# Demonstration of Erasure Conversion in a Molecular Tweezer Array

Connor M. Holland  Department of Physics, Princeton University, Princeton, New
Jersey 08544 USA  Yukai Lu  Department of Physics, Princeton University,
Princeton, New Jersey 08544 USA  Department of Electrical and Computer
Engineering, Princeton University, Princeton, New Jersey 08544 USA  Samuel J.
Li  Department of Physics, Princeton University, Princeton, New Jersey 08544
USA  Callum L. Welsh  Department of Physics, Princeton University, Princeton,
New Jersey 08544 USA  Lawrence W. Cheuk<EMAIL_ADDRESS>Department of
Physics, Princeton University, Princeton, New Jersey 08544 USA

###### Abstract

Programmable optical tweezer arrays of molecules are an emerging platform for
quantum simulation and quantum information science. For these applications,
reducing and mitigating errors that arise during initial state preparation and
subsequent evolution remain major challenges. In this paper, we present work
on site-resolved detection of internal state errors and quantum erasures,
which are qubit errors with known locations. First, using a new site-resolved
detection scheme, we demonstrate robust and enhanced tweezer array preparation
fidelities. This enables creating molecular arrays with low defect rates,
opening the door to high-fidelity simulation of quantum many-body systems.
Second, for the first time in molecules, we demonstrate mid-circuit detection
of erasures using a composite detection scheme that minimally affects error-
free qubits. We also demonstrate mid-circuit conversion of blackbody-induced
errors into detectable erasures. Our demonstration of erasure conversion,
which has been shown to significantly reduce overheads for fault-tolerant
quantum error correction, could be useful for quantum information processing
in molecular tweezer arrays.

##  I Introduction

Programmable optical tweezer arrays of ultracold molecules are an emerging
platform for quantum science Kaufman and Ni (2021). They combine the
microscopic detection and control capabilities offered by programmable optical
tweezer arrays Endres  _et al._ (2016); Labuhn  _et al._ (2016) with useful
properties of molecules such as rich internal structures and intermolecular
electric dipolar interactions between long-lived states Yan  _et al._ (2013);
Christakis  _et al._ (2023); Gregory  _et al._ (2024). In particular, the
internal structure of molecules provides additional ways to encode quantum
information and can be exploited for precision measurement experiments Carr
_et al._ (2009); Bohn  _et al._ (2017); Safronova  _et al._ (2018); Kozyryev
and Hutzler (2017); Anderegg  _et al._ (2023), while the intermolecular
electric dipolar interaction is the key ingredient for high-fidelity quantum
gates DeMille (2002); Ni  _et al._ (2018); Yu  _et al._ (2019) and simulation
of novel quantum many-body Hamiltonians Micheli  _et al._ (2006); Carr  _et
al._ (2009); Gadway and Yan (2016); Bohn  _et al._ (2017); Kaufman and Ni
(2021).

Starting with the first demonstrations of trapping and high-fidelity detection
of individual molecules in optical tweezer traps Anderegg  _et al._ (2019);
Zhang  _et al._ (2022); Holland  _et al._ (2023a); Ruttley  _et al._ (2023);
Vilas  _et al._ (2024), recent work has demonstrated highly coherent
rotational qubits in tweezers Burchesky  _et al._ (2021); Park  _et al._
(2023); Gregory  _et al._ (2024), observed coherent dipolar interactions
between individually held molecules, and demonstrated on-demand entanglement
and 2-qubit gates between rotational molecular qubits Holland  _et al._
(2023b); Bao  _et al._ (2023a). These works establish the building blocks for
quantum information processing and quantum simulation with molecular tweezer
arrays.

A major experimental challenge in any quantum science platform is reducing and
mitigating errors that occur during the initial preparation of a quantum
system and its subsequent evolution. In molecular tweezer arrays, one desires
to deterministically prepare tweezer sites such that they are occupied by
molecules from a single internal state. This preparation of molecular tweezer
sites is challenging, and has been a bottleneck for applications in quantum
simulation and information processing. In particular, the complex internal
structure of molecules combined with substantial light shifts inside tweezer
traps render high-fidelity optical pumping difficult Lu  _et al._ (2024); Bao
_et al._ (2023b), thereby limiting achievable internal state purities.
Concerning errors that can arise during quantum evolution, Raman scattering of
tweezer trapping light or blackbody radiation can excite molecules out of the
desired quantum states Holland  _et al._ (2023a). While directly reducing
these errors through better control is an important research area that has
attracted much interest, a complementary approach is to mitigate their effects
via measurement and feedback. In particular, for programmable tweezer arrays,
because local control is available, site-resolved error detection could in
some cases enable their subsequent removal.

Site-resolved detection of preparation errors has been used in atomic tweezer
arrays to increase internal state preparation fidelities Scholl  _et al._
(2023a), increase robustness to radiative decays of high-lying Rydberg states
Scholl  _et al._ (2023a); Ma  _et al._ (2023), and to lower motional
temperatures Scholl  _et al._ (2023b). In recent work with molecules that are
assembled from two atoms and trapped in tweezer arrays, this has been used to
enhance molecular formation efficiency and to reduce defects in molecular
arrays Picard  _et al._ (2024); Ruttley  _et al._ (2024).

Restricting to the specific case where information is stored in an array, an
information error whose location is known is called an erasure. For qubits,
such errors are called quantum erasures Grassl  _et al._ (1997); Bennett  _et
al._ (1997). Recently, it has been pointed out that leakage errors, where a
particle hosting a qubit exits its computational space, can be converted and
detected as an erasure Wu  _et al._ (2022). Notably, erasure conversion
substantially raises fault-tolerant quantum error correction thresholds to
practically achievable levels Wu  _et al._ (2022). This important insight has
led to several proposed erasure conversion schemes across a variety of qubit
platforms Wu  _et al._ (2022); Kubica  _et al._ (2023); Kang  _et al._ (2023);
Teoh  _et al._ (2023). Recently, the first demonstrations of quantum erasure
conversion were achieved in atomic tweezer arrays of alkaline-earth atoms Ma
_et al._ (2023); Scholl  _et al._ (2023a, b) and superconducting circuits Chou
_et al._ (2023); Levine  _et al._ (2024).

In this paper, we present new site-resolved detection schemes to mitigate both
state preparation errors and qubit leakage errors for laser-cooled molecular
arrays. First, we demonstrate a new site-resolved detection scheme that
enables robust and enhanced fidelities of preparing a tweezer site with a
molecule in a single internal state. We achieve a record-level tweezer site
preparation fidelity of $95.2(3)\%$, with internal state purity of
$99.5(1)\%$, substantially surpassing the previous best reported fidelities of
$\approx 80\%$ Holland  _et al._ (2023b); Lu  _et al._ (2024). Second, using a
new composite erasure detection scheme, we demonstrate quantum erasure
conversion in molecules for the first time. By utilizing a new hyperfine qubit
encoding that is both highly coherent and compatible with a previously
demonstrated 2-qubit gate Holland  _et al._ (2023b); Bao  _et al._ (2023a), we
demonstrate a mid-circuit detection scheme that minimally affects qubit
population and coherence. We further show mid-circuit erasure conversion of
leakage errors caused by blackbody radiation, achieving improved qubit
lifetimes and coherence times.

Our new preparation scheme overcomes a major challenge in molecular tweezer
arrays and opens the door to simulation of quantum many-body spin models with
low defect rates. Our demonstration of quantum erasure conversion adds an
important capability to the molecular quantum information processing toolkit,
and opens the door to mid-circuit quantum error correction in molecular
tweezer arrays.

##  II Framework for Site-Resolved Error Detection

In atomic and molecular tweezer array experiments, each tweezer site is either
empty or occupied by one particle. The local Hilbert space $\mathcal{H}$ at
each site thus consists of $\left|\emptyset\right\rangle$, the state
corresponding to an empty site, and the set of all internal states (Fig.
1(a)). (We will ignore motional states of the particle within a tweezer here,
although the same framework used here applies Scholl  _et al._ (2023b).) For a
particular application, only a target subset of these states $\mathcal{T}$ is
used. For example, in order to encode a qubit, at least two internal states
are needed.

For site-resolved error detection, one uses a set of detectable internal
states $\mathcal{F}$ to flag the presence of an error. The detection outcome
is binarized and converted into an error flag $\\{e_{i}\\}$ where $e_{i}=1$
($e_{i}=0$) indicates an error (error-free site) (Fig. 1(b)). $\\{e_{i}\\}$
could be used to remove errors mid-sequence, or to post-select for error-free
instances by excising error-flagged data.

When at least one bit of information is encoded in $\mathcal{T}$, errors with
a known location are called erasures Grassl  _et al._ (1997); Bennett  _et
al._ (1997). In particular, population leakage out of $\mathcal{T}$ into a set
of disjoint internal states $\mathcal{E}$ can be transferred to $\mathcal{F}$
and subsequently detected, a process known as erasure conversion Wu  _et al._
(2022).

For site-resolved error detection to be practically useful, it must not affect
error-free sites. Specifically, for quantum erasure detection, all populations
and coherences in $\mathcal{T}$ should be minimally affected by detection. We
will reserve the term mid-circuit to situations when $\mathcal{T}$ encodes a
qubit and is minimally affected by conversion or detection.

Figure 1:  Framework for Site-Resolved Error Detection. (a) $\mathcal{H}$ is
the Hilbert space on a tweezer site; $\mathcal{T}$ is the target space;
$\mathcal{F}$, the detection manifold, consists of detectable states used to
flag errors. Leakage errors transfer population from $\mathcal{T}$ to
$\mathcal{E}$, which can subsequently be converted to $\mathcal{F}$ for
detection. Loss of a particle results in an empty tweezer
($|\emptyset\rangle$). (b) Detection of total population in $\mathcal{F}$ is
binarized into a bitstring $\\{b_{i}\\}$, from which the error flag
$\\{e_{i}\\}$ is obtained.

##  III Our Platform: Programmable Optical Tweezer Arrays of Laser-Cooled
Molecules

Our experiment utilizes laser-cooled CaF molecules trapped in a programmable
optical tweezer array. A molecular beam of CaF molecules is created using a
cryogenic buffer gas cell Hutzler  _et al._ (2012), laser-slowed Truppe  _et
al._ (2017a), and trapped in a magneto-optical trap (MOT) Anderegg  _et al._
(2017); Truppe  _et al._ (2017b). Subsequently, the molecules are further
laser-cooled Cheuk  _et al._ (2018), compressed Li  _et al._ (2024), optically
trapped Anderegg  _et al._ (2018), and eventually transferred into a
programmable 1D array of optical tweezer traps Anderegg  _et al._ (2019);
Holland  _et al._ (2023a).

