pulses (yellow, dashed), we observe significantly enhanced
robustness for both protocols that we propose. Protocol A (brown, dotted)
retains $F\sim 0.996$ and B (in blue) retains $F>0.9995$ for the range of
variations plotted here. Inset – we now consider decay from $\ket{r}$ and look
at small Rabi variations – for $\delta\Omega/\Omega_{max}\sim 0.5\%$ both
protocols have higher fidelity than time-optimal pulses.

The protocols are differentiated by the implemented constraints, gate times
and states included in Eqns. 4 and 5. Protocol A implements the constraint
$|\Delta(t)|\leq 1.25\Omega_{\mathrm{max}}$, has a total pulse duration of
$T_{A}=10/\Omega_{\mathrm{max}}(\approx 0.23\mathrm{\mu s})$ and accounts for
robustness to deviations of detuning $\delta\Delta$ (see Sec. II.3). This time
scale is comparable to the time-optimal protocols
($T_{\mathrm{opt}}=7.61/\Omega_{\mathrm{max}}$) [14] – and thus benefits from
the associated mitigation of errors that propagate in time. This protocol is
intended to be implementable for current experiments. It places modest
requirements on the experimental controls and is near time-optimal.

For protocol B, we relax the constraints to $|\Delta(t)|\leq
3.25\Omega_{\mathrm{max}}$, set a pulse duration of
$T_{B}=15/\Omega_{\mathrm{max}}$ and do not include $d\Psi/d\Delta$ in the
optimisation. Protocol B is intended to probe the limits of our optimisation –
to explore how robust CZ gates can be made to $\delta\Omega$ deviations, while
still operating on a short timescale. During our optimisation, we noticed that
the optimiser tends to reach high values of $\Delta$. Hence, even for protocol
B we find it useful to set bounds on $|\Delta(t)|$ – significant robustness is
already achieved for protocol B with bounds set to
$|\Delta|<3.25\Omega_{\mathrm{max}}$, and we notice negligible improvements
over this protocol on relaxing the constraints further.

From Fig. 3, we note significant improvements in robustness of both protocols
A and B compared to the time-optimal pulses.

At $\delta\Omega=\pm 0.05\Omega_{\mathrm{max}}$, the time-optimal pulse
fidelity drops to $F_{\mathrm{opt}}=0.9896$, while the fidelity of protocol A,
$F_{A}=0.9964$, and the fidelity of protocol B, $F_{B}=0.9990$ – these values
also account for decay from $\ket{r}$. Hence, both protocols allow us to
achieve high-fidelity gates in the presence of significant $\delta\Omega$
deviations and decay.

We find that pulses A and B still offer a significant advantage for minor
deviations of $\Omega$ – at $\delta\Omega=5\times
10^{-3}\Omega_{\mathrm{max}}$ ($0.5\%$ of the maximum Rabi frequency) both
protocols A and B have higher fidelity $F\sim 0.999$ than the time-optimal
pulse (see inset of Fig. 3). The robustness of the protocols – in the presence
of decay from $\ket{r}$ – to $\delta\Omega$ and $\delta\Delta$ deviations is
studied in Fig. 4.

For the time-optimal gates, the average integrated Rydberg lifetime
$\overline{T}_{\ket{r}}^{\mathrm{opt}}=3.86/\Omega_{\mathrm{max}}$ [15].
Protocols A and B respectively spend 24%
($\overline{T}_{\ket{r}}^{A}=4.79/\Omega_{\mathrm{max}}$) and 45%
($\overline{T}_{\ket{r}}^{B}=5.61/\Omega_{\mathrm{max}}$) more time in the
Rydberg state, than the time-optimal pulses. This is a worthwhile comparison
as the time-optimal pulses also minimise $\overline{T}_{\ket{r}}$ [15] and
with that, the losses due to decay of $\ket{r}$.

Principal quantum number $n$ | 61 |
---|---|---
Rabi frequency $\Omega_{\mathrm{max}}$ | 2$\pi\times 6.8\;$ | MHz
vdW coefficient $C_{6}$ | 2$\pi\times-181$ | $\mathrm{GHz}\;\mu\mathrm{m}^{6}$
Rydberg lifetime $\tau$ | 96.5 |  $\mu$s
Interatomic distance $R$ | 3.5 |  $\mu$m
Table 1: Physical parameters considered for Sr

It follows that one trades a reduced lifetime in $\ket{r}$ for robustness –
however, we find that robust pulses with fidelities $F\sim 0.999$ can be
obtained over a wide range of control deviations for current Sr experiments.

