††thanks: A.J.P. and L.R.B.P contributed equally to this work.††thanks: A.J.P.
and L.R.B.P contributed equally to this work.

# Sub-millisecond Entanglement and iSWAP Gate between Molecular Qubits

Lewis R.B. Picard  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Department of Chemistry and Chemical Biology,
Harvard University, Cambridge, Massachusetts 02138, USA  Harvard-MIT Center
for Ultracold Atoms, Cambridge, Massachusetts 02138, USA  Annie J. Park
Department of Chemistry and Chemical Biology, Harvard University, Cambridge,
Massachusetts 02138, USA  Department of Physics, Harvard University,
Cambridge, Massachusetts 02138, USA  Harvard-MIT Center for Ultracold Atoms,
Cambridge, Massachusetts 02138, USA  Gabriel E. Patenotte  Department of
Physics, Harvard University, Cambridge, Massachusetts 02138, USA  Department
of Chemistry and Chemical Biology, Harvard University, Cambridge,
Massachusetts 02138, USA  Harvard-MIT Center for Ultracold Atoms, Cambridge,
Massachusetts 02138, USA  Samuel Gebretsadkan  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Department of Chemistry and
Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA
Harvard-MIT Center for Ultracold Atoms, Cambridge, Massachusetts 02138, USA
David Wellnitz  JILA, National Institute of Standards and Technology and
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado
80309, USA  Ana Maria Rey  JILA, National Institute of Standards and
Technology and Department of Physics, University of Colorado, Boulder,
Colorado 80309, USA  Center for Theory of Quantum Matter, University of
Colorado, Boulder, Colorado 80309, USA  Kang-Kuen Ni<EMAIL_ADDRESS>Department of Chemistry and Chemical Biology, Harvard University, Cambridge,
Massachusetts 02138, USA  Department of Physics, Harvard University,
Cambridge, Massachusetts 02138, USA  Harvard-MIT Center for Ultracold Atoms,
Cambridge, Massachusetts 02138, USA

###### Abstract

Quantum computation (QC) and simulation rely on long-lived qubits with
controllable interactions. Early work in quantum computing made use of
molecules because of their readily available intramolecular nuclear spin
coupling and chemical shifts, along with mature nuclear magnetic resonance
techniques [1, 2]. Subsequently, the pursuit of many physical platforms has
flourished. Trapped polar molecules have been proposed as a promising quantum
computing platform, offering scalability and single-particle addressability
while still leveraging inherent complexity and strong couplings of molecules
[3, 4, 5, 6, 7]. Recent progress in the single quantum state preparation and
coherence of the hyperfine-rotational states of individually trapped molecules
allows them to serve as promising qubits [8, 9, 10, 11, 12, 13, 14], with
intermolecular dipolar interactions creating entanglement [15, 16]. However,
universal two-qubit gates have not been demonstrated with molecules. Here, we
harness intrinsic molecular resources to implement a two-qubit iSWAP gate
using individually trapped $X^{1}\Sigma^{+}$ NaCs molecules. We characterize
the innate dipolar interaction between rotational states and control its
strength by tuning the polarization of the traps. By allowing the molecules to
interact for 664 $\mu$s at a distance of 1.9 $\mu$m, we create a maximally
entangled Bell state with a fidelity of 94(3)%, following postselection to
remove trials with empty traps. Using motion-rotation coupling, we measure
residual excitation of the lowest few motional states along the axial trapping
direction and find them to be the primary source of decoherence. Finally, we
identify two non-interacting hyperfine states within the ground rotational
level in which we encode a qubit. The interaction is toggled by transferring
between interacting and non-interacting states to realize an iSWAP gate,
which, together with single-qubit rotations, forms a universal set of gates.
We verify the gate performance by measuring its logical truth table.

##  I Introduction

Foundational work three decades ago used molecular systems with nuclear
magnetic resonances for proof-of-concept quantum computation [1, 2],
transforming theoretical ideas into early experiments [17, 18]. These
platforms enlisted the accessible complexity of molecules, using intrinsic
nuclear spins as qubits and natural spin-spin interactions to implement two-
qubit gates. However, the platform was difficult to scale and lacked true
entanglement [19] because the molecules were dissolved in a solvent at room
temperature and subjected to ensemble averaging. Subsequent developments in
quantum computing have shifted toward individually controllable systems with
simpler structures, such as trapped ions [20], neutral atoms [21], and
superconducting circuits [22]. Meanwhile, ultracold trapped polar molecules,
which have a complex internal structure spanning vibrational, rotational, and
hyperfine manifolds, were proposed as a new kind of qubit [3]. The properties
of molecules have been leveraged in recent precision measurements to place
orders-of-magnitude improved bounds on the electron electric dipole moment for
searches of physics beyond the standard model [23, 24]. These same molecular
resources are also expected to prove valuable in quantum science. In
particular, the long-range, tunable dipolar interactions between molecules
provide a way to perform quantum simulation of exotic quantum phases of matter
[25, 26, 27, 28]. For quantum information, the dipolar interaction can be used
to implement entangling gates and the rich structure of the molecules enables
dense encoding of information, which together may offer a new approach to
scalability and fault-tolerance [6, 29, 30].

Toward these goals, ultracold polar molecules have been prepared and isolated
in optical traps [31, 32, 33].In optical tweezer arrays, control and readout
of single quantum states of molecules have been achieved [34, 35, 36, 37], and
both hyperfine and rotational coherence have been extended well beyond the
dipolar interaction timescale [8, 9, 10, 11, 12, 13, 14]. Most recently,
entanglement between two isolated polar molecules has been realized [15, 16].
However, a universal two-qubit logic gate, which could apply to any arbitrary
initial state, has not yet been demonstrated.

In this work we demonstrate a sub-millisecond iSWAP gate, in the class of
universal logic gates, to entangle hyperfine states of pairs of individually
trapped NaCs molecules. The gate relies on the coherent electric dipole-dipole
interaction between rotational states of molecules. We characterize this
interaction by creating a two-qubit Bell state within 664 $\mu$s with a
measured fidelity of 94(3)%. The fidelity is limited by motion of molecules
along the weakly confined axial dimension of the traps, which we probe using
motion-rotation coupling to determine an axial ground state fraction of
32(5)%. We toggle the dipole-dipole interaction on and off by transferring
between rotational and nuclear-spin degrees of freedom, allowing us to encode
qubits in a non-interacting subspace. We measure the truth table of the iSWAP
gate, with entanglement inferred from the Bell state fidelity. We identify
hyperfine-changing pulses as the dominant source of gate infidelity, and offer
pathways for near-term improvements. Our results establish multi-level
molecules as a viable resources for universal quantum computation and advanced
quantum simulations.