We detect CaF molecules via fluorescence imaging; in particular, only
molecules in the $X^{2}\Sigma(v=0,N=1)$ rovibrational manifold can be cooled
and imaged, where $v$ ($N$) denote vibrational (rotational) quantum numbers.
For fluorescence imaging, light addressing the
$X^{2}\Sigma(v=0,N=1)\rightarrow A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ transition is
applied, and fluorescence emitted on the same transition is collected on a
camera. During each image, the molecules explore all 12 hyperfine states in
$X^{2}\Sigma(v=0,N=1)$.

In this paper, no tweezer rearrangement is utilized, but an initial non-
destructive image is taken to identify occupied tweezers. This process leaves
molecules in the set of detectable states $\mathcal{F}=X^{2}\Sigma(v=0,N=1)$
with a small probability of molecular loss (leakage into
$\left|\emptyset\right\rangle$). All measurements are conditioned on tweezer
sites that are initially identified to be occupied.

##  IV New Site-Resolved Detection Scheme with Rapid Resonant Imaging

Compared to detection requirements needed for preparing defect-free arrays,
our work here, which concerns enhancing internal state purities and quantum
erasure detection, has much more stringent detection requirements. Rather than
only needing to preserve the total population in $\mathcal{F}$, we instead
need to either preserve the population of a single quantum state or preserve
both population and coherence within a qubit subspace. These requirements
necessitate a new detection scheme, which we describe in this section.

We first discuss the choice of $\mathcal{F}$ and $\mathcal{T}$. Because a
measurement projects a qubit, the detection manifold $\mathcal{F}$ must be
disjoint from the target state(s) $\mathcal{T}$. In particular, detected
population in $\mathcal{F}$ will correspond to an error
($\\{e_{i}\\}=\\{b_{i}\\}$). In order for a measurement of $\mathcal{F}$ to
minimally affect $\mathcal{T}$, a viable approach is to choose internal states
for $\mathcal{F}$ and $\mathcal{T}$ such that they have widely separated
optical transitions compared to their optical linewidths. This general
approach has been proposed and demonstrated in alkaline-earth atomic tweezer
arrays Wu  _et al._ (2022); Ma  _et al._ (2023); Scholl  _et al._ (2023a, b),
where $\mathcal{F}$ and $\mathcal{T}$ use separate ground and metastable
electronic manifolds.

Here, instead of using different electronic manifolds, we make use of two
different long-lived rotational manifolds available in a molecule to encode
$\mathcal{F}$ and $\mathcal{T}$. Specifically, in our scheme for CaF
molecules, we use the set of optical cycling states $X^{2}\Sigma(v=0,N=1)$ as
$\mathcal{F}$, and states from the ground rovibrational manifold
$X^{2}\Sigma(v=0,N=0)$ to form the target subspace $\mathcal{T}$ (Fig. 2(a)).
The states in $\mathcal{T}$ are separated from the detection manifold states
$\mathcal{F}$ by a frequency of $\Delta\approx 2\pi\times 20\,\text{GHz}$,
which is much larger than the optical linewidth $\Gamma\approx 2\pi\times
10\,\text{MHz}$ of the relevant imaging transition used to detect
$\mathcal{F}$.

We next describe considerations for fluorescence detection of $\mathcal{F}$.
We desire to minimize errors for $\mathcal{T}$ when measuring population in
$\mathcal{F}$. These can occur due to off-resonant photon scattering of
imaging light that affects both populations and coherences in $\mathcal{T}$. A
useful figure of merit is the ratio $\eta$ of the off-resonant scattering rate
on $\mathcal{T}$ to the fluorescence rate of $\mathcal{F}$. For our choice of
$\mathcal{F}$ and $\mathcal{T}$, $\eta$ can be as low as
$9\Gamma^{2}/\Delta^{2}\sim 10^{-6}$ for on-resonant imaging Sup . Because we
scatter $\sim 100-1000$ imaging photons to achieve sufficient signal-to-noise,
our scheme has a minimum error of $\sim 10^{-3}$, which predominantly takes
the form of leakage into other molecular states. These leakage errors could in
principle be corrected with more complex schemes that involve error conversion
to $\mathcal{F}$. We note that our on-resonant imaging scheme is different
than that previously used for detection in CaF tweezer arrays, where the
primary requirement was to preserve molecular population. This was
accomplished using light detuned from resonance by $\sim 4\,\Gamma$ Cheuk  _et
al._ (2018); Anderegg  _et al._ (2019); Holland  _et al._ (2023a).

An additional consideration for error detection is crosstalk, where population
leaks between $\mathcal{F}$ and $\mathcal{T}$. In our scheme, crosstalk is
protected by selection rules. Because the optical transitions used are all E1
transitions, and $\mathcal{F}=X^{2}\Sigma(v=0,N=1)$ and $\mathcal{T}\subset
X^{2}\Sigma(v=0,N=0)$ are of opposite parity, no population leakage occurs
even in the presence of off-resonant scattering from imaging light.

We next describe how we implement on-resonant imaging and optimize its
performance. Specifically, to image molecules in $\mathcal{F}$, we apply light
resonant with all resolved hyperfine transitions of the
$X^{2}\Sigma(v=0,N=1)-A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ line. Laser light
addressing the $X^{2}\Sigma(v=1,2,3,N=1)\rightarrow
A^{2}\Pi_{1/2}(v^{\prime}=v-1,J=1/2,+)$ transitions acting as vibrational
repumpers is also applied. We additionally apply light addressing the
$X^{2}\Sigma(v=0,N=3)\rightarrow B^{2}\Sigma(v^{\prime}=0,N=2)$ transition,
which acts as a rotational repumper Holland  _et al._ (2023a). These beams
propagate perpendicular to the imaging axis, and cause no observable
background for typical imaging durations.

Figure 2:  Site-Resolved Detection using Rapid Resonant Imaging. (a) Imaging
Scheme. Molecules in the detection manifold $\mathcal{F}=X^{2}\Sigma(v=0,N=1)$
scatter light resonant with the $X^{2}\Sigma(v=0,N=1)\rightarrow
A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ transition (orange arrows). Vibrational (red
solid arrows) and rotational repumping light (blue solid arrow) is applied.
The imaging light is off-resonant for states in the target manifold
$\mathcal{T}\subset X^{2}\Sigma(v=0,N=0)$. (b) Error Identification
Performance. The false positive probability $\epsilon_{01}$ (red squares) and
false negative probability $\epsilon_{10}$ (green diamonds) are shown versus
the classification threshold $\theta_{\text{E}}$. Vertical dashed line marks
$\theta_{\text{E}}=\theta_{\text{E,0}}=4.8$, the value used in this work. (c)
Population $P_{0}$ in
$\left|{X^{2}\Sigma(v=0,N=0),J=1/2,F=1,m_{F}=-1}\right\rangle\in\mathcal{T}$
versus imaging duration $t$. An exponential fit gives a $1/e$ lifetime of
$\tau=790(60)\,\text{ms}$, much longer than the imaging duration of
$3\,\text{ms}$.

To optimize imaging for the purpose of identifying errors, we prioritize
minimizing the false negative probability ($\epsilon_{10}$), since it is the
probability that errors remain undetected. We note that for tweezer
preparation, the false positive probability ($\epsilon_{01}$) only affects the
data rate, since any site flagged as an error could be discarded during
subsequent rearrangement or post-selection. We therefore seek to optimize
$\epsilon_{10}$ with respect to the imaging light intensity
($X^{2}\Sigma(v=0,N=1)-A^{2}\Pi_{1/2}(v=0,J=1/2,+)$), the imaging duration,
and the tweezer depth. Specifically, we minimize $\epsilon_{10}$ at a fixed
threshold of $\theta_{\text{E}}=\theta_{\text{E,0}}=4.8$, and achieve a false
positive probability ($\epsilon_{10}(\theta_{\text{E,0}})=0.033(4)$) at an
imaging intensity of $I=4.5\,\text{mW/cm}^{2}$, imaging duration of
$3\,\text{ms}$, and tweezer depth of $k_{B}\times 930(20)\,\mu\text{K}$ (Fig.
2(b)). In principle, for detectable errors, these values allow reduction of
intrinsic error rates by up to a factor of $\sim 30$. Henceforth,
$\theta_{\text{E}}$ is fixed to $\theta_{\text{E,0}}$ unless noted. Because
the background is low enough to be largely independent of the imaging
duration, $\epsilon_{01}$, which determines the data rate, is independent of
imaging parameters and only depends on camera noise.

To verify that our site-resolved detection scheme minimally affects population
in $\mathcal{T}$, which is composed of states from $X^{2}\Sigma(v=0,N=0)$, we
prepare molecules in $|X^{2}\Sigma(v=0,N=0),J=1/2,F=1,m_{F}=-1\rangle$ and
measure the population loss per image (Fig. 2(c)). We find a loss rate of
$3.8(3)\times 10^{-3}$ per image, confirming that our scheme minimally affects
the population in $X^{2}\Sigma(v=0,N=0)$.

Figure 3:  Robust and Enhanced Tweezer Preparation Using Site-Resolved Error
Detection. (a) Our enhanced internal state preparation procedure involves: (i)
optical pumping to $|D^{\prime}\rangle$ (orange arrow), (ii) population
transfer to $\mathcal{T}=|-\rangle$ via a microwave pulse (grey arrow), and
(iii) error detection of left-over molecules in $\mathcal{F}$. (b) Verifying
Tweezer Preparation. A non-destructive image (top) identifies initially loaded
sites. The detection-enhanced internal state preparation procedure is applied,
resulting in an error image (middle) that flags sites with internal state
errors. For verification, we state-selectively transfer molecules from
$\mathcal{T}=\left|-\right\rangle$ to $\left|\uparrow\right\rangle$ for a
final destructive measurement (bottom). (c) The detection-conditioned
probability $\tilde{p}_{E}$ of a molecule appearing in the final image versus
optical pumping duration $t$, which is used to control the bare optical
pumping fidelity $f_{\text{OP}}$. $\tilde{p}_{E}$ for various error-detection
image classification thresholds $\theta_{E}=(4.6,5.6,6.6,\infty)$ (blue
circles, red triangles, green squares, gray diamonds) are shown. Colored bands
show corresponding theoretical predictions. (d) Error-excised tweezer fidelity
$f_{E}$ (probability of being occupied and in the target internal state)
versus bare tweezer fidelity $f$. Inset shows the error-excised internal state
purity $p_{E}$ versus the optical pumping fidelity $f_{\text{OP}}$.

##  V Robust and Enhanced Internal State Preparation Using Site-Resolved
Detection of Errors

We next make use of our site-resolved error detection scheme to achieve robust
and enhanced tweezer preparation fidelities. In previous work, the fidelity of
having a tweezer loaded with a molecule in a target internal state was
reported to be $\approx 80\%$, limited both by optical pumping fidelities and
imperfect microwave transfers Holland  _et al._ (2023b); Bao  _et al._
(2023a).