While two protocols were presented here, a large class of pulses can be
generated with the techniques presented in Sec. II.3. For example, we observed
that pulses with time $T_{\mathrm{opt}}<T<T_{B}$ – intermediate between the
time-optimal pulse and protocol B – can be found as well, with both the
robustness and $\overline{T}_{\ket{r}}$ intermediate between the two
protocols. This presents a way to tailor pulses to a given experimental setup
– thus minimising infidelity, conditional on the level of deviations present.

(a)

(b)

Figure 4: Contour plot of fidelity $F$ to investigate the robustness of the
proposed pulses against $\delta\Omega,\delta\Delta$.
(a) Protocol A is slightly more robust to $\delta\Delta$ than (b) protocol B,
whereas B is significantly more robust to $\delta\Omega$. Both pulses lead to
high-fidelity ($F\sim 0.999$) gates across a wide range of deviations.

##  IV Optimal Rydberg state choice

In this section we identify the optimal Rydberg state for our setup and show
that this can further minimise the infidelity of our gates. Sr atoms have long
been established for use in optical lattice clocks, and demonstrate accuracy
and precision above the Caesium standard [42, 43]. A technique crucial to the
operation of these clocks is _magic wavelength_ trapping [44]. This technique
aims to eliminate the differential light shift caused by the trap laser, by
identifying wavelengths that cause an identical shift on both clock transition
states. For the clock transition introduced in Sec. II.1, the magic wavelength
$\lambda_{m}=813.4$nm [45] is well understood, and often used in experiments.

Magic traps at this wavelength have also been employed in a quantum
information setting [25, 6], as this allows for long-lived qubits. For this
trap, the Rydberg state is anti-trapped [25], and this can lead to loss.
One strategy to avoid such loss is to switch off the trap for the duration of
the gate, then switch it on for recapture [6]. The anti-trapping behaviour is
then mitigated because the spatial wave function will evolve under a free
potential instead of a concave Gaussian potential. In this section, we adopt
this strategy and thus do not consider trap physics (see Appendix A for
further discussion).
In this section, we set again $\Omega_{n=61}=2\pi\times 6.8$MHz. For a fixed
power of the laser, $\Omega\propto(n^{*})^{-3/2}$ [46], and we use this
relation to obtain Rabi frequencies at other values of $n$. The fundamental
source of infidelity is the decay from $\ket{r}$ – characterised by the decay
rate $\Gamma$ (see Eqn. 11). Although $\Omega$ decreases with $n$, the ratio

$\frac{\Omega}{\Gamma}\propto\frac{(n^{*})^{-3/2}}{(n^{*})^{-3}+k(n^{*})^{-2}},$
(7)

for a constant $k$, increases with $n$. This ratio captures the two competing
time scales of our system – its increase should thus correspond to an increase
in $F$. Figure 5(a) reports fidelities of protocol A for the range
$n\in\\{40,\,\dots,\,120\\}$ and this increase is indeed observed. For
$n>120$, the fidelity increases slowly. At $n=200$, we obtain $F=0.9997$,
compared to $F=0.9995$ at $n=100$.

Stray electric fields in the experiment have been identified as a potential
source of infidelity [47, 48, 27, 13]. As the polarisability of
$\ket{r},\alpha_{s}\propto(n^{*})^{7}$ [31], the associated DC Stark shifts
scale in a dramatic manner with $n$ – this is further described in Appendix A.
Thus, there are two competing effects that scale with $n$ – while the lifetime
of $\ket{r}$ increases, reducing the infidelity due to decay, the infidelity
due to stray fields increases. This competition is illustrated in Fig. 5(a).