##  II Molecular structure

Figure 1: (a) Schematic of the geometry of a pair of molecules trapped in
neighbouring 1064 nm optical tweezers. A magnetic field of 864.1 G is applied
parallel to the $k$-vector of the tweezer arrays, which is along $z$, and $g$
indicates the direction of gravity. The tweezer polarization
$\varepsilon(\chi,\theta)$, defined in the $x-y$ plane by the ellipticity
$\chi$ and orientation $\theta$, is tuned to a magic ellipticity condition
$\varepsilon(\chi_{m},\theta)$ using a waveplate with a retardance of $\eta$
in the path of the initially horizontally polarized light. (b) Relevant
internal states are labeled by their dominant hyperfine states in the
rotational ground and first rotationally excited manifolds of NaCs. (c)
Illustration of the polarization ellipse in the x-y plane, perpendicular to
the $k$-vector of the tweezer. (d) Measured interaction strengths $J$ between
molecules separated by 1.9 $\mu$m at the magic ellipticity condition for two
different choices of waveplate retardance $\eta$. The black curve shows the
expected form of the interaction strength scaled to the maximum value of $J/h$
observed at $\eta=0.304~{}\lambda$.

Our experiment starts with a pair of individually trapped NaCs molecules
assembled from ground-state-cooled Na and Cs atoms following a procedure
detailed previously in refs. [38, 39, 40]. These molecules are held in “magic”
ellipticity optical tweezers [41] where the polarization ellipticity is chosen
to cancel first-order differential light shifts between the $N=0$ and one
sublevel ($\ket{e}$) of the $N=1$ rotational states, yielding long rotational
coherence times [13]. A schematic of the experimental setup is depicted in
Fig. 1(a). Three distinct molecular energy levels, labelled $\ket{0}$,
$\ket{1}$, and $\ket{e}$, are important for this work and are shown in Fig,
1(b). Molecules are initially assembled in $\ket{0}$ within the ground
rotational manifold and can be transferred using resonant microwaves to
$\ket{e}$ in the first rotationally excited manifold, which couples to
$\ket{0}$ via the dipolar interaction. Nuclear spin state $\ket{1}$ in the
ground rotational manifold is selected to encode a hyperfine qubit in the
molecule (see Appendix for full state labeling). For molecular detection, we
sequentially detect $\ket{0}$ and $\ket{e}$ or $\ket{0}$ and $\ket{1}$ within
a single experimental sequence by converting molecules back to atoms for
fluorescence imaging as described in Ref. [42]. This multi-state readout is
essential as it verifies the presence of a molecule in each tweezer
independent of its internal state, allowing us to identify cases where exactly
two molecules are detected in the system. The molecule pair production rate is
$\sim$3% of all experimental runs.

A key reason for the choice of NaCs molecules here is their large molecule-
frame dipole moment of $d=4.6$ Debye [43]. In the absence of an external
electric field, the lab-frame permanent dipole moment vanishes. Dipolar
interactions then lead to coherent exchange of rotational excitations between
pairs of molecules [44, 12, 15, 16] in the same hyperfine manifold at rate $J$
described by the Hamiltonian [26, 45, 6]

$\displaystyle\hat{H}_{DD}=\frac{J}{2}\quantity(\hat{s}_{1}^{+}\hat{s}_{2}^{-}+\hat{s}_{1}^{-}\hat{s}_{2}^{+})=\left(\frac{d}{\sqrt{3}}\right)^{2}\frac{1-3\cos^{2}{\theta}}{4\pi\epsilon_{0}R^{3}}\quantity(\ket{0e}\bra{e0}+\ket{e0}\bra{0e})$
(1)

where $R$ is the molecular separation and $\theta$ is the angle between the
array axis and the transition dipole moment $d/\sqrt{3}$ of the
$\ket{0}\rightarrow\ket{e}$ transition associated with the raising,
$\hat{s}^{+}_{i}=\ket{e}\bra{0}_{i}$, and lowering,
$\hat{s}^{-}_{i}=(\hat{s}^{+}_{i})^{\dagger}$, operators [46].

The dipolar interaction can be tuned by changing $R$ and $\theta$. Uniquely to
our system, the rotational eigenstates of molecules here are dominantly
quantized by the trapping light rather than static electric or magnetic
fields. Therefore, the orientation of the tweezer polarization ellipse (Fig.
1(c)) determines the angle $\theta$, providing a way to switch between
attractive ($J<0$) and repulsive ($J>0$) interactions. Experimentally, we
realize two different $\theta$s by choosing waveplates with a retardance
$\eta=0.225\lambda$ and $0.304\lambda$ while maintaining the “magic”
ellipticity polarization ($\chi_{m}$) condition (Fig. 1d). For the remainder
of this work, we use the latter waveplate to achieve $\theta=0$, maximizing
the magnitude of interaction strength. In the future, the strength could be
dynamically tuned using fast polarization control via optical interference of
multiple tweezer beams [47, 48].

##  III Dipole-dipole interaction and entanglement

Figure 2: (a) Populations of four possible two-qubit states of a pair of
molecules undergoing dipole-dipole mediated rotational exchange at a
separation of $R=1.9~{}\mu$m, postselected to include only cases where both
molecules are detected. The inset shows the spin-echo microwave pulse sequence
used to probe the interaction. Solid lines are theory curves from a full
quantum-mechanical simulation of the dipolar interaction and molecule motion,
with temperature and motion parameters determined from separate single-
molecule measurements (see Fig. 3), and tweezer separation and single molecule
dephasing chosen to match the data. (b) Scaling of the interaction strength
$J$ as function of distance, with the solid black line representing a single-
parameter fit with the function $a/R^{3}$ with best fit value
$a=5.14(4)~{}\textrm{kHz}~{}\mu\textrm{m}^{-3}$. (c) Parity oscillation signal
$\Pi=P_{00}+P_{ee}-P_{0e}-P_{e0}$ as a function of phase of the global
microwave $\pi/2$ pulse applied to the Bell state produced by the dipole-
dipole interaction. (d) Corresponding populations of the components of the
Bell state prior to applying the global microwave pulse.

Coherent dipolar interaction is the essential ingredient required to entangle
molecules with high fidelity. We characterize the coherent exchange of
rotational excitation between a pair of molecules using a spin echo Ramsey
sequence [49], illustrated in the inset of Fig. 2(a). Global microwave pulses
are used to address the $\ket{0}\rightarrow\ket{e}$ transition. The dipolar
exchange interaction results in an output state with amplitudes determined by
the interaction time:
$\ket{\psi(\tau)}=\frac{1}{2}(1-e^{-i\frac{J\tau}{2\hbar}})\ket{00}+\frac{1}{2}(1+e^{-i\frac{J\tau}{2\hbar}})\ket{ee}$.