Here, we present a scheme that allows internal state errors to appear as
detectable errors, which can be subsequently corrected mid-sequence. Our
scheme works as follows (Fig. 3(a)). Initially, molecules are distributed over
all 12 hyperfine states in $X^{2}\Sigma(v=0,N=1)$, the laser-coolable
manifold, which is also the detection manifold $\mathcal{F}$. We optically
pump the molecules into a single hyperfine state
$|D^{\prime}\rangle=|X^{2}\Sigma(v=0,N=1),J=3/2,F=2,m_{F}=-2\rangle\in\mathcal{F}$.
Subsequently, we “shelve” them into the ground rotational manifold
$X^{2}\Sigma(v=0,N=0)$. Specifically, we use a microwave pulse with a
frequency of $\approx 20.5\,\text{GHz}$ to state-selectively transfer
$|D^{\prime}\rangle$ to a single internal state
$|-\rangle=|X^{2}\Sigma(v=0,N=0),J=1/2,F=1,m_{F}=-1\rangle$, which is our
target state ($\mathcal{T}=\\{|-\rangle\\}$). All internal state preparation
errors leave the molecule in $\mathcal{F}$, that is,
$\mathcal{E}=\mathcal{F}$. Specifically, optical pumping errors leave a
molecule in $(\mathcal{F}-\left|D^{\prime}\right\rangle)\subset\mathcal{F}$,
and imperfect microwave transfer leaves a molecule in
$\left|D^{\prime}\right\rangle\in\mathcal{F}$. Internal state preparation
errors thus appear directly as detectable errors. They can then be identified
by the site-resolved detection scheme described in Section IV, which minimally
affects population in $\mathcal{T}=\\{|-\rangle\\}$ on error-free sites.

We next implement our scheme and examine its performance. Following internal
state preparation into $\mathcal{T}$, we perform one error detection image.
Subsequently, we transfer molecules from the target state
$\left|-\right\rangle$ to
$|{\uparrow}\rangle=\left|X^{2}\Sigma(v=0,N=1),J=1/2,F=0,m_{F}=0)\right\rangle$,
which is part of the detection manifold $\mathcal{F}$ (Fig. 3(b)). Finally, we
destructively measure the total population in $\mathcal{F}$.

To evaluate the robustness of our scheme, we intentionally vary the amount of
optical pumping errors by varying the optical pumping duration $t$. For each
optical pumping duration, which serves as a proxy for the optical pumping
fidelity, we extract $\tilde{p}_{E}$, the probability that we detect a
molecule in the final image conditioned upon no identified error. In (Fig.
3(c)), we show $\tilde{p}_{E}$ for various error detection thresholds
$\theta_{E}$. As expected, identifying errors more aggressively by lowering
$\theta_{E}$ (smaller false-negative probability $\epsilon_{10}$ and larger
false-positive probability $\epsilon_{01}$) improves $\tilde{p}_{E}$.

To show the gains of our scheme directly, we compare the internal state purity
conditioned upon no errors, $p_{\text{E}}$, with the optical pumping fidelity
$f_{\text{OP}}$. We determine the experimentally achieved $f_{\text{OP}}$ for
each pumping duration $t$ using $\tilde{p}_{E}$ and the error imaging
infidelities $\epsilon_{01}$ and $\epsilon_{10}$ Sup . As shown in Fig. 3(d,
inset), error excision allows significantly improved internal state purity,
especially at low optical pumping fidelities. Notably, we achieve an internal
state purity of $p_{E}=0.995(1)$ when the achieved bare optical pumping
fidelity is $f_{\text{OP}}\approx 0.96$. Notably, $p_{E}$ remains high even
when $f_{OP}$ is degraded. For example, at an optical pumping fidelity of
$f_{\text{OP}}=0.46$, the error-excised internal state purity remains at
$p_{E}\approx 0.96$. Finally, we extract the error-excised tweezer preparation
fidelity $f_{E}$, defined as the probability that a tweezer site is occupied
with a molecule in the target internal state when conditioned upon an absence
of a detected error. This is the practically useful metric for preparing low-
defect arrays using tweezer rearrangement. In Fig. 3(d), we show $f_{E}$
versus the bare tweezer preparation fidelity $f$, which takes into account
occupation errors. As expected, we find that error excision provides
significant improvement and robustness. Notably, we achieve a tweezer
preparation fidelity of $f_{E}=0.952(3)$, with an estimated reduction in data
rate of only $7.2(2)\%$. This significantly improves upon previous reported
values ($f\approx 80\%$) Holland  _et al._ (2023b); Bao  _et al._ (2023a); Lu
_et al._ (2024); Bao  _et al._ (2023b) and importantly, our method is robust
to day-to-day fluctuations in $f_{\text{OP}}$.

We note although all error-conditioned measurements in this work are done via
post-selection, where we excise sites that are error-flagged, our measurements
indicate that our detection scheme minimally affects correctly prepared
molecules. Thus, in combination with the ability to perform low-loss
rearrangement (previously demonstrated to have losses $\sim 0.1\%$ Holland
_et al._ (2023b)), our scheme can be used with another round of tweezer
rearrangement to achieve similar final fidelities. This will allow the
creation of arrays with low defect rates both in spatial configuration and
internal state purity, which is crucial to quantum computation and simulation
applications with large-scale molecular tweezer arrays. In particular, the
ability to create arrays with few-percent level errors opens the door to
simulating interacting quantum spin systems with low defect rates.
Specifically, these low defect rates would allow practical post-selection for
perfect arrays (possible with full state-resolved detection) for up to 10
sites, with data rates reduced by only $\sim 40\%$.

##  VI A Quantum Erasure Detection Scheme For CaF Molecules

Figure 4:  Coherent Control and Coherence Times of New Hyperfine Qubit. (a)
Previous rotational qubit encoding
$\left|{\downarrow}\right\rangle$-$\left|{\uparrow}\right\rangle$ (boxed in
blue) is not compatible with our error detection scheme, since
$\left|\uparrow\right\rangle$ is in the detection manifold $\mathcal{F}$. Our
new qubit encoding $\left|{\downarrow}\right\rangle$-$\left|0\right\rangle$
uses two hyperfine states in $X^{2}\Sigma(v=0,N=0)$ that are outside of
$\mathcal{F}$ (boxed in green). (b) Coherent manipulations of
$|{\downarrow}\rangle$-$|0\rangle$ is performed using a two-photon Raman
processes. Our two schemes rely on light near-detuned to the $X$-$A$ and
$X$-$B$ optical transitions. (c) Molecules are prepared in
$\left|\downarrow\right\rangle$ and driven using Raman light for a variable
duration $t$. The top (bottom) panels show Rabi oscillations of the
$\left|\downarrow\right\rangle$ population when driven by $X$-$A$ ($X$-$B$)
Raman light. Solid lines show fits to damped sinusoids. (d) Hyperfine qubit
coherence versus hold time is measured via a Ramsey sequence (top panel). Data
points show measured Ramsey fringe amplitudes versus hold time $t$, with solid
lines showing fits to exponential decay curves. Data without (with) a spin-
echo pulse are shown by blue circles (red triangles); data with XY8 dynamical
decoupling pulses using $X$-$B$($X$-$A$) light are shown by green squares
(orange diamonds). The fitted $1/e$ lifetimes are $19.5(7)\,\text{ms}$,
$288(15)\,\text{ms}$, $630(50)\,\text{ms}$, and $1100(70)\,\text{ms}$,
respectively. The dashed green (orange) line shows the estimated population
decay due to off-resonant scattering for the $X$-$B$ ($X$-$A$) Raman
configuration Sup .

In the second part of our work, we demonstrate erasure detection for the first
time in molecules. Compared to enhancing tweezer preparation using site-
resolved detection, where only the total population in $\mathcal{F}$ needs to
be preserved, erasure detection is more challenging since both population and
coherence in error-free qubits should be minimally affected.

###  VI.1 A New Hyperfine Qubit Encoding: Coherent Control and Qubit Coherence

To implement quantum erasure detection, we first seek a qubit encoding that is
compatible with our site-resolved error detection scheme described in Section
IV. In previous work with CaF molecules, qubits were encoded using two states
from neighboring rotational manifolds, namely,
$|{\uparrow}\rangle=|X^{2}\Sigma(v=0,N=1),J=1/2,F=0,m_{F}=0\rangle$ and
$|{\downarrow}\rangle=|X^{2}\Sigma(v=0,N=0),J=1/2,F=1,m_{F}=0\rangle$
Burchesky  _et al._ (2021); Holland  _et al._ (2023b); Bao  _et al._ (2023a).
This rotational encoding was used to demonstrate a 2-qubit iSWAP gate, but is
incompatible with our detection scheme because $|{\uparrow}\rangle$ is part of
the detection manifold $\mathcal{F}$. The detection of $\mathcal{F}$, which is
a projective measurement of its population, will destroy qubit coherence
between $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$.

Here, we use a new qubit encoding that makes use of two hyperfine states
within the ground rovibrational manifold $X^{2}\Sigma(v=0,N=0)$, which are
dark to the imaging light used for site-resolved error detection.
Specifically, the hyperfine encoding uses the states
$|{\downarrow}\rangle=|X^{2}\Sigma(v=0,N=0),J=1/2,F=1,m_{F}=0\rangle$ and
$|0\rangle=|X^{2}\Sigma(v=0,N=0),J=1/2,F=0,m_{F}=0\rangle$ (Fig. 4(a)). These
states are predicted to have long coherence times because of the absence of
tensor ac Stark shifts and their low magnetic moments. In the framework of
Section II, the detection manifold $\mathcal{F}$ remains the same
($\mathcal{F}=X^{2}\Sigma(v=0,N=1)$), while the target subspace
$\mathcal{T}=\\{|{\downarrow}\rangle,|{0}\rangle\\}$ now consists of the two
hyperfine qubit states.

We first demonstrate coherent control over this hyperfine qubit, and that it
has long coherence times. The two qubit states are connected by an M1
transition at $\sim 122\,\text{MHz}$. We coherently manipulate the qubit with
two-photon Raman transitions rather than directly driving the M1 transition.
Specifically, we use Raman light near-detuned to either the $X-A$ or $X-B$
optical transitions (Fig. 4(b)). The two frequencies in the Raman light co-
propagate and drive $\sigma^{+}$ transitions. To demonstrate coherent control,
we prepare molecules in $|{\downarrow}\rangle$, and apply Raman light for a
variable duration. As shown in Fig. 4(c), we observe Rabi oscillations with
Rabi frequencies as high as $\Omega_{R}\approx 2\pi\times 10\,\text{kHz}$ for
both Raman schemes.