(a)

(b)

Figure 5: (a) The simulated fidelity $F$ of protocol A (see Fig. 2(a)) plotted
against the principal quantum number $n$, for E = 0 mV/cm (black, diamond), E
= 1 mV/cm (violet, plus) and E = 5 mV/cm (red, stars). As expected from
theory, $F$ increases monotonically for the case with no background fields.
(b) plots the maximum fidelity (blue), and the $n$ at which this fidelity is
obtained (green, plus), against the magnitude of the electric field
$E/(\mathrm{mVcm^{-1}})\in[0.4,10.0]$. $F$ is also plotted at $n=61$ (orange).

Figure 5(a) also plots fidelities in the presence of residual fields with
magnitude $E=5\mathrm{mV/cm}$ – this corresponds to significant field
cancellations that have been experimentally demonstrated [47]. In this case,
$F_{\mathrm{max}}=0.9990$ is obtained at $n=62$. Looking at another case, with
$E=1\mathrm{mV/cm}$, $F_{\mathrm{max}}=0.9995$ at $n=97$.

Figure 5(b) plots $F_{\mathrm{max}}$ and the optimal $n$ for protocol A,
against the residual field strength $E$. Here, we also plot the fidelity of
the $n=61$ state $F_{n=61}$, and note the intersection between the two
fidelity curves at $E_{\mathrm{int}}=5.3$mV/cm. At $E<E_{\mathrm{int}}$,
$F_{n=61}$ is limited by the lifetime of $\ket{r}$, and at
$E>E_{\mathrm{int}}$ by the DC Stark shifts (see Eq. 13).

We conclude that the characterisation of residual electric fields in an
experiment allows the identification of the optimal Rydberg state for
operation. On a NISQ experiment, this allows for $F\sim 0.999$. Higher
background field cancellation can achieve $F\sim 0.9995$, and a combination of
higher laser power and weaker fields would be required for $F\sim 0.9999$.

##  V Conclusion

In this study, we have devised fast robust CZ gates – with time scales much
shorter than the lifetime of the Rydberg states involved, and comparable to
the time-optimal gates – and investigated the associated errors. We found that
fidelities $F>0.999$ are achievable over a wide range of deviations in the
laser field coupling $\delta\Omega$ and the detuning $\delta\Delta$. We
further identified optimal Rydberg states to maximise fidelities on a NISQ
system.

To the best of our knowledge, this work is the first to carry out a detailed
analysis towards the identification of optimal Rydberg states – we also showed
in Sec. IV that such an analysis can lead to significant gains in two-qubit
fidelities, when combined with experimental mitigation of residual electric
fields.

These two results pave the way towards realising gates with higher fidelities
in a Sr experimental setup. The techniques used in our analysis can also be
extended to experiments with other neutral atoms.

Future studies could look at identification of optimal Rydberg states for
different computation schemes – such as VQOC [40], which would require a
careful investigation of the trapping of Rydberg states in the tweezer (over
the duration of the quantum circuit). Optimising gates to be robust against
deviation of the blockade strength is also a worthwhile area of research [14],
and the approach presented in our work can be adapted to consider such
optimisations. Additionally, it would be interesting to look at the generation
of larger multiqubit gates using these techniques. In the interest of
tailoring pulses to specific experiments, rise times of the coupling laser
field could be directly incorporated into the optimisation framework.

On the experimental side, further techniques to characterise and mitigate
background fields could lead to the use of higher-lying Rydberg states,
towards even higher fidelities $F\sim 0.9999$ that are conducive to quantum
error correction. The verification of our findings on an experimental setup
would also greatly benefit the substance of this work – for instance, through
an experiment where laser intensity noise can be controlled, in order to
demonstrate the robustness of our pulses.

_During our research, we became aware of related work being carried out to
demonstrate robust pulses for two-qubit gates on the Rydberg platform[49]_.

###### Acknowledgements.

We acknowledge valuable discussions with Ivo Knottnerus, Alex Urech, Jasper
Postema and Sven Jandura. Various computational libraries across Python, C++,
Fortran and Julia made this work possible – notable are QuTiP [50] and
Altro.jl [38] for the variational optimisation and simulation. ARC3.0 [41],
FGH [51], NUMER [52] and the MQDT library from C. Vaillant [32] were used for
the computation of atomic properties. This research has received funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement number 955479. S.K. and R.K.
acknowledge funding from the Dutch Ministry of Economic Affairs and Climate
Policy (EZK) as part of the Quantum Delta NL programme, and from the
Netherlands Organisation for Scientific Research (NWO) under Grant No.
680-92-18-05.