In Fig. 2(a), we show the evolution of the populations in each of the possible
states, $\ket{00}$, $\ket{0e}$, $\ket{e0}$, and $\ket{ee}$, of a pair of
molecules separated by 1.9 $\mu$m as a function of the total interaction time
$\tau$. The data are postselected to include only cases in which molecules
were detected in both tweezers. The populations represent a maximum likelihood
estimate of the probability of obtaining each state and the shaded regions are
1$\sigma$ confidence intervals. We analyze the data with the aid of a master
equation simulation of the interaction with decoherence from which we extract
an interaction rate of $J/h=715(2)$ Hz (see Appendix). We observe that the
decoherence is dominated by noise on the interaction strength, which can be
seen from Fig. 2(a) by comparing the symmetric decay of the oscillations
between $\ket{00}$ and $\ket{ee}$ with the slower rise of the $\ket{0e}$ and
$\ket{e0}$ populations. We model this decoherence using a full quantum
mechanical simulation of the motion of the molecules. The solid lines
represent the decaying oscillations predicted by this model, with parameters
extracted from independent single-molecule measurements shown in Fig. 3.

We performed the same experiment at separations of 2.2 and 2.5 $\mu$m and
extracted $J$ for each distance from the master equation fit (see Extended
Data Fig. E.2). In Fig. 2(b) we plot the interaction rate against the
separation and perform a single parameter fit to the function $a/R^{3}$. We
observe good agreement with the expected $1/R^{3}$ scaling of the dipole-
dipole interaction, and extract a distance-independent interaction rate of
$\frac{J}{hR^{3}}=5.14(4)~{}\textrm{kHz}~{}\mu\textrm{m}^{-3}$. This rate is
faster by 17% than expected from the dipole moment of NaCs [43], which could
be explained by a systematic error of 4% in the distance calibration. The
measured dipolar strength $J$ is the fastest observed in work on ultracold
molecules and magnetic atoms, even considering the closer spacings in optical
lattice systems [44, 12, 50].

The dipole-dipole interaction natively generates entanglement between the
rotational states of the molecules [6]. Applying the spin-echo pulse sequence
with an interaction time $h/2J$ maps the input $\ket{00}$ product state to the
maximally entangled state $\frac{1}{\sqrt{2}}(\ket{00}-i\ket{ee})$, which is
equivalent up to a global rotation to the $\ket{\Phi^{\pm}}$ Bell states. This
was demonstrated experimentally with CaF molecules recently with a fidelity of
$<$0.9, benefiting from motional narrowing due the disparate timescales of the
dipolar interaction and motion, despite a large motional excited state
population [15, 16]. We expect our molecules with a large motional ground-
state fraction to do significantly better. We prepare this maximally entangled
state at a molecule separation of 1.9 $\mu$m, using an interaction time of 664
$\mu$s. In order to verify the entanglement of the generated state, we measure
the off-diagonal coherences of the density matrix by applying an additional
global analysis microwave pulse with a variable phase to the pair of
molecules. At each phase we measure the parity-signal
$\Pi=P_{00}+P_{ee}-P_{0e}-P_{e0}$, which we expect to oscillate with the
applied phase $\phi$ as $\Pi=C\sin(2\phi)$, where the amplitude, $C$, of the
parity oscillation is the coherence [51]. The total entanglement fidelity is
determined by the coherence and the diagonal populations,
$F=\frac{1}{2}(C+P_{00}+P_{ee})$. In Fig. 2 (c) we show this measured parity
oscillation signal, along with the populations of all the two qubit states
prior to the analysis pulse in (d). We extract a coherence $C=0.93(3)$ using a
weighted least squares fit, and populations $P_{00}=0.47(1)$ and
$P_{ee}=0.48(1)$. The total entanglement fidelity, conditioned on the
detection of both molecules at the end of the sequence, is thus $F=0.94(3)$.
This is consistent to within 1$\sigma$ with a maximum fidelity of 0.97
predicted given the currently observed decoherence of the dipole-dipole
interaction.

##  IV Motion-rotation coupling

Figure 3:  (a) Measured Ramsey coherence as a function of pulse separation
$\tau$ between XY-8 $\pi$-pulses. The left plot shows three different trap
depths (inferred from theoretical polarizability of NaCs [52, 13]), at which
we apply three XY-8 repetitions, totaling 24 $\pi$-pulses over an
interrogation time of 24$\tau$. By altering the tweezer depth during the
measurement, we can modify the trapping frequency $\omega_{\textrm{ax}}$,
which shifts the $\tau$ that produces maximal decoherence. Dashed lines are a
Gaussian fit serving as a guide to the eye. The right plot shows a detailed
spectrum of the noise peak at the depth of 3.5 MHz at which we perform
interactions, probed with six XY-8 repetitions, from which we extract an axial
trap frequency of 4.95(1) kHz. (b) Illustration of the motional wavefunction
of a single molecule following excitation from $\ket{0}$ to $\ket{e}$ when the
$\ket{e}$ trapping potential is displaced. (c) Observed oscillations in
$\ket{0}$ at the axial trapping frequency as a function of free evolution time
$\tau_{\textrm{free}}$ during spin-echo sequence. (d) Population in $\ket{0}$
under continuous microwave drive along the -Y direction, where the drive Rabi
frequency of $\Omega/2\pi=5.18(3)$ kHz is chosen to be close to the axial
trapping frequency. The solid lines in a, c and d are generated by the same
theoretical model with temperature, trap displacement, and dephasing
parameters chosen to match all of the data simultaneously.

The dominant dephasing mechanism of the two-body dipolar interaction is noise
on the interaction strength. Such noise can be induced by relative motion of
interacting molecules, which leads to fluctuation of $R$ and $\theta$ in
Equation 1. To investigate the effect of motion, we first focus on the
rotational coherence between $\ket{0}$ and $\ket{e}$ of a single molecule in a
tweezer. While a common way to remove noise and extend single-particle
coherence is by applying dynamical decoupling [12, 15, 16, 13], we found
empirically that the coherence is significantly reduced by an XY-8 pulse
sequence when the $\pi$-pulse spacing $\tau$ is chosen to decouple noise on
the interaction timescale $h/2J$. In particular, we observe a dip in the
coherence as a function of $\tau$, the position of which shifts with the trap
depth (Fig. 3(a)). We identify this dip as corresponding to the axial trapping
frequency of the molecules, originating due to parametric heating of the
molecules by pulses spaced by half the trap oscillation period. From a fine
scan of this dip, we can extract an axial trap frequency of 4.95(1) kHz at the
trap depth at which we perform dipolar interactions.

To explain the parametric heating there has to be a physical coupling between
rotation and motional state, implying a deviation from ideal “magic” trapping.
This could be caused either by a change in the trap depth or a displacement of
the trap center between the $\ket{0}$ and $\ket{e}$ states. Because the
tweezer polarization is empirically tuned to match the trap depths for the
$\ket{0}$ and $\ket{e}$ states [13] we can eliminate the former, and turn
towards characterizing the displacement.We construct a minimal model with a
state-dependent displacement of the trapping potential as illustrated in Fig.
3(b) described by

$\displaystyle\hat{H}_{\rm motion-rotation}$ $\displaystyle=\hbar\omega_{\rm
ax}\hat{a}^{\dagger}\hat{a}-\frac{\hbar\zeta\omega_{\rm
ax}}{2}\ket{e}\bra{e}\quantity(\hat{a}+\hat{a}^{\dagger})\,,$ (2)

and attribute the origin of the coupling to optical aberrations, specifically
astigmatism. Here, $\zeta$ is the displacement in units of harmonic oscillator
length $\sqrt{\hbar/(2m\omega_{\rm ax})}=80$nm, with $m=156$a.u. the mass of
NaCs and $\omega_{\rm ax}$ the axial trapping frequency. $\hat{a}$ is the
lowering operator of the axial trap in the harmonic approximation.