We next measure the qubit coherence time using a Ramsey spectroscopy sequence
consisting of two Raman $\pi/2$ pulses separated by a variable duration. In
order to minimize the effect of inhomogeneous light shifts among the tweezer
sites, we operate at a tweezer depth of $U_{\text{hf}}=k_{B}\times
39(1)\,\mu\text{K}$. By measuring the decay rate of the amplitude of the
Ramsey fringe, we find a bare coherence time of
$T_{2}^{*}=19.5(7)\,\text{ms}$, which can be increased to
$T_{2}=288(15)\,\text{ms}$ with a single spin-echo. The coherence time can be
further extended by applying dynamical decoupling. Specifically, using a XY8
pulse sequence (pulse separation of 33.3 ms), the coherence times increase to
$630(50)\,\text{ms}$ and $1100(70)\,\text{ms}$ for the $X$-$B$ Raman scheme
and $X$-$A$ Raman scheme, respectively (Fig. 4(d)).

In addition to long coherence times, we note that our hyperfine qubit encoding
is also compatible with the previously used rotational qubit encoding
Burchesky  _et al._ (2021); Holland  _et al._ (2023b); Bao  _et al._ (2023a).
We have verified that the three states involved can remain simultaneously
coherent, and that we can coherently transfer information encoded in a
rotational qubit ($|{\uparrow}\rangle,|{\downarrow}\rangle$) to a hyperfine
encoding ($|{\downarrow}\rangle,|{0}\rangle$) Sup .

Figure 5:  Mid-Circuit Quantum Erasure Detection. (a) Our composite erasure
detection scheme comprises a tweezer depth increase, an error detection image,
and a mid-detection $X$-$B$ Raman $\pi$-pulse. (b) Ramsey fringe offset versus
the number of composite erasure detections $N_{\text{rep}}$. (c) Ramsey fringe
amplitude versus $N_{\text{rep}}$. For (b,c), green circles (orange triangles)
are data obtained using the $X$-$B$ ($X$-$A$) Raman scheme.(d) Population loss
measurements from combinations of tweezer ramp (T), error detection light (E),
and $v=1$ repump (V). The plot shows population $P_{\uparrow}$ versus
$N_{\text{rep}}$ cycles of TEV (blue circles), EV (red triangles), T (purple
squares). Brown diamonds are measured at a shallow tweezer depth without TEV.
The brown, purple, and red arrows indicate contributions from background
vacuum and blackbody loss, tweezer ramp, and erasure detection. Inset: The sum
of these measured loss rate contributions (colored bars summing to gray
pentagon) is consistent with the total loss (blue bar and circle).

###  VI.2 Composite Erasure Detection Sequence to Preserve Qubit Coherence

Having established long coherence times and coherent control of the hyperfine
qubit, we next investigate whether qubit coherence is preserved after an error
detection image. Compared to the tweezer depth $U_{\text{hf}}$ used to obtain
long hyperfine qubit coherence times, we find that a much higher tweezer depth
of $U_{\text{ED}}=k_{B}\times 930(20)\,\mu\text{K}$ is needed to obtain high
error detection fidelities.

Therefore, for erasure detection, the tweezer trap needs to be ramped up by a
factor of $\sim 30$ from $U_{\text{hf}}$ to $U_{\text{ED}}$. At the deeper
depth, the qubit frequency shifts by $2\pi\times 0.40(6)\,\text{kHz}$, which
on average leads to additional phase accumulation in the hyperfine qubit. This
phase accumulation is inconsequential if the tweezers had uniform depth.
However, our tweezer array has depth inhomogeneity of $\sim 10\%$, giving rise
to an effective phase variation of $\approx 1\,\text{rad}$ at $U_{\text{ED}}$
over the imaging duration. To counteract this inhomogeneous dephasing, we
implement a “composite erasure detection” scheme consisting of four steps
(Fig. 5(a)): 1) the tweezer depth is ramped from $U_{\text{hf}}$ to
$U_{\text{ED}}$, 2) local errors are detected using rapid resonant imaging, 3)
a $\pi$-pulse is applied to the qubit halfway through the image, and 4) the
tweezer depth is returned to $U_{\text{hf}}$. The $\pi$-pulse echos out the
effects of inhomogeneous light shifts due to tweezer depth variations, along
with Stark shifts arising from the imaging light.

To determine how well the qubit population and coherence are preserved during
composite erasure detection, we repeatedly apply composite erasure detection
between two $\pi$/2 pulses in a Ramsey sequence. We find that a Ramsey fringe
is observable even after multiple images. The offset of the Ramsey fringe is
proportional to the total population, and the amplitude is proportional to
both the total population and coherence of the qubit states. Per image, we
determine a fractional population reduction of $\varepsilon_{p}=3.3(2)\times
10^{-2}$ (Fig. 5(b)) and a fractional amplitude reduction of
$\varepsilon_{c}=4.6(3)\times 10^{-2}$ (Fig. 5(c)). These measurements
indicate that our composite erasure detection sequence affects the hyperfine
qubit population and coherence only at the few percent level. To our
knowledge, this is the first demonstration of erasure detection in molecules.

We note that for the purposes of composite erasure detection, we only use
$X-B$ Raman light, which has much shorter population lifetimes and coherence
times compared to our $X-A$ Raman scheme due to higher off-resonant
scattering. This is because our $X-A$ Raman light (due to constraints on laser
availability) is resonant with the detection manifold
$\mathcal{F}=X^{2}\Sigma(v=0,N=1)$. It resonantly heats a molecule in
$\mathcal{F}$ out of a tweezer, and is therefore not compatible with erasure
detection. This limitation can easily be overcome with a $X-A$ Raman light
source at a more optimal detuning. As we describe in detail in the next
section, we anticipate that errors at the $10^{-3}$ level are achievable.

###  VI.3 Technical and Fundamental Limits of Our Erasure Detection Scheme

To understand the origin of the $\sim 10^{-2}$ level errors per erasure image,
we independently measure the contributions of each step in our composite
erasure detection sequence. First, we quantify the effect of off-resonant
photon scattering on the qubit states caused by the spin-echo $\pi$-pulse. We
perform a similar Ramsey measurement with only $\pi$ pulses applied (without
tweezer ramps and imaging light). This gives a population error per image of
$\varepsilon_{p,R}=2.7(2)\times 10^{-2}$ and a coherence error per image of
$\varepsilon_{c,R}=4.4(4)\times 10^{-2}$. Comparing these with the total
composite erasure errors $\varepsilon_{p}$ and $\varepsilon_{c}$, we conclude
that our present scheme is limited by off-resonant scattering of Raman light
by the qubit states in $\mathcal{T}$.

Off-resonant scattering can be significantly improved by a more optimal Raman
scheme. In particular, we expect that coherence and population errors could be
reduced to the $10^{-3}$ level. To quantify the off-resonant scattering rate,
we define a Raman $\pi$-pulse quality factor
$Q_{\pi}=\Omega_{R}/\pi\Gamma_{sc}$, where $\Omega_{R}$ is the two-photon Rabi
frequency, and $\Gamma_{sc}$ is the photon scattering rate. Because off-
resonant scattering predominantly leads to leakage from the qubit subspace,
$\Gamma_{sc}$ can be directly inferred from population loss rates. $Q_{\pi}$
is therefore directly measurable. In our present Raman scheme ($X$-$B$), the
light is detuned halfway between the $X^{2}\Sigma(N=0)\rightarrow
B^{2}\Sigma(N=1,J=1/2)$ and $X^{2}\Sigma(N=0)\rightarrow
B^{2}\Sigma(N=1,J=3/2)$ transitions. This results in a single-photon detuning
of $\Delta_{XB}\sim 2\pi\times 500\,\text{MHz}$ for optical transitions of the
qubit states in $\mathcal{T}$. $Q_{\pi}$ for this scheme is measured to be
37(3).

To partially investigate a better Raman scheme, we performed similar Ramsey
measurements using an $X$-$A$ Raman scheme, where the Raman light is detuned
by $\Delta_{XA}\sim 2\pi\times 20\,\text{GHz}$ from the optical transitions of
the hyperfine qubit. This light cannot be used for composite erasure detection
because it is resonant with the detection manifold $\mathcal{F}$.
Nevertheless, we can quantify its effect on the hyperfine qubit. A Ramsey
measurement reveals significantly lower population errors and coherence errors
per $\pi$-pulse. Specifically, $\varepsilon_{p,R}=3.4(3)\times 10^{-3}$ and
$\varepsilon_{c,R}=10(1)\times 10^{-3}$. When this scheme is employed in a
composite erasure detection sequence (with tweezer ramps but no erasure image
due to technical limitations), we obtain overall population and coherence
errors of $\varepsilon_{p}=5.2(2)\times 10^{-3}$ and
$\varepsilon_{c}=11(1)\times 10^{-3}$ per image, respectively (Fig. 5(b,c)).
These improvements agree with predictions due to off-resonant scattering Sup .

Our current $X$-$A$ Raman scheme could thus be improved significantly and made
compatible with composite erasure detection simply by using a different
detuning. In particular, by detuning halfway between the
$X^{2}\Sigma(v=0)\rightarrow A^{2}\Pi_{1/2}(v=0)$ and the
$X^{2}\Sigma(v=0)\rightarrow A^{2}\Pi_{3/2}(v=0)$ transitions ($\sim
1.1\,\text{THz}$ from each), the Raman light will no longer be resonant with
the detection manifold $\mathcal{F}$. The number of off-resonantly scattered
photons for a Raman $\pi$-pulse would then decrease by a factor of 30 compared
to our current $X$-$A$ scheme. Conservatively, we estimate that population and
coherence errors due to Raman light at an optimal detuning will be suppressed
to $\varepsilon_{p,R}\approx 1\times 10^{-4}$ and $\varepsilon_{c,R}\approx
6\times 10^{-3}$.

When the off-resonant Raman scattering errors are reduced to these levels, we
must examine the remaining error sources. These include the detection light,
the tweezer depth ramps, and loss to higher vibrational states due to
blackbody radiation. We measure their individual contributions by comparing
population loss from $|{\downarrow}\rangle$ with and without various
combinations of detection light, tweezer ramps, and $v=1$ vibrational
repumping light (Fig. 5(d)). From these measurements, we isolate a population
loss per image of $3.0(3)\times 10^{-3}$ due to detection light, and
$2.6(3)\times 10^{-3}$ due to the tweezer ramps. The background loss in the
absence of detection light and tweezer ramps is $1.2(1)\times 10^{-3}$,
consistent with the predicted excitation rate to higher vibrational states due
to blackbody radiation at room temperature. Therefore, by using optimally
detuned Raman light, we estimate that total population (coherence) errors of
$7\times 10^{-3}$ ($1.4\times 10^{-2}$) per image can be achieved.

