# Analyzing and Predicting Low-Listenership Trends in a Large-Scale Mobile
Health Program: A Preliminary Investigation

Arshika Lalan  Google ResearchIndia<EMAIL_ADDRESS>,  Shresth Verma
Google ResearchIndia<EMAIL_ADDRESS>,  Kumar Madhu Sudan  ARMMANIndia
<EMAIL_ADDRESS>,  Amrita Mahale  ARMMANIndia<EMAIL_ADDRESS>,  Aparna
Hegde  ARMMANIndia<EMAIL_ADDRESS>,  Milind Tambe  Google
ResearchIndia<EMAIL_ADDRESS>and  Aparna Taneja  Google ResearchIndia
<EMAIL_ADDRESS>
###### Abstract.

Mobile health programs are becoming an increasingly popular medium for
dissemination of health information among beneficiaries in less privileged
communities. Kilkari is one of the world’s largest mobile health programs
which delivers time sensitive audio-messages to pregnant women and new
mothers. We have been collaborating with ARMMAN, a non-profit in India which
operates the Kilkari program, to identify bottlenecks to improve the
efficiency of the program. In particular, we provide an initial analysis of
the trajectories of beneficiaries’ interaction with the mHealth program and
examine elements of the program that can be potentially enhanced to boost its
success. We cluster the cohort into different buckets based on listenership so
as to analyze listenership patterns for each group that could help boost
program success . We also demonstrate preliminary results on using historical
data in a time-series prediction to identify beneficiary dropouts and enable
NGOs in devising timely interventions to strengthen beneficiary retention.

Time-Series Prediction, Mobile Health

††conference: Make sure to enter the correct conference title from your rights
confirmation emai; August 06–10, 2023; Longbeach, CA††booktitle: KDD ’23,
Workshop on Data Science for Social Good, 2023, Longbeach, CA

##  1\. Introduction

Mobile Health (mHealth) programs make use of mobile phone devices to deliver
healthcare services while raising awareness about critical information and
knowledge that contributes to optimal health outcomes. They play a key role in
making healthcare more accessible for the less privileged (McCool et al.,
2022; Wood et al., 2019; Guo et al., 2020).

In this paper, we focus on our collaboration with ARMMAN (ARMMAN, 2019), an
India-based non-profit organization which conducts mHealth programs in India
to increase awareness during antenatal and postpartum care for pregnant women
and mothers. It has adopted mHealth interventions to assist in bringing down
maternal and child mortality rates. mMitra, an initiative by ARMMAN, is a free
mobile call service that delivers organized preventive care information to
enrolled women throughout their pregnancy and child infancy, on a weekly/bi-
weekly basis. mMitra has successfully deployed SAHELI (Verma et al., 2023), a
system to efficiently utilize the limited capacity of healthcare resources to
boost engagement with the program.

In 2016, the Ministry of Health & Family Welfare (MoHFW) launched the Kilkari
program, a free mobile health (mHealth) education service that sends women
preventive care information during pregnancy and child infancy. Kilkari is an
IVR service designed to deliver weekly pre-recorded, stage-specific audio
messages to pregnant women and mothers with children under the age of 1 year.
Currently operational in 18 states and Union Terriroties, Kilkari has reached
over 30 million women and their children to date, and has 3 million active
subscribers. ARMMAN is a technical, content & creative production, and
implementation partner to the MoHFW in making Kilkari available pan-India.

Like most mHealth programs (Madanian et al., 2019; Aranda-Jan et al., 2014),
Kilkari continues to evolve with improvements in technology and
infrastructure. We discuss some unexplored questions based on these new
developments in Kilkari. Our key contributions are :

  * •

Analyzing listenership patterns of different beneficiary buckets: We segment
our cohort of beneficiaries into four buckets based on listenership (High
Pickup Rate - High Engagement Rate, High Pickup Rate - Low Engagement Rate,
Low Pickup Rate - High Engagement Rate, Low Pickup Rate - Low Engagement Rate
) and analyse each bucket individually. This segmentation helps us analyze the
problem of listenership using two metrics - pickup and engagement - which we
show are equally important. While previous works have only used pickup rates
as an indication of listenership, we show that high pickup rates could be
coupled with low engagement rates and thus should be taken into account to
bolster program success.