Such motion-rotation coupling can transfer motional excitation to internal
rotational states [53], and vice-versa, allowing us to quantify the various
motional effects that contribute to the two-body decoherence seen in Fig. 2a.
In a spin-echo Ramsey experiment (Fig. 3(d) inset), motion will lead to
coherent dephasing and re-phasing with half the trap frequency, as the spatial
wavepackets of the two internal states $\ket{0}$ and $\ket{e}$ separate and
recombine. In harmonic approximation, the functional form is given by
$1-\exp{\frac{-2\zeta^{2}\sin^{4}(\omega_{\rm ax}t/4)}{\tanh[\hbar\omega_{\rm
ax}/(2k_{B}T)]}}$ [54]. As a consequence, this measurement is sensitive to the
joint parameter $\zeta^{2}/\tanh[\hbar\omega_{\rm ax}/(k_{B}T)]$, where $T$ is
the molecule temperature and $k_{B}$ is the Boltzmann constant.

We can couple motion and rotation more directly by applying a continuous drive
with a Rabi frequency $\Omega=\omega_{\rm ax}$, which generates dressed
rotational states $\ket{\pm}=(\ket{0}\pm\ket{e})/\sqrt{2}$ with an energy
separation $\hbar\Omega$. In the dressed state picture, the rotation-motion
coupling in Eq. (2) enables resonant excitation transfer between motion and
rotation driven by $\hat{H}_{\mathrm{eff}}=\zeta\omega_{\rm
ax}/4\,(\ket{+}\bra{-}\hat{a}+\ket{-}\bra{+}\hat{a}^{\dagger})$, where
rotating terms at frequency $\omega_{\rm ax}$ have been neglected. We prepare
the initial state as the lower dressed state $\ket{-}$ by applying a $\pi/2$
pulse with a phase +90° with respect to the drive such that excitation can
initially only be transferred from motion to rotation. A final $\pi/2$ pulse
with the same phase maps the dressed states back to the $\ket{0}$, $\ket{e}$
basis for readout. Since this transfer only occurs if initial motional
excitation is present, the oscillation amplitude is directly related to the
motional ground state fraction. We perform this experiment using a Rabi
frequency of $\Omega/2\pi=5.18(3)$ kHz, slightly detuned from $\omega_{ax}$,
resulting in oscillation at the generalized motion-dependent Rabi frequency
$\sqrt{\hat{a}^{\dagger}\hat{a}\,\zeta^{2}\omega_{\rm
ax}^{2}/4+(\Omega-\omega_{\rm ax})^{2}}$ with decreased amplitude.

To fit the data, we use a discrete variable representation to model the trap
generated by a tweezer with astigmatism from first principles (see Appendix).
The astigmatism is directly related to the displacement $\zeta$. We compute
theory curves for a grid with varying astigmatism, temperature, and motional
dephasing, and extract expectation values and error bars from parametric
bootstrapping [55]. We find $\hbar\omega_{\rm ax}/(k_{B}T)=0.39(8)$
corresponding to an axial ground state fraction of 32(5)%, and
$\zeta=0.062(5)$ from an astigmatism of 0.13(1)$\lambda$. This ground state
fraction is consistent with a previously measured lower bound on the molecule
temperature using thermometry of atoms after dissociation [42], but is lower
than expected given the measured Feshbach molecule production efficiency under
the assumption of identical trapping frequencies for both atoms [56].

With these parameters, we can model the two-molecule contrast oscillations in
Fig. 2(a). We use the discrete variable representation for single-molecule
computations and project dipole interactions into the motional subspace (see
Appendix for details). We include single-molecule decoherence with a
decoherence time of $\tau=80$ ms to match the rise of $\ket{e0}$ and
$\ket{0e}$ population and set the distance to $1.79$ $\mu$m to match the
observed oscillation frequency. We find good quantitative agreement between
theory and data.

For a larger astigmatism of 0.16$\lambda$ at a temperature of
$\hbar\omega_{\rm ax}/(k_{B}T)=0.4$, our model predicts a Bell state
infidelity of 3%, with a contribution of around 1% due to single-molecule
dephasing (see Appendix). Looking ahead, without single-molecule dephasing and
assuming a ground state fraction of 80%, our model predicts a Bell state
infidelity of 0.5% limited by aberrations. Finally, by eliminating
astigmatism, the infidelity could be decreased to $2\times 10^{-4}$, with
approximately equal contributions from residual thermal motion and interaction
back-action onto motion [57]. This back-action comes from the mechanical force
that the dipoles exert on each other, which even at zero temperature can lead
to off-resonant population transfer to motionally excited states. This effect
can ultimately be reduced by increasing tweezer separation or confinement, to
either reduce the force $\sim 1/R^{4}$ or make population transfer more off-
resonant.

##  V Quantum logic gates between hyperfine qubits

Figure 4: (a) Microwave pulse sequence for implementing an iSWAP gate between
two hyperfine qubits. (b) Time-evolution of resonant dipolar exchange for a
pair of molecules initially prepared in the two-qubit state $\ket{01}$. Solid
lines are theory curves from a master equation model, using parameters
extracted from the data in 2. (c) Measured truth tables for the full iSWAP
gate sequence. Left shows the probabilities of measuring each two-qubit output
state given each input state, relative to the probability of detecting the
$\ket{00}$ state with no pulses applied. Losses here are dominated by leakage
to the $\ket{e}$ state due to imperfect hyperfine-state changing pulses both
during state-preparation and measurement and in the pulses which start and end
the gate. Right shows the probabilities postselected to cases where two
molecules were detected in the $(\ket{0},\ket{1})$ manifold, which represents
the truth table for the entangling interaction with leakage errors caused by
the single-qubit gate removed.

The dipolar exchange interaction can naturally realize an iSWAP gate, which is
a universal two-qubit gate, when the duration of the interaction is precisely
controlled [6]. The Bell state creation protocol demonstrated thus far limited
the duration of the interaction using global pulses applied to a particular
initial state, but it cannot be generalized to arbitrary initial states and
multiple gates. To realize an iSWAP gate for any arbitrary input within the
computational basis, we need to switch the dipolar interactions on and off and
show the desired logic outcomes.

Here, we introduce a third state, $\ket{1}$ (Fig. 1b), within the ground
rotational manifold, which we use to implement the iSWAP gate with a qubit
that does not natively interact [6]. The qubit is encoded in a non-interacting
hyperfine degree of freedom, which has been shown to be insensitive to
external fields and capable of maintaining long coherence times [8, 9, 10].
The state $\ket{1}$ is weakly coupled to $\ket{e}$ by the nuclear quadrupole
moment [58], and is chosen among all the accessible hyperfine states because
it is the most detuned from $\ket{0}$, allowing for easier individual
manipulation. The interaction is toggled by transferring between $\ket{1}$ and
the rotationally excited state $\ket{e}$. We note that $\ket{1}$ is strongly
coupled to the non-hyperfine changing excited state $\ket{e^{\prime}}$, which
would be an important consideration for the gate implementation, discussed
below.