The population loss rates can be further improved. First, improving the photon
collection efficiency of the imaging system allows shorter image durations and
therefore lower population loss. Second, the population loss appears primarily
as internal state leakage errors into the $X^{2}\Sigma(v=0,N=2)$ manifold and
the $\left|X^{2}\Sigma(v=0,N=0),J=1/2,F=1,m_{F}=\pm 1\right\rangle$ states.
These errors can be erasure-converted to $\mathcal{F}$ via additional
microwave and optical pulses. Thus, we estimate that the population error due
to detection light can be suppressed to well below $10^{-3}$. Furthermore,
blackbody induced errors can be erasure-converted, as we will show, or
eliminated in a cryogenic setup. Depending on the population loss mechanism of
the tweezer ramps, which we did not explore, the corresponding population
errors could potentially be converted and corrected. Therefore, population
errors at the low $10^{-3}$ level could potentially be achievable.

Figure 6:  Erasure Conversion of Blackbody Errors. (a) Erasure Conversion
Scheme. Blackbody radiation (black wavy arrow) predominantly excites molecules
from $\mathcal{T}$ to $\mathcal{E}$. We erasure convert to $\mathcal{F}$ using
$v=1$ repumping light (red arrow). (b) Robustness of $\mathcal{F}$ to
blackbody leakage errors. Blackbody leakage errors (black wavy) primarily
populate $X^{2}\Sigma(v=0,N=3)$ states, which are converted back to
$\mathcal{F}$ by the rotational repumping light present during erasure
detection (solid green). (c) Hyperfine qubit population $P_{0}$ versus hold
time $t$. Data without erasure excision (gray squares), with a single final
error detection image (red triangles), and with excision using five mid-
sequence images (blue circles) are shown; solid lines show fits to exponential
decay curves. Inset: fitted $1/e$ loss rates $\gamma_{p}$ are shown, with red
band ($\pm 1\sigma$) indicating prediction from separately measured rates Sup
. No dynamical decoupling Raman pulses are applied. (d) Ramsey fringe
amplitude of the hyperfine qubit versus hold time with (red triangles) and
without (gray squares) error excision. Dynamical decoupling with Raman pulses
is applied during the hold time. Upper Inset: Amplitude loss rate $\gamma_{c}$
without error excision (gray square) is consistent with the expected combined
loss rate from the $X$-$B$ Raman beams and the $N=0$ population loss (green
dashed line). Lower inset shows the loss rate improvement $\gamma_{c}$ when
excising errors. The improvement for the amplitude loss rate (green hexagon)
is consistent with the improvement in population loss (purple diamond).

##  VII Quantum Erasure Conversion and Detection of Blackbody Errors

We next demonstrate that leakage errors induced by blackbody radiation can be
converted and detected as erasures. In CaF, blackbody radiation at room-
temperature predominantly drives $\Delta v=1$ vibrational-changing E1
transitions, and leads to population leakage from the vibrational ground
manifold $X^{2}\Sigma(v=0)$ to the first excited vibrational manifold
$X^{2}\Sigma(v=1)$ at a rate of $\Gamma_{01}\approx 0.3\,\text{s}^{-1}$ Hou
and Bernath (2018). Consequently, the hyperfine qubit states
$\mathcal{T}=\\{|{\downarrow}\rangle,|0\rangle\\}\subset X^{2}\Sigma(v=0,N=0)$
predominantly leak to $\mathcal{E}=X^{2}\Sigma(v=1,N=1)$. As we will show,
these errors can be erasure converted by using $v=1$ vibrational repumping
light, which excites molecules from $\mathcal{E}$ to
$A^{2}\Pi_{1/2}(v=0,J=1/2,+)$. The excited molecules decay back to
$\mathcal{F}$ with $98\%$ probability (Fig. 6(a)).

We note that $\mathcal{E}$ spontaneously decays back to $X^{2}\Sigma(v=0)$ at
a rate $\Gamma_{10}\approx 5.3\,\text{s}^{-1}$, which is much quicker than the
blackbody excitation rate $\Gamma_{01}$. If such a decay occurs, conversion
via $v=1$ vibrational repumping light is no longer possible. Therefore, in our
scheme, one must erasure convert $\mathcal{E}$ to $\mathcal{F}$ at a rate much
faster than $\Gamma_{10}$. It is not necessary, however, to detect
$\mathcal{F}$ faster than $\Gamma_{10}$.

We next discuss the effect of blackbody radiation on the detection manifold
$\mathcal{F}$. Absorption of a blackbody photon brings a molecule from
$\mathcal{F}$ to $X^{2}\Sigma(v=1,N=0,2)$. With a $1/e$ decay time of
$1/\Gamma_{10}\approx 190\,\text{ms}$, these states spontaneously decay back
to $\mathcal{F}$ and $X^{2}\Sigma(v=0,N=3)$. Molecules in the latter manifold
can be pumped back to $\mathcal{F}$ via rotational repumping light, which is
already present during fluorescence imaging. Therefore, even in the presence
of blackbody radiation, there is minimal undetectable leakage from
$\mathcal{F}$ (Fig. 6(b)).

Experimentally, we first confirm that population in $\mathcal{F}$ is robust to
blackbody induced leakage errors. As expected, when a rotational repump is
applied during detection, we find that the population lifetime
($\tau=13.5(5)\,\text{s}$) exceeds the blackbody-limited $X^{2}\Sigma(v=0)$
lifetime by a factor of $\approx 4$. We attribute residual loss to background
gas collisions and double blackbody excitation events into $N\geq 5$, which
are not rotationally repumped during detection Sup .

Next, we investigate detection of population leakage errors by preparing
molecules in each of the two hyperfine levels in $\mathcal{T}$ and comparing
population loss rates with and without excising errors. Specifically, we hold
the molecules for a variable duration, and perform mid-sequence conversion by
applying $v=1$ repumping light every $50\,\text{ms}$. Subsequently, we perform
site-resolved error detection, and destructively detect population remaining
in the initially prepared hyperfine state. Without error excision, we measure
a population loss rate of $\gamma_{p}=0.30(1)\,\text{s}^{-1}$, consistent with
the predicted blackbody excitation rate. With error excision, the loss rate
decreases to $\gamma_{p,E}=0.13(1)\,\text{s}^{-1}$ (Fig. 6(c)).

Using a loss rate model incorporating vacuum loss, blackbody excitation,
erasure conversion efficiency (estimated to be $\approx 90\%$), identification
errors, and the internal state preparation fidelity, we obtain loss rates
consistent with the measurements in Fig. 6(c). We use this model and the
measured lifetime to infer that $\approx 80\%$ of blackbody errors are
converted and detected correctly Sup . With near-term technical improvements,
efficiencies approaching $95\%$ should be possible Sup .

Having explored conversion of blackbody population errors followed by a single
error detection image, we next explore mid-sequence detection. We measure the
erasure-excised loss rate with five images equally spaced during a variable
hold time and measure a loss rate of $\gamma_{p,E5}=0.12(1)\,\text{s}^{-1}$,
which is lower than that obtained with one final erasure image by
$0.014(15)\,\text{s}^{-1}$. This shows that the interspersed detection images
do not lead to additional observable loss. In fact, the data hints that
because mid-sequence detection reduces population leakage from $\mathcal{F}$
to higher vibrational and rotational states ($N\geq 5$), the lifetime of the
detection states $\mathcal{F}$ is increased. We note that for these population
loss investigations, we neither apply dynamical decoupling nor the spin-echo
pulse in composite erasure detection, and are therefore free from the
technical limitations of our Raman scheme (see Section VI.3).

Lastly, we demonstrate that coherence loss due to blackbody leakage errors can
be reduced by mid-circuit erasure conversion. Using a Ramsey sequence, we
measure the hyperfine qubit coherence with and without erasure excision. For
these measurements, we erasure-convert blackbody excitations mid-circuit, and
perform error detection at the end of the sequence. During the Ramsey hold
time, XY8 dynamical decoupling using the $X$-$B$ Raman configuration is
applied. Without excision, the coherence loss rate is
$1.64(4)\,\text{s}^{-1}$, consistent with the hyperfine qubit coherence in the
$X$-$B$ Raman scheme, blackbody loss, and vacuum loss. In particular, off-
resonant scattering from the Raman light contributes at least $\approx 50\%$
of the decoherence, and can be substantially reduced (see discussion in
Section VI.3). By excising data with erasures, we see a small but
statistically significant ($\approx 3\sigma$) decrease in the decoherence rate
of $0.17(6)\,\text{s}^{-1}$ (Fig. 6(d)), consistent with the improvement the
measured population loss $(0.162(15)\,\text{s}^{-1})$. Taken together, our
measurements show that blackbody radiation primarily causes population loss
but not qubit dephasing. Although the improvement is small compared to the
total coherence loss rate, which is limited by population loss due to the XY8
Raman pulses, we expect that these losses can be substantially decreased by up
to a factor of 60 to $0.1\,\text{s}^{-1}$ with a more optimal Raman light
source (see Section VI.3). At that point, blackbody radiation will become a
non-negligible source of error that can be erasure-converted and detected.

##  VIII Conclusion and Outlook

In summary, we have demonstrated 1) a site-resolved detection scheme that
enables robust and record-level fidelities for preparing tweezers loaded with
molecules occupying a single internal state, 2) an erasure detection scheme
for molecules for the first time, with error rates at the few percent level,
and 3) mid-circuit erasure conversion of blackbody induced leakage errors.
These results open several possibilities. First, our achieved tweezer
preparation fidelity of $\approx 95\%$ opens access to quantum many-body
simulation of spin models with few percent-level defects. Second, our erasure
conversion scheme adds a powerful capability that could be useful for quantum
error correction. Third, our work on erasure conversion of blackbody errors
shows that they do not fundamentally impose an upper limit in circuit depth in
molecular tweezer arrays, since they can be corrected mid-circuit when
combined with tweezer reloading and state initialization Singh  _et al._
(2022); Shaw  _et al._ (2023).

In addition to the possibilities above, our development of a new hyperfine
qubit that is simultaneously coherent with a previously demonstrated
rotational encoding opens the door to applications requiring 3-level systems,
such as quantum information processing with qutrits, simulation of $S=1$
quantum spin models, bosonic $t$-$J$ models Homeier  _et al._ (2023), and
lattice gauge theory models Halimeh  _et al._ (2023). Separately, our work on
detecting blackbody excitations could be relevant for investigations of
molecule-based temperature standards Norrgard  _et al._ (2021). Lastly, our
work on enhanced tweezer preparation and catching blackbody errors could also
aid molecular tweezer-based precision measurement experiments Kozyryev and
Hutzler (2017); Anderegg  _et al._ (2023).

###### Acknowledgements.

We thank Jeff Thompson, Waseem Bakr, and their groups for fruitful
discussions. This work is supported by the National Science Foundation under
Grant No. 2207518. C. M. H. acknowledges support from a Joseph Taylor Graduate
Student Fellowship. S. J. L. and C. L. W. acknowledge support from Princeton
Quantum Initiative Graduate Student Fellowships. L. W. C. acknowledges support
from the Alfred P. Sloan Foundation under Grant No. FG-2022-19104.