  * •

Analyzing the impact of time slot on call pickup and call engagement: For our
analysis, we construct seven time slots in the interval from 8 AM to 10 PM,
divided into two-hour intervals. All calls, including reattempts, are put into
one of these time slots. Our findings show that the beneficiaries in different
buckets tend to exhibit a preference for some time slots over the others,
which can be useful in boosting listenership

  * •

Predicting low-listeners: Our final contribution is predicting engagement
rates (the rate at which beneficiary listens to a call for more than 30
seconds) along with pickup rates (the rate at which beneficiary picks up
calls) of the entire cohort using preliminary time-series modelling. We
showcase how to make such predictions using only listenership trajectories and
no demographic features, which can be useful in programs with sensitive and
limited beneficiary information, a key characteristic of most mHealth
programs. Additionally, predicting low-listenership of beneficiaries from
historical data can assist NGOs in planning timely interventions to improve
beneficiary retention.

Through the study of Kilkari as a real-world use case, our analysis offers
insights that can be extended to other large-scale mHealth programs.

##  2\. Related Works And Background

Adherence monitoring in healthcare is an extensively studied problem for
diseases like HIV (Tuldrà et al., 1999), cardiac problems (Son et al., 2010),
Tuberculosis (Killian et al., 2019; Pilote et al., 1996), etc. In mobile
health programs as well, techniques such as sequential modelling (Nishtala et
al., 2021) and markov decision process models (Mate et al., 2022; Killian et
al., 2019) have been used to improve beneficiaries adherence to audio
messages.

Kilkari is the largest maternal mHealth messaging program in the world
(ARMMAN, [n. d.]). Given the scale of Kilkari’s outreach, it has been a
subject of multiple research studies in the past (Mohan et al., 2022;
Chamberlain et al., 2021). Infrastructural and technological investments have
led to the successful resolution of some of the key challenges discussed in
previous works, while others have become less significant in the course of
time.

(Chamberlain et al., 2021) address the process of scaling Kilkari successfully
across geographies. At scale, Kilkari was redesigned to make calls throughout
the day as opposed to based on subscriber preference to increase cost-
efficiency. An unintentional consequence was that sometimes calls were made at
times that could prove unsuitable to beneficiaries. (Chamberlain et al., 2021)
also highlight certain considerations that need to be taken into account in
achieving success at scale among the rural population, such as sim churn and
gender gap in mobile phone access and digital literacy. (Mohan et al., 2021)
investigate the impact of variations in phone ownership levels among different
social strata, such as income and education. The paper highlights that a
successfully picked call may not guarantee that it was answered by the
intended recipient, as it may have been answered by another unaware family
member. Consequently, it would be beneficial to analyse call engagement
(duration of call listened to) in conjunction with pickup rate to measure the
outcome of the program, as opposed to analysing pickup rates exclusively. Our
work represents the first attempt in predicting both pickup rates and
engagement rates for program beneficiaries.

The key contribution of this paper centers around the challenges that Kilkari
faces presently, in particular emphasizing predicting and contacting
beneficiaries in their preferred time slots, as well as prioritizing program
engagement rates along with pickup rates. Finally, we use predictive time-
series modelling to predict engagement rates and pickup rates of the cohort.
(Shah et al., [n. d.]) also uses a predictive model for low listenership
prediction. However, our work uses both engagement and call pickup as metrics
for low listenership, which our secondary analysis from bucketing
beneficiaries shows are equally important metrics. We thus learn separate
binary classification models for both these target variables. Moreover, our
secondary analysis also underscores factors such as time slot of placing calls
and technical success rate of calls which can be important for boosting pickup
and engagement rates for the low listener groups, which has not been
highlighted by (Shah et al., [n. d.]).

##  3\. Problem Formulation

###  3.1. Data Description

The Kilkari mobile health program is an outbound service that makes periodic
automated voice calls to pregnant women and new mothers, starting from the
second trimester of pregnancy until the time the child is one year old. One
voice message is scheduled for every beneficiary in each of these 72 weeks,
which contains information on topics such as maternal and child health,
immunization and family planning.

Each beneficiary receives a maximum of 9 call attempts over 4 days, until they
pick up the phone to increase the probability of them picking up the call. For
every call attempt, the time and day of the attempt as well as the technical
status is noted in the call records database. The technical status gives
information on whether the call was picked up or the line was busy, switched
off or out of network coverage.