##  Appendix A Calculation of polarisabilities and lifetimes

In this appendix we detail the calculation of atomic properties as introduced
in Sec. II.1.

First, we employ quantum defect theory (QDT) [28] towards the calculation of
the lifetimes of the ${}^{3}\mathrm{S}_{1}$ Rydberg states of Sr. Our
treatment is inspired by the work of Vaillant _et al._ [32] – we point the
reader to this reference for an in-depth description of QDT towards computing
various atomic properties.

We will use atomic units in this section. $L,S,J$ refer to the total orbital
angular momentum, the total spin and the total angular momentum of an atom –
the corresponding lowercase letters $l,s,j$ refer to the momentum of
individual electrons.

The starting point of QDT is the Coulomb approximation: for high lying states,
the wavefunction of the active electron is far extended and the potential it
experiences can be regarded as a Coulombic potential for large electron
distances from the core, $x$. QDT then provides a numerical recipe to
calculate wavefunctions, and thus to compute the dipole matrix elements – that
capture the atomic physics relevant to our problem, as discussed in the main
text.

Dipole matrix elements can be split into angular and the radial dipole matrix
elements (RDMEs). A dipole matrix element between two states with quantum
numbers $n_{1},l_{1},m_{1}$ and $n_{2},l_{2},m_{2}$ (for $m$ the magnetic
quantum number) respectively can be written as

$\bra{n_{1}l_{1}m_{1}}e\mathbf{x}\ket{n_{2}l_{2}m_{2}}=\Braket{l_{1}m_{1}}{\frac{\mathbf{x}}{x}}{l_{2}m_{2}}\bra{n_{1}l_{1}}ex\ket{n_{2}l_{2}},$
(8)

where $x$ is the magnitude of the atomic distance vector of the electron
$\mathbf{x}$, $e$ the electron charge. One begins by calculating RDMEs,
following which the angular components can be incorporated.

For the calculation of the lifetime of the Rydberg state, one must consider
both spontaneous emission and black body radiation (BBR) – at 300K in our
setup. For transitions between states $a$ and $b$, the spontaneous decay width
is [32]

$\Gamma_{a,b}^{sp}=\frac{4\alpha}{3c^{2}}|\omega_{a,b}|^{3}|\bra{\Psi_{b}}e\mathbf{x}\ket{\Psi_{a}}|^{2},$
(9)

where $\omega_{a,b}=E_{b}-E_{a}$ is the difference in energies, $\alpha$ is
the fine-structure constant and $c$ the speed of light in vacuum.

The lifetime due to spontaneous emission for a state $a$, then, is given by

$\tau_{a}^{sp}=\sum_{b}\frac{1}{\Gamma_{a,b}^{sp}}.$ (10)

The effects of blackbody radiation can be incorporated with the following
equation [53],

$\displaystyle\Gamma_{a}=\sum_{b,E_{a}>E_{b}}\Gamma_{a,b}^{sp}\left(1+\frac{1}{\mathrm{e}^{|\omega_{a,b}|/k_{B}T}-1}\right)$
(11)
$\displaystyle+\sum_{b,E_{a}<E_{b}}\frac{\Gamma_{a,b}^{sp}}{\mathrm{e}^{|\omega_{a,b}|/k_{B}T}-1},$

where the first term accounts for the states lying below $a$ and the second
term for the higher lying states, for which only BBR contributes to the width.
The total lifetime is then $\tau_{a}=\Gamma_{a}^{-1}$.

This treatment is in general valid for alkali atoms, and the situation for Sr
is complicated by the presence of two valence electrons in the outermost shell
– for instance, the lifetimes of the ${}^{1}S_{0}$ and ${}^{1}D_{2}$ series in
Sr are strongly quenched because of the mixing of doubly-excited low lying
states (termed _perturbers_) with the desired singly-excited Rydberg states
[32]. Such perturbers can lead to reduced Rydberg state lifetimes and energy
shifts, and thus a careful approach is required in the calculation of physical
properties.

For such a general treatment one needs to consider multichannel quantum defect
theory (MQDT), as treated in the main reference, Vaillant _et al._ [32]. In
this context, a channel refers to a specific configuration of the core and the
active electron, and perturbers imply a contribution of multiple channels in
the atomic properties.