## References

  * Kaufman and Ni (2021) A. M. Kaufman and K.-K. Ni, Nature Physics 17, 1324 (2021).
  * Endres  _et al._ (2016) M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletić, M. Greiner, and M. D. Lukin, Science 354, 1024 (2016).
  * Labuhn  _et al._ (2016) H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Nature 534, 667 (2016).
  * Yan  _et al._ (2013) B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature 501, 521 (2013).
  * Christakis  _et al._ (2023) L. Christakis, J. S. Rosenberg, R. Raj, S. Chi, A. Morningstar, D. A. Huse, Z. Z. Yan, and W. S. Bakr, Nature 614, 64 (2023).
  * Gregory  _et al._ (2024) P. D. Gregory, L. M. Fernley, A. L. Tao, S. L. Bromley, J. Stepp, Z. Zhang, S. Kotochigova, K. R. A. Hazzard, and S. L. Cornish, Nature Physics 20, 415 (2024).
  * Carr  _et al._ (2009) L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, New Journal of Physics 11, 055049 (2009).
  * Bohn  _et al._ (2017) J. L. Bohn, A. M. Rey, and J. Ye, Science 357, 1002 (2017).
  * Safronova  _et al._ (2018) M. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Reviews of Modern Physics 90, 025008 (2018).
  * Kozyryev and Hutzler (2017) I. Kozyryev and N. R. Hutzler, Physical review letters 119, 133002 (2017).
  * Anderegg  _et al._ (2023) L. Anderegg, N. B. Vilas, C. Hallas, P. Robichaud, A. Jadbabaie, J. M. Doyle, and N. R. Hutzler, Science 382, 665 (2023).
  * DeMille (2002) D. DeMille, Physical Review Letters 88, 067901 (2002).
  * Ni  _et al._ (2018) K.-K. Ni, T. Rosenband, and D. D. Grimes, Chem. Sci. 9, 6830 (2018).
  * Yu  _et al._ (2019) P. Yu, L. W. Cheuk, I. Kozyryev, and J. M. Doyle, New Journal of Physics 21, 093049 (2019).
  * Micheli  _et al._ (2006) A. Micheli, G. K. Brennen, and P. Zoller, Nature Physics 2, 341 (2006).
  * Gadway and Yan (2016) B. Gadway and B. Yan, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 152002 (2016).
  * Anderegg  _et al._ (2019) L. Anderegg, L. W. Cheuk, Y. Bao, S. Burchesky, W. Ketterle, K.-K. Ni, and J. M. Doyle, Science 365, 1156 (2019).
  * Zhang  _et al._ (2022) J. T. Zhang, L. R. B. Picard, W. B. Cairncross, K. Wang, Y. Yu, F. Fang, and K.-K. Ni, Quantum Science and Technology 7, 035006 (2022).
  * Holland  _et al._ (2023a) C. M. Holland, Y. Lu, and L. W. Cheuk, Phys. Rev. Lett. 131, 053202 (2023a).
  * Ruttley  _et al._ (2023) D. K. Ruttley, A. Guttridge, S. Spence, R. C. Bird, C. R. Le Sueur, J. M. Hutson, and S. L. Cornish, Phys. Rev. Lett. 130, 223401 (2023).
  * Vilas  _et al._ (2024) N. B. Vilas, P. Robichaud, C. Hallas, G. K. Li, L. Anderegg, and J. M. Doyle, Nature 628, 282 (2024).
  * Burchesky  _et al._ (2021) S. Burchesky, L. Anderegg, Y. Bao, S. S. Yu, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Phys. Rev. Lett. 127, 123202 (2021).
  * Park  _et al._ (2023) A. J. Park, L. R. B. Picard, G. E. Patenotte, J. T. Zhang, T. Rosenband, and K.-K. Ni, Phys. Rev. Lett. 131, 183401 (2023).
  * Holland  _et al._ (2023b) C. M. Holland, Y. Lu, and L. W. Cheuk, Science 382, 1143 (2023b).
  * Bao  _et al._ (2023a) Y. Bao, S. S. Yu, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Science 382, 1138 (2023a).
  * Lu  _et al._ (2024) Y. Lu, S. J. Li, C. M. Holland, and L. W. Cheuk, Nature Physics , 1 (2024).
  * Bao  _et al._ (2023b) Y. Bao, S. S. Yu, J. You, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, arXiv:2309.08706v1 (2023b).
  * Scholl  _et al._ (2023a) P. Scholl, A. L. Shaw, R. B.-S. Tsai, R. Finkelstein, J. Choi, and M. Endres, Nature 622, 273 (2023a).
  * Ma  _et al._ (2023) S. Ma, G. Liu, P. Peng, B. Zhang, S. Jandura, J. Claes, A. P. Burgers, G. Pupillo, S. Puri, and J. D. Thompson, Nature 622, 279 (2023).
  * Scholl  _et al._ (2023b) P. Scholl, A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, J. Choi, and M. Endres, arXiv:2311.15580 (2023b).
  * Picard  _et al._ (2024) L. R. B. Picard, G. E. Patenotte, A. J. Park, S. F. Gebretsadkan, and K.-K. Ni, PRX Quantum 5, 020344 (2024).
  * Ruttley  _et al._ (2024) D. K. Ruttley, A. Guttridge, T. R. Hepworth, and S. L. Cornish, PRX Quantum 5, 020333 (2024).
  * Grassl  _et al._ (1997) M. Grassl, T. Beth, and T. Pellizzari, Phys. Rev. A 56, 33 (1997).
  * Bennett  _et al._ (1997) C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin, Phys. Rev. Lett. 78, 3217 (1997).
  * Wu  _et al._ (2022) Y. Wu, S. Kolkowitz, S. Puri, and J. D. Thompson, Nature communications 13, 4657 (2022).
  * Kubica  _et al._ (2023) A. Kubica, A. Haim, Y. Vaknin, H. Levine, F. Brandão, and A. Retzker, Phys. Rev. X 13, 041022 (2023).
  * Kang  _et al._ (2023) M. Kang, W. C. Campbell, and K. R. Brown, PRX Quantum 4, 020358 (2023).
  * Teoh  _et al._ (2023) J. D. Teoh, P. Winkel, H. K. Babla, B. J. Chapman, J. Claes, S. J. de Graaf, J. W. O. Garmon, W. D. Kalfus, Y. Lu, A. Maiti, K. Sahay, N. Thakur, T. Tsunoda, S. H. Xue, L. Frunzio, S. M. Girvin, S. Puri, and R. J. Schoelkopf, Proceedings of the National Academy of Sciences 120, e2221736120 (2023).
  * Chou  _et al._ (2023) K. S. Chou, T. Shemma, H. McCarrick, T.-C. Chien, J. D. Teoh, P. Winkel, A. Anderson, J. Chen, J. Curtis, S. J. de Graaf, _et al._ , arXiv:2307.03169 (2023).
  * Levine  _et al._ (2024) H. Levine, A. Haim, J. S. C. Hung, N. Alidoust, M. Kalaee, L. DeLorenzo, E. A. Wollack, P. Arrangoiz-Arriola, A. Khalajhedayati, R. Sanil, H. Moradinejad, Y. Vaknin, A. Kubica, D. Hover, S. Aghaeimeibodi, J. A. Alcid, C. Baek, J. Barnett, K. Bawdekar, P. Bienias, H. A. Carson, C. Chen, L. Chen, H. Chinkezian, E. M. Chisholm, A. Clifford, R. Cosmic, N. Crisosto, A. M. Dalzell, E. Davis, J. M. D’Ewart, S. Diez, N. D’Souza, P. T. Dumitrescu, E. Elkhouly, M. T. Fang, Y. Fang, S. Flammia, M. J. Fling, G. Garcia, M. K. Gharzai, A. V. Gorshkov, M. J. Gray, S. Grimberg, A. L. Grimsmo, C. T. Hann, Y. He, S. Heidel, S. Howell, M. Hunt, J. Iverson, I. Jarrige, L. Jiang, W. M. Jones, R. Karabalin, P. J. Karalekas, A. J. Keller, D. Lasi, M. Lee, V. Ly, G. MacCabe, N. Mahuli, G. Marcaud, M. H. Matheny, S. McArdle, G. McCabe, G. Merton, C. Miles, A. Milsted, A. Mishra, L. Moncelsi, M. Naghiloo, K. Noh, E. Oblepias, G. Ortuno, J. C. Owens, J. Pagdilao, A. Panduro, J.-P. Paquette, R. N. Patel, G. Peairs, D. J. Perello, E. C. Peterson, S. Ponte, H. Putterman, G. Refael, P. Reinhold, R. Resnick, O. A. Reyna, R. Rodriguez, J. Rose, A. H. Rubin, M. Runyan, C. A. Ryan, A. Sahmoud, T. Scaffidi, B. Shah, S. Siavoshi, P. Sivarajah, T. Skogland, C.-J. Su, L. J. Swenson, J. Sylvia, S. M. Teo, A. Tomada, G. Torlai, M. Wistrom, K. Zhang, I. Zuk, A. A. Clerk, F. G. S. L. Brandão, A. Retzker, and O. Painter, Phys. Rev. X 14, 011051 (2024).
  * Hutzler  _et al._ (2012) N. R. Hutzler, H.-I. Lu, and J. M. Doyle, Chem. Rev. 112, 4803 (2012).
  * Truppe  _et al._ (2017a) S. Truppe, H. J. Williams, N. J. Fitch, M. Hambach, T. E. Wall, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt, New Journal of Physics 19, 022001 (2017a).
  * Anderegg  _et al._ (2017) L. Anderegg, B. L. Augenbraun, E. Chae, B. Hemmerling, N. R. Hutzler, A. Ravi, A. Collopy, J. Ye, W. Ketterle, and J. M. Doyle, Phys. Rev. Lett. 119, 103201 (2017).
  * Truppe  _et al._ (2017b) S. Truppe, H. J. Williams, M. Hambach, L. Caldwell, N. J. Fitch, E. A. Hinds, B. E. Sauer, and M. R. Tarbutt, Nature Physics 13, 1173 (2017b).
  * Cheuk  _et al._ (2018) L. W. Cheuk, L. Anderegg, B. L. Augenbraun, Y. Bao, S. Burchesky, W. Ketterle, and J. M. Doyle, Phys. Rev. Lett. 121, 083201 (2018).
  * Li  _et al._ (2024) S. J. Li, C. M. Holland, Y. Lu, and L. W. Cheuk, Physical Review Letters (2024), 2311.05447v1 .
  * Anderegg  _et al._ (2018) L. Anderegg, B. L. Augenbraun, Y. Bao, S. Burchesky, L. W. Cheuk, W. Ketterle, and J. M. Doyle, Nature Physics 14, 890 (2018).
  * (48) See Supplemental Material.
  * Hou and Bernath (2018) S. Hou and P. F. Bernath, Journal of Quantitative Spectroscopy and Radiative Transfer 210, 44 (2018).
  * Singh  _et al._ (2022) K. Singh, S. Anand, A. Pocklington, J. T. Kemp, and H. Bernien, Physical Review X 12, 011040 (2022).
  * Shaw  _et al._ (2023) A. L. Shaw, P. Scholl, R. Finklestein, I. S. Madjarov, B. Grinkemeyer, and M. Endres, Physical Review Letters 130, 193402 (2023).
  * Homeier  _et al._ (2023) L. Homeier, T. J. Harris, T. Blatz, U. Schollwöck, F. Grusdt, and A. Bohrdt, arXiv preprint arXiv:2305.02322 (2023).
  * Halimeh  _et al._ (2023) J. C. Halimeh, L. Homeier, A. Bohrdt, and F. Grusdt, arXiv preprint arXiv:2305.06373 (2023).
  * Norrgard  _et al._ (2021) E. B. Norrgard, S. P. Eckel, C. L. Holloway, and E. L. Shirley, New Journal of Physics 23, 033037 (2021).