In our analysis, we consider beneficiaries enrolled in the Kilkari program in
the Indian state of Orissa between the year 2020-2022. Particularly, we take
the set of 240K beneficiaries registered in Orissa and who received a call in
the first week of 2022. We then create a dataset of their call history
trajectory containing information on call attempts number, date and time of
call, gestational age of beneficiary at the time of call, technical status of
call and duration of the call.

###  3.2. Time Slot Analysis

Every week, a beneficiary receives the first call attempt on the same day as
the day of their Last Menstrual Period date. If the beneficiary doesn’t pick
up the call, the call is attempted again.

However, the time of receiving a call in a day is currently randomized. In our
analysis, we demonstrate the value in predicting and utilizing a favourable
time slot for beneficiaries.

###  3.3. Low-Listenership Prediction

Low-listenership of beneficiaries can be characterised using various metrics.
Based on discussions with domain experts, we consider two definitions of low-
listenership:

  1. (1)

low-pickup rate: beneficiaries who have picked up less than 3 calls within a 6
week time window

  2. (2)

low-engagement rate: beneficiaries who have engaged with a call less than 3
times within a 6 week time window. Engagement is defined as listening to a
call for more than 30 seconds (average message length is typically around 90
seconds).

Based on these definitions, we consider the following problem: Can a
prediction be made in advance on when a beneficiary will become a low-
listener. In such a case, early intervention by the NGO can help keep
beneficiaries engaged with the health-information calls in the long run.

This low-listenership prediction problem can be formulated as a time-series
prediction task. Starting at some point in time, we use history of
beneficiaries’ listenership trajectory for $N_{features}$ weeks to predict
low-listenership $N_{offset}$ weeks in the future. To convert the raw dataset
described previously for the time series prediction task, we split
listenership trajectory of all beneficiaries into multiple rolling windows.
Specifically, each window is of length $N_{features}+N_{offset}+6$ weeks where
the last 6 weeks are used for defining the low-listenership binary flag. For
each beneficiary, we create time series of duration of calls every week,
number of attempts, status of calls, and the date and time of calls. Given
these temporal features, we try to predict whether a beneficiary would have
low-listenership. Note that beneficiaries enter and exit the Kilkari program
at different points in time. Thus, we would get different number of time-
windows for different beneficiaries.

Finally, we split the beneficiaries into train ($80\%$) and test ($20\%$)
sets, and thus create the train and test time series datasets. Splitting on
beneficiaries rather than temporally allows us to learn a prediction model
that can predict low-listenership for new beneficiaries which enter the system
and have at least $N_{features}$ weeks of call history.

##  4\. Experiments

###  4.1. Measuring efficacy of each successive attempt

The IVR System makes a maximum of 9 attempts to a beneficiary in a week. Fig.1
shows the percentage of beneficiaries reached with each successive attempt.
The magnitude of the bars indicates the effectiveness of an attempt in
reaching beneficiaries. We see a steady decrease in the percentage of
beneficiaries that are reached with each successive attempt. The blue line
indicates the cumulative percentage of beneficiaries reached. It can be seen
that on average, 23% beneficiaries are not reached despite 9 attempts. Given
the scale of the program, this is a significant number.

Figure 1. Efficacy of each successive attempt

###  4.2. Analysing Listenership Patterns of Beneficiary buckets

Based on call pickup and call engagement behaviour, the beneficiaries were
divided into four buckets: High Pickup Rate - High Engagement Rate, High
Pickup Rate - Low Engagement Rate, Low Pickup Rate - High Engagement Rate, Low
Pickup Rate - Low Engagement Rate. To avoid noisy patterns caused due to ad
hoc listenership, two extreme set of beneficiaries were picked from the
original data - those that have a high listening trajectory of greater than 50
weeks and those who have a lower listening trajectory of less than 20 weeks.
These sets were constructed for the purpose of preliminary analysis, and we
plan to expand them further in the future to include more beneficiaries. The
previously mentioned four buckets were then constructed from the union of
these two sets of beneficiaries. The distribution of beneficiaries per bucket
is as per Table 1.