The ${}^{3}\mathrm{S}_{1}$ series that we use as the Rydberg state is
effectively unperturbed [32, 54, 55] – there is a negligible mixture of low-
lying doubly excited states – and thus we do not expect a significant
quenching of lifetimes. This also simplifies considerably the theoretical
treatment of the lifetimes.

Consider the MQDT equation (derived in [32]) for the natural width of a state
$a$,

$\begin{aligned}
\Gamma_{a}^{sp}=\frac{4\alpha}{3c^{2}}\sum_{b}|\omega_{a,b}|^{3}\Bigr{[}\sum_{i,j}(-1)^{l_{1,a}^{(i)}+\mathrm{max}(l_{2,a}^{(i)},l_{2,b}^{(j)})+S_{a}^{(i)}}\overline{A}_{ai}\overline{A}_{bj}\\\
\times\sqrt{\mathrm{max}(l_{2,a}^{(i)},l_{2,b}^{(j)})(2L_{b}^{(j)}+1)(2J_{b}^{(j)}+1)(2L_{a}^{(i)}+1)}\\\
\times\begin{Bmatrix}J_{b}^{(j)}&1&J_{a}^{(i)}\\\
L_{a}^{(i)}&S_{a}^{(i)}&L_{b}^{(j)}\end{Bmatrix}^{2}\begin{Bmatrix}L_{b}^{(j)}&1&L_{a}^{(i)}\\\
l_{2a}^{(i)}&l_{1a}^{(i)}&l_{2b}^{(j)}\end{Bmatrix}^{2}R_{b,a}^{(ij)}\Bigr{]}^{2}\end{aligned},$
(12)

where $i,j$ are summation indices over the considered channels for states
$a,b$ respectively, $\overline{A}_{ai}$ and $\overline{A}_{bj}$ are the
respective _channel fractions_ (that quantify the mixing of the perturbed
states), the curly brackets represent Wigner $6j$ symbols and $R_{b,a}^{(ij)}$
is the RDME corresponding to states $a$ and $b$, in channels $i$ and $j$.

Figure 6: Lifetimes (including spontaneous emission and BBR at 300K) for Sr as
computed with quantum defect theory (QDT). We find excellent fits to a
function of the form $f(n)=1/(A(n^{*})^{-2}+B(n^{*})^{-3})$ (for effective
principal quantum number $n^{*}$) as is expected from scaling laws [31].

The lifetime due to spontaneous emission of state $a$ is then
$\tau_{a}^{sp}=(\Gamma_{a}^{sp})^{-1}$. Given that the ${}^{3}\mathrm{S}_{1}$
state is effectively unperturbed, it is worth asking whether QDT (the ’single-
channel’ equivalent of MQDT) can be used. Further, the states that the
${}^{3}\mathrm{S}_{1}$ state decays to – the ${}^{3}\mathrm{P}_{0,1,2}$ states
– are still perturbed, but lack enough experimental data to carry out a full
MQDT analysis [32].

We argue that a QDT approach in this case provides a lower bound on the total
lifetimes. This is because of three reasons. Firstly, quenching of the
lifetimes for divalent atoms arises due to significant radial matrix elements
between the doubly-excited states – as the ${}^{3}\mathrm{S}_{1}$ states do
not have these states mixed in, such contributions to the width are
negligible. Secondly, the single-excitation wavefunctions can be calculated –
as for Rubidium in [32], to excellent agreement with experimental lifetimes –
with QDT. Thirdly, $R_{b,a}$ between a doubly-excited state (mixed in the
${}^{3}\mathrm{P}_{0,1,2}$ state) and the singly-excited states is zero. Thus,
we use QDT to obtain the lower bounds on the lifetimes in Fig. 6– further
studies would have to involve both theory and experiment to obtain MQDT
characterisations of the ${}^{3}\mathrm{S}_{1}$ state lifetimes.