We engineer the interaction between qubits using the pulse sequence shown in
Fig. 4(a). We initialize a pair of molecules in the computational basis at a 5
$\mu$m trap separation with site-selective $\ket{0}$ to $\ket{e}$ transfer
[42] and a 310 $\mu$s $\pi$-pulse from $\ket{e}$ to $\ket{1}$ (Extended Data
Fig. E.3). At 1.9 $\mu$m trap separation, we toggle the interaction with the
same hyperfine state changing pulse ($\ket{1}\leftrightarrow\ket{e}$), and
apply a spin echo pulse on the $\ket{0}\leftrightarrow\ket{e}$ transition in
the middle of the interaction period.

For detection, we transfer $\ket{1}$ to $\ket{e}$ at a 5 $\mu$m trap
separation and read out the $\ket{0}$ and $\ket{e}$ states. In Fig. 4(b), we
show the dipolar interaction between two qubits initialized in the $\ket{01}$
state. Here, the data are postselected for trials where a pair of molecules
are both detected within the qubit subspace. The exchange interaction leads to
oscillation between the $\ket{01}$ and $\ket{10}$ states. The full exchange
occurs 330 $\mu$s after the end of the toggling pulse, shorter than if the
pulses were infinitely fast, because the interaction occurs already during the
duration of the toggling pulse.

At the full exchange time, the sequence realizes the iSWAP gate up to a global
rotation, originating from the spin echo pulse. The total gate time, composed
of the full exchange time and the duration of both toggling pulses, is 960
$\mu$s. The unitary corresponding to the ideal implementation in the
$\left(\ket{0},\ket{1}\right)$ basis is given by

$U=\begin{bmatrix}0&0&0&-1\\\ 0&-i&0&0\\\ 0&0&-i&0\\\ -1&0&0&0\end{bmatrix}$
(3)

Under the application of this gate, $\ket{01}$ and $\ket{10}$ obtain a non-
trivial phase of $-i$ from the dipolar exchange, which is the same phase we
verified via the parity oscillation of the Bell state in Fig. 2(c). We
characterize the logic outcomes of the iSWAP gate for each computational basis
state and show the resulting population truth table in Fig. 4(c). For trials
where two molecules are detected in the qubit subspace, an average of the
probabilities for each input state to transform into the intended output state
yields a truth table fidelity [59] of $0.92^{+0.05}_{-0.09}$.

The ease of choosing three states $\ket{0}$, $\ket{1}$, $\ket{e}$ showcases
the versatile structure of molecules, but the dense structure also risks
causing leakage outside the computational basis [6]. To take leakage into
account, we instead normalize the detected outcomes to the population measured
in the initial state $\ket{00}$ before performing site-selective state
preparation and the iSWAP gate. The corresponding truth table is shown in Fig.
4(d) and the resulting truth-table fidelity is $0.39^{+0.1}_{-0.09}$.

The current gate fidelity is primarily limited by leakage during four
hyperfine changing pulses, which are needed for state preparation, the iSWAP
gate, and readout. Rotational transitions that change nuclear spin are two
order of magnitude weaker than those that conserve nuclear spin at a magnetic
field of 864 G. The strong microwave pulse required to drive the hyperfine-
changing transition introduces off-resonant Stark shifts from nearby
transitions making the transfer pulse sensitive to microwave amplitude noise.
The Stark shifts specifically from non-“magic” rotational states fluctuate
with trap intensity noise and further decohere the transition. Due to these
decoherence sources, the current one-way transfer efficiency is limited to
89%. In the future, the hyperfine-changing transfer can be made more coherent
with active microwave amplitude stabilization and polarization control. The
latter also mitigates depopulation of the $\ket{0}$ state, allowing faster
driving of the transition. Another source of leakage is due to interactions
during the toggling pulses. We anticipate that 9% of population leaves the
computational basis with perfect hyperfine-changing transfer at the current
toggling duration. Nevertheless, this leakage may be reduced given the
approximately quadratic decrease with pulse duration. Other methods could
reduce such leakage errors, including faster optical or microwave Raman
transfers, a lower magnetic field to enhance hyperfine changing coupling for a
faster transfer, and moving traps to toggle the interaction. Both leakage
sources described here lead to populations remaining $\ket{e}$ and
$\ket{e^{\prime}}$, the latter occurring due to the spin-echo pulse (see
Appendix). In future, these populations could be detected without disturbing
the computational basis by state-selective transfer to the $N=2$ rotational
states for readout. Such detectable errors may be correctable using erasure
conversion schemes.

##  VI Conclusions and outlook

We demonstrate a molecular iSWAP gate, a key requirement for universal quantum
computation with polar molecules. The gate is based on the native coherent
interaction between molecules, such that noise on the interaction is due more
to imperfect state preparation than to variations in external fields. We
characterize the interaction by measuring a two-qubit Bell state entanglement
fidelity of 94(3)% with error primarily due to imperfect motional state
preparation (32(5)% axial ground state fraction). A modest improvement in the
axial ground state fraction to 80% and elimination of tweezer astigmatism and
single-particle dephasing would allow achieving a fidelity exceeding $0.999$,
at which point more fundamental decoherence mechanisms dominate. The iSWAP
gate is mainly limited by imperfect transfer between hyperfine states due to
the density of nearby molecular states with strong transitions which can
induce AC Stark shifts and population leakage. This could be improved through
finer microwave polarization control, opening up the complex molecular
structure for multi-level applications. Excitations to multiple rotational
levels, paired with coherent dipolar interactions, will enable quantum
simulation of complex many-body Hamiltonians [60, 61], and encoding of robust
logical qubits [29] and qudits [30]. In parallel, efforts to pursue efficient
molecule production and detection remain important for the utility of the
molecular platform, for example, in a hybrid molecule-Rydberg system with
orders of magnitude faster gates and mid-circuit readout of molecular states
[62, 63, 64, 65]. The coherent control of motion-rotation coupling offers a
new sensitive probe of molecular temperature and confinement, adding an
addional degree of freedom to the molecule toolbox. This promises new
opportunities for quantum simulation and metrology based on spin-motion
entanglement [53, 66, 67, 68].