# Comment on “Coexistence of superconductivity and topological aspects in
beryllenes”, Materials Today Physics 38, 101257 (2023)

Mikhail Petrov  Department of Mechanical Engineering, Tufts University,
Medford, MA 02155, USA  Milorad V. Milošević<EMAIL_ADDRESS>Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020
Antwerp, Belgium

###### Abstract

In a recent publication by Li et al. [1], two phases of beryllene - $\alpha$
and $\beta$ \- were predicted to be single-gap superconductors with critical
temperatures of 9.9 K and 12.6 K respectively. Moreover, the
$\alpha$-beryllene was shown to host type-I Dirac fermions with the existence
of nontrivial edge states. We observe significantly weaker superconducting
properties of both beryllene configurations. We argue that the superconducting
gap evolution with temperature, as shown in Figure 5 (b and d) of Ref. [1],
exhibits clearly unphysical trends with increasing temperature, leading to
significantly overestimated values of the critical temperature and erroneous
conclusions concerning the two-gap superconducting nature of
$\beta$-beryllene. On a positive note, we report the value of the gap in the
Dirac cone of the topological states of interest that exceeds the temperature
range of superconductivity in $\alpha$-beryllene, supporting the coexistence
of topological features and superconductivity in this material.

## Computational details

Electronic, vibrational and structural properties of the beryllenes were
calculated with density functional (perturbation) theory as implemented in
Quantum Espresso (QE) version 7.2 [2, 3], using the Perdew-Burke-Ernzerhof
(PBE) functional and norm-conserving pseudopotentials from the PseudoDojo
project [4]. Kinetic energy cutoffs for wavefunctions and for charge density
and potential were set to 100 Ry and 400 Ry respectively. More than
$15$\mathrm{\SIUnitSymbolAngstrom}$$ of vacuum was allowed in the
$z$-direction to simulate 2D monolayers (independent from its periodic images
in $z$-direction). We used 24 $\times$ 24 $\times$ 1 $k$-point grid and 12
$\times$ 12 $\times$ 1 $q$-point grid for electronic and vibrational
properties respectively. A range of electronic smearing values - 0.02, 0.025,
0.03 and 0.04 Ry - was considered for both beryllene phases in order to obtain
vibrational dispersion without numerical instabilities. After this screening,
electronic smearing of 0.04 Ry was selected optimal for $\beta$-beryllene, and
of 0.03 Ry for $\alpha$-beryllene.

Wannier90 (W90) package [5] was used to compute maximally-localized Wannier
functions (MLWFs) for the beryllenes. We used 500 hundred steps for both
disentanglement and minimization procedures. The initial projections were
chosen to be $s$ and $p$ orbitals. The resulting spreads of the MLWFs were
below $2$\mathrm{\SIUnitSymbolAngstrom}$^{2}$. Band structures obtained in W90
were ensured to match those obtained in QE. Based on the obtained Wannier
functions, topological properties were determined by computing the Z2
invariant via evolution of Wannier charge centers (WCC) [6], as implemented in
the WannierTools package [7].

Superconducting properties of the beryllenes were determined by solving fully
anisotropic Migdal-Eliashberg equations, as implemented in the Electron-Phonon
Wannier (EPW) package version 5.7 [8, 9, 10]. The screened Coulomb repulsion
parameter $\mu^{*}$ was chosen to be 0.1, same as in Ref. [1]. Interpolated
fine meshes of 200 $\times$ 200 $\times$ 1 $k$\- and $q$-points were used for
solving Migdal-Eliashberg equations. The width of the Fermi surface window
(fsthick in EPW) was set to 0.6 eV which is around 6 times larger than the
maximal vibrational frequency of the beryllenes. The cutoff for the fermion
Matsubara frequencies (wscut in EPW) of 0.2 eV was used. Smearing in the
energy-conserving delta functions (degaussw in EPW) was set to 0.1 eV. In
order to accurately evaluate multigap nature of the superconducting state in
$\beta$-beryllene, superconducting density of states (SDOS) was computed for a
dense grid of energies (controlled by nqstep in EPW) containing 5000 points.

## Structural, electronic and topological properties

Figure 1: (a,b) Crystal structures and (c,d) electronic band structures of
$\alpha$ and $\beta$-beryllenes respectively. The inset of (c) shows the zoom-
in on the electronic dispersion in the vicinity of the Dirac cone, computed
with and without spin-orbit coupling. With spin-orbit coupling, a gap of
$\Delta$ = 1 meV opens in the cone.

The corresponding structures of the two beryllene phases are shown in Fig.
1(a,b). $\alpha$-Beryllene has a hexagonal lattice with the Be-Be interatomic
distance of 2.13 $\mathrm{\SIUnitSymbolAngstrom}$. $\beta$-Beryllene
constitutes a non-planar hexagonal lattice with two atoms per unit cell. This
structure can be seen as two $\alpha$-beryllene-like layers separated by 1.91
$\mathrm{\SIUnitSymbolAngstrom}$ in $z$-direction, stacked on top of each
other with a small relative shift. Within each $\alpha$-like layer, Be-Be
interatomic distance is 2.17 $\mathrm{\SIUnitSymbolAngstrom}$. These results
are in good agreement with prior reports [11] and Ref. [1].

In terms of electronic properties, the two beryllene phases are rather
similar. There are two types of states crossing the Fermi level [cf. Fig. 1
(c,d)]. States of the first type are located in the vicinity of the $\Gamma$
point and are predominantly of $p_{\mathrm{z}}$ orbital character. States of
the second type are located on both sides of the K points and stem from
$p_{\mathrm{x,y}}$ orbitals. Similarity of the band structure close to the
Fermi level is also well represented by the corresponding Fermi surfaces
[shown in Fig. 3] which in both cases consist of a circular sheet around the
$\Gamma$ point and differently shaped pockets around the K point. However, the
density of states (DOS) at the Fermi level is significantly higher in
$\beta$-beryllene compared to the $\alpha$ phase (0.28 and 0.18 states/eV
respectively).

$\alpha$-Beryllene phase possesses a type-I Dirac band crossing located around
2.5 eV above the Fermi level [zoomed on within Fig. 1 (c)]. By taking spin-
orbit coupling (SOC) into account in the calculation, we find that a gap
$\Delta$ of around 1 meV opens in the Dirac cone. Such energy corresponds to a
temperature $\Delta$/$k_{\mathrm{B}}$ of roughly 12 K. To confirm the non-
trivial topology, we computed the Z2 topological invariant [6]. The resulting
Z2 is 1 confirming the non-trivial topology of the Dirac crossing.

| $\lambda$ | Isotropic Tc [K]
---|---|---
$\alpha$-beryllene with 0.02 Ry | 0.53 | 4.35
$\alpha$-beryllene with 0.03 Ry | 0.46 | 2.78
$\alpha$-beryllene with 0.04 Ry | 0.43 | 2.18
$\beta$-beryllene with 0.02 Ry | 0.53 | 6.15
$\beta$-beryllene with 0.04 Ry | 0.52 | 6.09
Table 1: Comparison of the total electron-phonon coupling $\lambda$ and the
isotropic critical temperatures for the two beryllene phases (using the Allen-
Dynes formula), computed with different values of the electronic smearing set
in QE for electronic and vibrational properties calculations. The underlined
cases are selected for further anisotropic Migdal-Eliashberg calculations of
superconductivity.

## Phonon dispersions, electron-phonon coupling and superconducting properties

Figure 2: Phonon dispersion with mode-resolved electron-phonon coupling (EPC)
$\lambda_{\mathbf{q}\nu}$ indicated by both color and dot size (left panels),
and the isotropic Eliashberg function $\alpha^{2}F$ with isotropic EPC
function $\lambda$ (right panel) for (a) $\alpha$-beryllene and (b)
$\beta$-beryllene. The colorbar as shown in (a) serves for both beryllene
phases. The inset in (a) reflects the effect of electronic smearing on phonon
dispersion in the region marked by dashed rectangle.

Both beryllene phases are dynamically stable as can be seen from the absence
of negative frequencies in their phonon dispersions [Fig. 2 (a,b)]. However,
we found that the dispersion of the lowest acoustic modes in the vicinity of
the $\Gamma$ point is sensitive to the electronic smearing (degauss parameter
in QE, not to be confused with the smearing variables implemented in EPW). To
investigate the effect that the electronic smearing value has on the predicted
superconducting properties, we performed a complete sequence consisting of
structure optimization, electronic/phonon and EPW calculations for each
smearing value we chose. For $\alpha$-beryllene, smaller values of electron
smearing result in an oscillatory behavior of the lowest acoustic phonon mode
[as shown in inset of Fig. 2(a)]. As the value of the smearing is increased,
phonon dispersion becomes more stable and the smearing of 0.03 Ry is set as
optimal for the $\alpha$ configuration. For $\beta$-beryllene, smaller values
of the smearing result in small negative energies in the lowest acoustic mode
around the $\Gamma$ point, which persist for the values of smearing lower than
0.04 Ry. Thus, the smearing value of the 0.04 Ry is set as optimal for the
$\beta$ configuration. Phonon band structures shown in Fig. 2 (a,b) were
computed with the smearing values as defined above. While instabilities do not
have a profound impact on the EPC in $\beta$-beryllene, they do increase it
significantly in $\alpha$-beryllene (nearly twice, cf. Table 1), which is the
case because the smearing-sensitive lowest acoustic mode exhibits one of the
most profound contributions to the total EPC.

The calculated values of EPC are not particularly large, but the coupling is
sufficiently strong for both beryllene configurations to be conclusively
superconducting. To map out the origins of superconductivity from the
available electronic states, we consistently solved the fully anisotropic
Migdal-Eliashberg equations. In Fig. 3 (a)-(d) we show the distribution of the
EPC and the superconducting gap plotted on the Fermi surface, for both studied
phases of beryllene.