(a)

(b)

Figure 7: Scalar polarisabilities $\alpha_{s}$ for the
$5\mathrm{s}n\mathrm{s}\;^{3}\mathrm{S}_{1}$ series. (a) The ranges
$n\in\\{60,\,\dots,\,120\\}$ and
$|\mathbf{E}|/(\mathrm{mV\;cm^{-1}})\in[0.0,10.0]$ are plotted. The values
obtained by diagonalisation (green, circles) show agreement with the fitting
form $f(n)=A(n^{*})^{7}$ (black, line), as expected from theory [25]. (b) DC
Stark shifts for the $n=81$ state (circled in red on (a) ), from which
$\alpha_{s}$ can be deduced – see Eq. 13.

In Fig. 7 we diagonalise the Hamiltonian in the presence of electric fields,
to obtain the DC Stark shifts and corresponding polarisabilities for the
$5\mathrm{s}n\mathrm{s}^{3}\mathrm{S}_{1}$ Rydberg series. For the considered
ranges, we find excellent agreement to models of scalar polarisability – that
is,

$\frac{\delta\Delta_{DC}}{2\pi}=-\frac{1}{2}\alpha_{S}|\mathbf{E}|^{2},$ (13)

and we extract the polarisability from fits of the DC Stark shift to the
electric field ranges in the inset of Fig. 7. The ARC3.0 library [41] was used
for these calculations.

We now briefly discuss the role of optical dipole traps during the gate
operation. Although the trap can be switched off during the pulse (see Sec.
III), the atom will still expand under the free potential and eventually leave
the trap for a long enough duration [56]. There are also other challenges when
switching off the trap. The blinking on and off of the traps can lead to
heating of the atoms and thus heat the entire qubit array. On a sequential
gate-based platform, this limits the depth of the gate circuit.

Another possibility is to simply leave the trap on for the Rydberg excitation.
It is often assumed that the anti-trapping will then lead to exponential loss
of the atoms in time [15], however, experimentally it is found that this is
not the case [6, 57, 49]. More research in recapture rates depending on the
status of the trap needs to be done to shed light on this discussion.

##  Appendix B QOC: State vectors and metrics

In this appendix, we clarify the variational implementation of the gate
dynamics problem with ALTRO [38]. The complete augmented state vector is given
by

$\mathbf{x}=\begin{pmatrix}\Psi,&\partial_{\Omega}\Psi,&\partial_{\Delta}\Psi,&\Omega,&\Delta,&\phi,&P_{\mathrm{tot}},&\overline{T}_{\ket{r}}\end{pmatrix}^{\intercal},$
(14)

where $\mathbf{a}^{\intercal}$ refers to the transpose of a vector
$\mathbf{a}$. A reduced vector $\hat{\mathbf{x}}$ was presented in the main
text. $\mathbf{x}$ includes all the terms that the controls $\mathbf{c}$ can
influence over the pulse duration, and $\hat{\mathbf{x}}$ consists of the
terms in $\mathbf{x}$ penalised in the cost function of Eq. 4.

We choose a _multi-state transfer_ approach [39] – the states are evolved
individually under the same pulse, which ALTRO then optimises. The population
of a state $\beta$, $P_{\beta}=\bra{\beta}\Psi\ket{\beta}$ and we define the
total population as

$P_{\mathrm{tot}}=P_{00}+P_{01}+P_{10}+P_{11}=2P_{01}+P_{11}+1,$ (15)

where $P_{00}=1$, and $P_{01}=P_{10}$ (see Sec. II.3). We set the constraint
$P_{\mathrm{tot}}=4$ at the final time step $N$ to ensure that the
computational states are returned to themselves at the end of the protocol.

The time-integrated probability of being in the Rydberg state $\ket{r}$, with
an input computational state $\beta$ [15],

$T_{\ket{r}}^{\beta}=\int_{0}^{T_{\mathrm{gate}}}(\Psi^{\beta})^{*}\,n\,\Psi^{\beta}dt,$
(16)

where $n=\ket{r}\bra{r}\otimes I+I\otimes\ket{r}\bra{r}$ and $\Psi^{\beta}$
the time-dependent wavefunction associated with $\beta$ (see Sec. II.3). The
average integrated Rydberg lifetime, then,

$\overline{T}_{\ket{r}}=\frac{1}{3}(2T_{\ket{r}}^{01}+T_{\ket{r}}^{11}),$ (17)

where the $\ket{00}$ state is not included, as it does not contribute to
infidelity in our protocol.

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