##  VII Appendix

External fields and quantization

We define the quantization axis to be along the magnetic field of 863.9 G,
which is kept constant during the experiment and is parallel to the $k$-vector
of the optical tweezer. The large magnetic field independently aligns the
nuclear spins of the two atoms and decouples them from molecular rotation. The
rotational sublevels are split by the AC electric field of the optical
tweezer, which is elliptically polarized in the plane perpendicular to the
quantization axis. We label three states relevant to this work: the state in
which assembled molecules are initially prepared,
$\ket{0}\equiv\ket{N=0,m_{N}=0,m_{I_{{\rm Na}}}=3/2,m_{I_{{\rm Cs}}}=5/2}$
[35]; the magic state in the first rotationally excited manifold with the same
nuclear hyperfine quantum numbers, $\ket{e}\equiv\ket{N=1,m_{I_{{\rm
Na}}}=3/2,m_{I_{{\rm
Cs}}}=5/2}\otimes\frac{1}{\sqrt{2}}\left(\ket{m_{N}=1}-\ket{m_{N}=-1}\right)$
[13]; and an additional nuclear hyperfine state in the rotational ground
manifold that we use as one of our qubit states,
$\ket{1}\equiv\ket{N=0,m_{N}=0,m_{I_{{\rm Na}}}=-1/2,m_{I_{{\rm Cs}}}=5/2}$.

Molecule preparation and detection

The procedure for preparing single ultracold NaCs molecules in optical
tweezers is detailed in our previous work [56, 35, 39, 40]. Briefly, we begin
by Raman sideband cooling Na and Cs atoms in separate optical tweezer arrays.
We then merge the arrays, magnetoassociate the atoms into molecules at a
Feshbach resonance, and perform a two-photon detuned Raman transfer to the
rovibrational ground state. Specifically, we prepare the molecules in the
$\ket{0}$ state, which has the largest coupling to the weakly bound Feshbach
molecule given our choice of $\sigma^{+}$ polarization for both the Pump and
Stokes lasers used in two-photon transfer.

After preparing the molecules in an 8-site array, we detect and remove any
remaining unassociated Cs atoms and rearrange the molecules to a higher
density region at one end of the array. All dipole-dipole interaction data
reported in this work includes only the two sites with the highest molecule
density at the end of the array. To detect the molecule population, we
transfer from $\ket{0}$ back to the weakly bound Feshbach molecule,
magnetodissociate at the Feshbach resonance, and image the resultant Cs atoms.
Since the two-photon transfer is far off-resonant for the $\ket{e}$ state, we
can perform this process sequentially: first measuring $\ket{0}$ in one image
and then transferring remaining population from $\ket{e}$ to $\ket{0}$ before
repeating the process. This allows us to read out both $\ket{0}$ and $\ket{e}$
in each shot of the experiment. To initially prepare a pair of molecules in
different internal state combinations starting from the $\ket{00}$ state, we
employ site-selective excitation of molecules using dynamic ramping of tweezer
depths combined with microwave shelving pulses [42].

Microwave pulse sequences

To probe dipole-dipole interactions between $\ket{0}$ and $\ket{e}$, we begin
with a pair of molecules in the $\ket{00}$ state and apply a $\pi/2$-pulse to
prepare the product state
$\frac{1}{\sqrt{2}}(\ket{0}+\ket{e})_{i}\otimes\frac{1}{\sqrt{2}}(\ket{0}+\ket{e})_{j}$.
The phase of this first pulse defines rotation about the x-axis of the Bloch
sphere. We then allow the system to evolve under $H_{DD}$ for time $\tau$
before applying another $\pi/2$ pulse. To cancel out any phase accumulation
due to finite detuning of the microwave pulse from resonance or slow drifts in
the resonance frequency, we apply a spin-echo $\pi$-pulse in the middle of the
interaction period, with a phase of +90° relative to the initial pulse,
realizing an effective y-rotation. This sequence is illustrated in the inset
of Fig. 2. The $\pi$-pulses have a total duration of 12.83 $\mu$s, and for all
pulses addressing this transition we use a truncated Gaussian pulse shape with
a width of 0.3 times the pulse duration to reduce off-resonant coupling to
other rotational states [69]. To measure the parity oscillation signal for an
entangled Bell pair produced using this sequence, we apply an additional
$\pi/2$ pulse with a variable phase to the Bell pair.

To investigate the impact of molecule motion, we perform noise-spectroscopy
via dynamical decoupling. At a large molecule separation, where there is no
significant dipole-dipole interaction on relevant timescales, we perform a
Ramsey sequence composed of a $\pi/2$ pulse on
$\ket{0}\leftrightarrow\ket{e}$, a series of dynamical decoupling pulses, and
a final $\pi/2$ pulse for readout. We use the XY-8 decoupling pulse sequence,
consisting of evenly spaced $\pi$-pulses along the x and y axes, illustrated
in Fig. 3(a), which cancels the effect of any pulse area imperfections [49].
To observe coherent effects of motional dephasing and rephasing we use the
same spin-echo pulse sequence as described above, as well as a continuous
drive sequence, described in full in the main text.

To perform an iSWAP gate between hyperfine qubits, we toggle the interaction
on and off using 310 $\mu$s square $\pi$-pulses on the
$\ket{1}\leftrightarrow\ket{e}$ transition. At the midpoint of the interaction
period we apply a single spin-echo $\pi$-pulse on
$\ket{0}\leftrightarrow\ket{e}$ with a duration of 12.83 $\mu$s and a
truncated Gaussian shape, as before.

Hyperfine spectroscopy

To identify a suitable hyperfine state to act as the $\ket{1}$ state of the
hyperfine storage qubit, we spectroscopically investigate multiple nuclear-
spin-changing transitions from $\ket{e}$. Transitions which change nuclear
spin are mediated by the electric quadrupole interaction and are significantly
weaker than those which do not. All the identified nuclear spin states are
listed in Extended Data Table 1. The state $\ket{1}$ was chosen because it is
the state furthest detuned from $\ket{0}$ which still has significant coupling
to $\ket{e}$, such that off-resonant coupling to the excited rotational states
of the same hyperfine manifold as $\ket{1}$ is suppressed. The full hyperfine
structure of NaCs is shown in Extended Data Fig. E.1, along with a simulated
spectrum of the couplings of the $\ket{1}$ and $\ket{e}$ states under a strong
microwave drive.

Postselection and error analysis

To study the interactions between molecules, we postselect on cases where two
molecules are detected at the readout stage, which is enabled by the fact that
we measure both $\ket{0}$ and $\ket{e}$ during each experimental run.
Postselection allows us to eliminate background in the interaction signal due
to shots where one or both traps are empty, at the expense of discarding shots
in which both molecules were present but one or both were not detected due to
measurement error. The only remaining background on the postselected signal is
due to the small 0.0038 false positive rate of the imaging step, giving a
negligible contribution of $10^{-5}$ to the two-body signal (assuming
uncorrelated false positives between sites) [42]. The outcomes of each trial
following post-selection are described by a binomial distribution,

$p(S|\theta,T)=\begin{pmatrix}T\\\ S\end{pmatrix}\theta^{S}(1-\theta)^{T-S},$

where the number of binomial trials, $T$, is the total number of molecules
detected and the number of binomial successes, $S$, is the number detected in
a particular state combination with a corresponding population $\theta$. Given
a measurement of $S$, the maximum likelihood estimator of the population,
$\hat{\theta}$, and associated 1$\sigma$ confidence interval can be computed
by inverting the distribution using the Clopper-Pearson method [70].