Figure 3: Electron-phonon coupling (EPC) plotted on the Fermi surface and the
normalized distribution of EPC $\rho(\lambda)$, for (a) $\alpha$-beryllene and
(b) $\beta$-beryllene. The $\rho(\lambda)$ plot also serves as the colorbar
for the EPC on the Fermi surface. (c,d) The superconducting gap plotted on the
Fermi surface, for $\alpha$-beryllene at 1 K and $\beta$-beryllene at 2 K
respectively. (e,f) Anisotropic superconducting gap spectrum as a function of
temperature, for $\alpha$-beryllene and $\beta$-beryllene respectively. The
weighted averages of different domes are traced with solid lines.

In order to find the critical temperature, we solved the anisotropic Migdal-
Eliashberg equations for a range of temperatures, where the critical
temperature is determined as the lowest temperature for which there were no
solutions of the equations for the superconducting gap. As shown in Fig. 3
(e), $\alpha$ \- beryllene is a single-gap superconductor with the critical
temperature around 3.8 K, which is slightly above the predicted Tc of 2.78 K
from the isotropic Allen-Dynes formula (given in Table 1). The superconducting
gap spectrum is clearly anisotropic and spread over a range of energies. In
particular, $p_{\mathrm{z}}$ states located around the $\Gamma$ point show
smaller values of the superconducting gap $\Delta$ compared to the
$p_{\mathrm{x,y}}$ states.

Even though $\beta$-beryllene does not exhibit significantly stronger EPC, its
critical temperature obtained from the anisotropic Migdal -Eliashberg
calculations reaches significantly higher - around 8.5 K [Fig. 3 (f)].
However, it is also only slightly higher than the predicted 6 K in the
isotropic Allen-Dynes limit. Higher critical temperature of the $\beta$ phase
can be at least partly attributed to the higher electronic density of states
at the Fermi level, which is about 50% larger than that of the
$\alpha$-beryllene. The superconducting gap distribution is highly
anisotropic, with two distinct domes. The lower dome stems from the
$p_{\mathrm{z}}$ states located around the $\Gamma$ point and a smaller Fermi
sheet located around K points whereas the higher dome stems predominantly form
the larger Fermi sheet around K points. The resulting gap spectrum
distribution suggests two separate gaps, but the separation between the domes
associated with those gaps is small and effectively vanishes at higher
temperatures [cf. Fig. 3 (f)]. Nevertheless, by carefully evaluating
superconducting density of states (SDOS) on a dense grid of energies (see
Computational Details), we confirm the two-gap nature of the superconducting
state of $\beta$-beryllene [see Fig. 4]. The SDOS results presented in Figure
S5 of the supplementary material to Ref. [1] erroneously suggest the single-
gap behavior, likely due to the lack of resolution at the energies where two
domes in the gap spectrum are located.

Figure 4: Quasiparticle density of states in the superconducting state (SDOS),
relative to the DOS in the normal state
$N_{\textrm{S}}$($\omega$)/$N_{\textrm{N}}$($\omega$) for $\beta$-beryllene,
computed at 4 K and 6 K. Dashed lines mark the weighted average values of
different domes in the anisotropic superconducting gap spectrum at the
corresponding temperatures [cf. Fig. 3 (f)].

## Summary

Our results concerning the structural and electronic properties of beryllenes
are in good agreement with Li et al. [1]. An actual opening of the gap in the
Dirac cone upon taking into account spin-orbit coupling was not shown in Ref.
[1], neither was the gap energy value indicated. We point out that the gap
exceeds 1 meV and suggests its robustness in the temperature range where
superconductivity is expected. We agree with Ref. [1] on the Z2 invariant of
that Dirac crossing, thus confirming the existence of the non-trivial states
there.

For vibrational properties, we observed high sensitivity of the lowest
acoustic phonon mode dispersion in the close proximity of the $\Gamma$ point
to the electronic smearing. In order to obtain convincingly stable phonon
dispersion, we had to increase the smearing values to 0.03 and 0.04 Ry for
$\alpha$ and $\beta$ beryllenes respectively. Ref. [1] reports stable phonons
already for lower 0.02 Ry smearing.

Despite the differences in vibrational dispersion, we observed near perfect
agreement of the EPC in our calculations and in Ref. [1]. Moreover, taking the
values of $w_{\textrm{log}}$ and $\lambda$ obtained in EPW by Li et al. (Table
2 of Ref. [1]) and substituting them into the Allen-Dynes formula, yielded
excellent agreement between our results and Ref. [1] for the isotropic
critical temperature.

However, the most striking difference arose in the anisotropic calculations.
As expected, we found a monotonic decrease of the superconducting gap energy
with temperature for both beryllene phases, with the critical temperature
roughly corresponding the BCS relation to the gap at zero temperature.
Surprisingly, Ref. [1] shows an initial upward trend of $\Delta(T)$, thus
unphysically pushing the critical temperature to much higher values.
Additionally, for $\alpha$-beryllene, at the lowest considered temperature of
1 K, we observed the superconducting gap spectrum reaching at most 0.7 meV
(cf. Fig. 3 (e)). In contrast, Li et al. showed the gap values reaching above
0.9 meV for 2 K even before the onset of the upward trend in $\Delta(T)$
(Figure 5(b) of Ref. [1]). As a consequence, the Migdal-Eliashberg anisotropic
critical temperature for $\alpha$-beryllene in our work is only about 40% of
the value reported in Ref. [1] (3.8K vs. 9.9K). The same arguments can be
applied to the $\beta$-beryllene, where our Migdal-Eliashberg result for the
critical temperature is about 65% of the value reported in Ref. [1] (8.5K vs.
12.6K). Lastly, we dispute the claimed single-gap behavior of
$\beta$-beryllene in Ref. [1]. $\beta$-Beryllene is a two-gap superconductor,
which is confirmed by careful evaluation of SDOS in our work.

## NOTE ON SCIENTIFIC ETHICAL CONDUCT

This comment was sent to authors and the journal in question in an effort to
make a correction in the public record, without any malicious intent. The
authors did not opt to respond in any way or make the correction accordingly,
and the journal does not publish corrections or comments. The Editorial Board
of the journal was in contact with the authors and has decided that correction
is not warranted.

## Acknowledgments

This work was supported by the Research Foundation-Flanders (FWO). The
computational resources and services were provided by the VSC (Flemish
Supercomputer Center), funded by the FWO and the Flemish Government –
department EWI.

## References

  * [1] Jingyu Li, Muchun Guo, Jianguo Si, Lanting Shi, Xianbiao Shi, Jiang-Jiang Ma, Qinyong Zhang, David J. Singh, Peng-Fei Liu, and Bao-Tian Wang.  Coexistence of superconductivity and topological aspects in beryllenes.  Materials Today Physics, 38:101257, 2023.
  * [2] Paolo Giannozzi, Stefano Baroni, Nicola Bonini, Matteo Calandra, Roberto Car, Carlo Cavazzoni, Davide Ceresoli, Guido L Chiarotti, Matteo Cococcioni, Ismaila Dabo, Andrea Dal Corso, Stefano de Gironcoli, Stefano Fabris, Guido Fratesi, Ralph Gebauer, Uwe Gerstmann, Christos Gougoussis, Anton Kokalj, Michele Lazzeri, Layla Martin-Samos, Nicola Marzari, Francesco Mauri, Riccardo Mazzarello, Stefano Paolini, Alfredo Pasquarello, Lorenzo Paulatto, Carlo Sbraccia, Sandro Scandolo, Gabriele Sclauzero, Ari P Seitsonen, Alexander Smogunov, Paolo Umari, and Renata M Wentzcovitch.  Quantum espresso: a modular and open-source software project for quantum simulations of materials.  Journal of Physics: Condensed Matter, 21(39):395502, sep 2009.
  * [3] P Giannozzi, O Andreussi, T Brumme, O Bunau, M Buongiorno Nardelli, M Calandra, R Car, C Cavazzoni, D Ceresoli, M Cococcioni, N Colonna, I Carnimeo, A Dal Corso, S de Gironcoli, P Delugas, R A DiStasio, A Ferretti, A Floris, G Fratesi, G Fugallo, R Gebauer, U Gerstmann, F Giustino, T Gorni, J Jia, M Kawamura, H-Y Ko, A Kokalj, E Küçükbenli, M Lazzeri, M Marsili, N Marzari, F Mauri, N L Nguyen, H-V Nguyen, A Otero de-la Roza, L Paulatto, S Poncé, D Rocca, R Sabatini, B Santra, M Schlipf, A P Seitsonen, A Smogunov, I Timrov, T Thonhauser, P Umari, N Vast, X Wu, and S Baroni.  Advanced capabilities for materials modelling with quantum espresso.  Journal of Physics: Condensed Matter, 29(46):465901, oct 2017.
  * [4] M.J. van Setten, M. Giantomassi, E. Bousquet, M.J. Verstraete, D.R. Hamann, X. Gonze, and G.-M. Rignanese.  The pseudodojo: Training and grading a 85 element optimized norm-conserving pseudopotential table.  Computer Physics Communications, 226:39–54, 2018.
  * [5] Arash A. Mostofi, Jonathan R. Yates, Young-Su Lee, Ivo Souza, David Vanderbilt, and Nicola Marzari.  wannier90: A tool for obtaining maximally-localised wannier functions.  Computer Physics Communications, 178(9):685–699, 2008.
  * [6] Alexey A. Soluyanov and David Vanderbilt.  Wannier representation of z2 topological insulators.  Phys. Rev. B, 83:035108, Jan 2011.
  * [7] QuanSheng Wu, ShengNan Zhang, Hai-Feng Song, Matthias Troyer, and Alexey A. Soluyanov.  Wanniertools: An open-source software package for novel topological materials.  Computer Physics Communications, 224:405–416, 2018.
  * [8] E. R. Margine and F. Giustino.  Anisotropic migdal-eliashberg theory using wannier functions.  Phys. Rev. B, 87:024505, 2013.
  * [9] S. Poncé, E.R. Margine, C. Verdi, and F. Giustino.  Epw: Electron–phonon coupling, transport and superconducting properties using maximally localized wannier functions.  Computer Physics Communications, 209:116 – 133, 2016.
  * [10] F. Giustino, M. L. Cohen, and S. G. Louie.  Electron-phonon interaction using wannier functions.  Phys. Rev. B, 76:165108, 2007.
  * [11] Minglei Sun, Yuan Yan, and Udo Schwingenschlögl.  Beryllene: A promising anode material for na- and k-ion batteries with ultrafast charge/discharge and high specific capacity.  The Journal of Physical Chemistry Letters, 11(21):9051–9056, 2020.