Master equation model of decoherence

We perform initial analysis of the strength and decoherence of the dipole-
dipole interaction using a Lindblad master equation with three
phenomenological decay channels corresponding to dephasing of the dipole-
dipole interaction itself, relative detuning noise between the two molecules,
and global detuning noise on both molecules. These processes are represented,
respectively, by the jump operators:

$\displaystyle
L_{J}=\sqrt{\gamma_{J}}\left(\ket{0e}\bra{e0}+\ket{e0}\bra{0e}\right)$ (4)
$\displaystyle
L_{\delta}=\sqrt{\gamma_{\delta}}\left(\ket{e0}\bra{e0}-\ket{0e}\bra{0e}\right)$
(5) $\displaystyle
L_{\Delta}=\sqrt{\gamma_{\Delta}}\left(\ket{ee}\bra{ee}-\ket{00}\bra{00}\right)$
(6)

We fit the master equation model to the data by minimizing the weighted least
squares residual of the model and experimental data, using the Nelder-Mead
algorithm to optimize the rates of the decoherence processes along with the
interaction strength, $J$, and a static relative detuning, $\delta$, between
the molecules. We implement the master equation model using the Julia Quantum
Optics toolbox [71].

Numerical Simulations of motion

To simulate the dynamics with rotation-motion coupling, we use a combination
of discrete variable representation (DVR) and exact diagonalization. We
compute the local electromagnetic field of a single tweezer in the presence of
astigmatism following Ref. [72]. Of the primary aberrations, astigmatism and
coma produce displacement of the $\ket{0}$ and $\ket{e}$ states in the axial
and radial directions, respectively, by altering the ellipticity of the
tweezer polarization at the focus. The local energy of the two states was
determined from their polarizabilities $\alpha_{0}$ and
$\alpha_{0}+\alpha_{2}\frac{1-3\cos(2\chi)}{10}$, where $\alpha_{0}$ and
$\alpha_{2}$ are the scalar and tensor polarizabilities of NaCs in a 1064 nm
tweezer. We identify the position of minimal energy by gradient descent, which
corresponds to the point of maximum intensity for $\ket{0}$.

This gives us a DVR of the potential energy, from which we get the DVR
Hamiltonian $\hat{H}_{1m,0/e}$ for each rotational state following Ref. [73].
We numerically compute the $n_{\rm lvl}$ lowest motional eigenenergies and
corresponding eigenstates of each rotational state. We choose a tweezer depth
of $h\times 2.45$ MHz to match the numerically extracted level spacing to the
experimentally measured value of $h\times 4.95$ kHz.

To compute the two-body Hamiltonian of dimension $4n_{\rm lvl}^{2}$, we
project the dipolar interactions described by the Hamiltonian

$\displaystyle\hat{H}_{\mathrm{dip}}$ $\displaystyle=\frac{1}{2}\int
d^{3}\vec{r}d^{3}\vec{r}^{\prime}\frac{1-3\cos^{2}(\theta)}{4\pi\epsilon_{0}\absolutevalue{\vec{r}-\vec{r}^{\prime}}^{3}}\quantity[\hat{\psi}_{e}^{\dagger}(\vec{r})\hat{\psi}_{0}^{\dagger}(\vec{r}^{\prime})\hat{\psi}_{e}(\vec{r}^{\prime})\hat{\psi}_{0}(\vec{r})+h.c.]$
(7)

into the low-energy eigenstates, where we replace integrals with sums over
discrete grid values. Analogously, we also project the pulse/drive Hamiltonian

$\displaystyle\hat{H}_{\mathrm{pulse}}$
$\displaystyle=\frac{\Omega}{2}\ket{0}\bra{e}+\frac{\Omega^{*}}{2}\ket{e}\bra{0}+\Delta\ket{e}\bra{e}$
(8)

into the DVR basis, which couples different motional levels according to their
state overlap. We choose $\Delta$ such that the microwave pulse is exactly in
resonance
$\Delta=-\bra{\psi_{0}}\hat{H}_{1m,e}\ket{\psi_{0}}+\bra{\psi_{0}}\hat{H}_{1m,0}\ket{\psi_{0}}$,
where $\ket{\psi_{0}}$ is the ground state of $\hat{H}_{1m,0}$. We numerically
verify that residual detuning $\lesssim h\times 1$ kHz is removed by spin-
echo.

Finally, we include single-molecule dephasing at rate $\gamma_{\mathrm{deph}}$
described by the Lindblad operators

$\displaystyle\hat{L}_{i}$
$\displaystyle=\sqrt{\frac{\gamma_{\mathrm{deph}}}{2}}\quantity(\ket{e}\bra{e}_{i}-\ket{0}\bra{0}_{i})\,.$
(9)

For the single-molecule simulations, we further include motional dephasing
modeled by the diagonal matrix

$\displaystyle\hat{L}_{\mathrm{m},i}$ $\displaystyle=\sqrt{2\gamma_{\rm
m}}\begin{pmatrix}1&0&\quad\cdots&0\\\ 0&2&\quad\cdots&0\\\
\vdots&\vdots&\quad\ddots&\vdots\\\ 0&0&\quad\cdots&n_{\rm
lvl}\end{pmatrix}\,.$ (10)

We then propagate the resulting master equation in real time to reproduce the
experimental data. We ignore any dephasing that occurs during the pulses for
simplicity, which is expected to be insignificant for pulse durations $\sim
10\mu$s, but keep dipolar interactions on during the pulses.

To make the two-particle simulations more efficient, we ignore off-diagonal
elements between motional states whose energies differ by more than $h\times
200$ Hz when computing the lines for Fig. 2. Since the coupling between these
manifolds due to dipolar interactions is $<h\times 10$ Hz, this only affects
dynamics during pulses. It removes small oscillations with a few percent
amplitude smaller than experimental error bars from the theory curves.

In all simulations, we set $\Omega=h\times 38$ kHz,
$\gamma_{\mathrm{deph}}=1/$(80 ms), and $R=$ 1.79 $\mu$m, slightly smaller
than the experimentally measured value, to match the observed interaction rate
for a molecule frame dipole moment $d=4.6$ D. For the single-molecule
simulations we set $n_{\rm lvl}=20$, for the two-molecule simulations we set
$n_{\rm lvl}=25$.

Parametric bootstrapping for fit parameter uncertainty

To determine the best fit parameters to match the data, we first fix the trap
frequency using the XY-8 scan in Fig. 3(a) by generating simulated curves for
a linear scan of trap frequency values, computing the weighted squared
residuals between the curve and the data for each point, and fitting a
parabola to determine the best fit trap frequency. To fit the remaining
parameters to the XY-8, spin-echo and continuous drive data simultaneously, we
generate curves for a 3D grid of the temperature, astigmatism and motional
dephasing parameters. We then approximate the best fit value as the grid point
which gives the lowest weighted squared residual for the combined data.

To estimate the uncertainties in the optimized parameters of the master
equation and motion-rotation models, we apply the parametric bootstrap method
[55]. For every dataset which we fit, we generate new bootstrap samples using
the estimator $\hat{\theta}$ for each two-qubit state at each point, drawing
from the distribution $p(S|\hat{\theta},T)$. We then repeat the fitting
procedure for each bootstrapped sample, generating a distribution of optimal
fit parameters. We generate 300 bootstrap samples for each experimental
dataset for the master equation fit, and 500 for the motion-rotation model.
The uncertainties in the fit parameters represent the standard deviation of
their distribution.

Tweezer separation

We estimate the tweezer separation from the frequencies of the RF tones
applied to the acousto-optic deflector used to generate the tweezers and the
expected propagation of beams through the 4f optical system [39]. This does
not account for any systematic deviations of the tweezers from ideal Gaussian
beams, such as could be caused by aberrations or by the overlap of the
tweezers at small separations.

Data Availability

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

##  VIII Acknowledgement

We thank Till Rosenband, Guido Pupillo, Sven Jandura, Matteo Bergonzoni, and
Bihui Zhu for discussion. This work is supported by AFOSR (FA9550-23-1-0538),
NSF (PHY-2110225 & PFC-PHY-2317149), and AFOSR-MURI (FA9550-20-1-0323 &
FA9550-21-1-0069).

Note - Ref. [74] is a related molecule work leveraging hyperfine states as
qubits.

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Transition | Frequency - 2B (kHz) | Relative Rabi frequency (a.u.)
---|---|---
$\ket{0}\leftrightarrow\ket{3/2,5/2}\otimes\ket{1,0}$ | 691.7(2) | 0.77(5)
$\ket{0}\leftrightarrow\ket{e}$ | 0.2(1) | 1
$\ket{0}\leftrightarrow\ket{3/2,5/2}\otimes\left(\frac{\ket{1,1}+\ket{1,-1}}{\sqrt{2}}\right)$ | -710.3(2) | 0.84(2)
$\ket{0}\leftrightarrow\ket{3/2,3/2}\otimes\left(\frac{\ket{1,1}-\ket{1,-1}}{\sqrt{2}}\right)$ | 474.6(2) | 0.01838(7)
$\ket{3/2,7/2}\otimes\ket{0,0}\leftrightarrow\ket{e}$ | 466.1(3) | 0.0053(1)
$\ket{3/2,1/2}\otimes\ket{0,0}\leftrightarrow\ket{e}$ | -927.1(3) | 0.0218(2)
$\ket{1}\leftrightarrow\ket{e}$ | -1902.2(1) | 189(4)
Table 1: Measured hyperfine transitions in NaCs. Eigenstates are labeled by their dominant nuclear spin and rotational quantum numbers, $\ket{m_{I_{Na}},m_{I_{Cs}}}\otimes\ket{N,m_{N}}$. The first three rows correspond to transitions from $\ket{0}$ to different rotational sublevels of $N=1$. The fourth row corresponds to a hyperfine-changing transition starting from $\ket{0}$, and the last three rows to hyperfine-changing transitions starting from $\ket{e}$. In all cases, the trap depth and magnetic field are fixed. Rabi frequency is defined relative to that of the $\ket{0}\leftrightarrow\ket{e}$ transition and corrected for the different microwave amplitudes used to probe each transition. Figure E.1: (a) Rotational-hyperfine structure of NaCs in the experimental regime. Color-shaded regions denote states of the same rotational sub-level. The x-axis represents the combined nuclear-spin contribution to the Zeeman shift, and is chosen to make the rotational structure apparent. (b) Estimated impurity of the states $\ket{1}$ and $\ket{e}$ when dressed by the microwave field used for the $\ket{1}\leftrightarrow\ket{e}$ transition, which depicts all available transitions from these states. The hyperfine-changing transition, labeled by 3, is red-detuned from $\ket{0}\leftrightarrow\ket{e}$, labelled 4, by 1.9 MHz and comparably narrow. Off-resonant coupling is primarily due to the transitions 1, 2, and 4. Figure E.2: Dipole-dipole interactions using the same sequence as illustrated in Fig. 2 for distances of 2.2 and 2.5 $\mu$m. Separation ($\mu$m) |  $J/h$ (Hz) |  $\delta/h$ (Hz) |  $\gamma_{J}$ (Hz) |  $\gamma_{\delta}$ (Hz) |  $\gamma_{\Delta}$ (Hz)
---|---|---|---|---|---
1.9 | 715(2) | 17(17) | 107(10) | 25(3) | 13(2)
2.2 | 477(3) | 214(111) | 125(29) | 31(31) | 16(16)
2.5 | 314(6) | 159(159) | 114(32) | 53(53) | 35(35)
Table 2: Fit values and bootstrapped uncertainty estimates for master equation
model of interactions, for the three distances shown in Fig. 2 and Extended
Data Fig. E.2. The bootstrapped distributions of fit values for the master
equation dephasing rates and detuning $|\delta|$ are in some cases consistent
with zero to within 1$\sigma$, and are truncated below zero due to the
restriction that values must be positive. In these cases we report the value
as 1$\sigma$ above zero, but note that the distribution deviates from Normal
due to the truncation. Figure E.3: (a) Hyperfne-changing microwave pulse from
$\ket{e}$ to $\ket{1}$. Fit is a $\cos^{2}$ function with symmetric
exponential decay, giving a $\pi$-time of 0.307(9) ms and $1/e$ decay time of
1.2(4) ms. Figure E.4: Full outcomes of truth table measurements for hyperfine
gate: (a) Populations measured in each two-qubit state following state-
preparation in that state, relative to population in $\ket{00}$, representing
the relative SPAM fidelity of each state. (b) Populations measured in each
two-qubit state as a fraction of the total detected molecule population in the
$(\ket{0},\ket{1})$ manifold for each of the input states to the gate. This
corrects for leakage to the $\ket{e}$ state during state-preparation and
measurement, as well as during the hyperfine gate itself. (c) Populations
measured in each two-qubit state following application of the iSWAP gate
sequence, relative to the initial population in $\ket{00}$ in (a),
illustrating the amount lost to leakage. Figure E.5: Bell state infidelity at
the optimal time predicted as a function of inverse temperature. The solid
purple curve illustrates the prediction of the full model including
astigmatism of 0.16$\lambda$. The green curve illustrates the corresponding
result without any astigmatism. The two dashed curves show what happens
without single-molecule decoherence $\gamma_{\rm deph}=0$, for both cases with
(orange) and without (blue) astigmatism. Numbers quoted in the main text at
80% ground state fraction correspond to $\hbar\omega_{\rm ax}/(k_{B}T)=1.6$.
Computed with $n_{\rm lvl}=15$ ($n_{\rm lvl}=10$ for $\hbar\omega_{\rm
ax}/(k_{B}T)\geq 2$), leading to errors on the 0.1% level at the lowest
temperatures, and negligible errors for $\hbar\omega_{\rm ax}/(k_{B}T)>1$.


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