Pickup Listenership | High | Low
---|---|---
High | 1212 | 2651
Low | 1421 | 6087
Table 1.  Distribution of beneficiary in each bucket

Three major conclusions can be drawn from the analysis:

  * •

High pickup rate need not translate to high engagement.

  * •

Contacting beneficiaries in their preferred time slots could increase
listenership.

  * •

Technical failures reduce pickup rates and could potentially lead to dropouts.

Some external factors could also affect beneficiary pick up rates and
engagement. For example, there was an observed decrease in listenership of
specific beneficiaries during the New Year period, which could be a possible
phenomenon during most major festivities. Fig.2 illustrates this for one
beneficiary. The first graph indicates which attempt day the call was picked
on. The red hashed bars indicate weeks when the call was not picked up. The
second plot shows whether the beneficiary engaged with the call (green bar) or
not (red bar). Finally, the last graph indicates the week of the year at each
Kilkari message index. We see the period of new year’s starts around message
index 45, around which the beneficiary doesn’t pick up calls.

Figure 2. Listenership pattern of a beneficiary around new year - the red
hashed bars indicate that the beneficiary did not pick up calls around new
years

For figures 3, 4, 5 and 6, the six graphs display various characteristics of
these aggregate groups. The first left graph shows the average pickup rate.
The first right graph shows on average, what attempt day did the beneficiaries
pick up the call in (’blue’ if attempt day 1, ’green’ if attempt day 2,
’brown’ if attempt day 3, ’red’ if attempt day 4). The second left graph
indicates the average technical success ratio (a green dot indicates a
technical success ratio greater than 0.5, a red dot will indicate otherwise).
The second right graph indicates which time slot did most of the beneficiaries
pick up in out of the 7 time slots . Time Slot 0 is the earliest time slot
(8AM-10PM) and Time Slot 6 is the latest (8PM-10PM) The third left graph
indicates the call duration on average for every message index (’red’ i.e non-
engaging if call duration $<=$ 30.0 seconds else ’green’ i.e engaging). The
third right graph indicates which day of the week did most of the
beneficiaries pick up in. Day 0 indicates a Monday and Day 6 a Sunday. The
color-coded scatter plots of time slot and day of the week are for easy
visualisation of which points are in abundance.

####  4.2.1. Bucket 1: High Pickup Rate - High Engagement Rate beneficiaries

On average, this group of beneficiaries (Fig. 3) picked up the call on the
first attempt day. The average technical success rate for these beneficiaries
was extremely high, around 90%. Thus, high pickup rates seem to be indicative
of a good technical success ratio. Most of these beneficiaries seem to pick up
across diverse call time slots, with a slight preference for time slot 5 (6 PM
to 8 PM).

Figure 3. Listenership patterns of beneficiaries with high pick up rates and
high listenership

####  4.2.2. Bucket 2: High Pickup Rate - Low Engagement Rate beneficiaries

The second bucket (Fig.4) comprises of beneficiaries that highlight one of the
key focus points of this paper -  engagement in the program is a better metric
than pickup rates to measure program success. Despite a high average technical
success rate, the engagement in the program remains low. While a predictive
model for pickup rates will prefer this set of beneficiaries over
beneficiaries with lower pick up rates, a predictive model for call engagement
will ensure that beneficiaries more receptive to the program are given their
due attention.

Figure 4. Listenership patterns of beneficiaries with high pick up rates and
low listenership

####  4.2.3. Bucket 3: Low Pickup Rate - High Engagement Rate beneficiaries

The third bucket (Fig.5) of beneficiaries tends to highly engage with the
program, despite lower pickup rates. A possible reason for lower pick up rates
could be the low average technical success rates of calls. These beneficiaries
also display a strong preference for the morning and evening slots, with most
calls being picked up during the first two and last two slots of the day.
Thus, providing a higher number of attempts to address technical failures and
rearranging calls for beneficiaries in this bucket for early or late in the
day can potentially improve their performance in the program .

Figure 5. Listenership patterns of beneficiaries with low pick up rates and
high listenership

####  4.2.4. Bucket 4: Low Pickup Rate - Low Engagement Rate beneficiaries

The average technical success rate for these beneficiaries (Fig. 6) is the
lowest out of all the buckets, and is a meager 20%. This indicates that most
calls made to these beneficiaries do not go through. This could be
contributory to their low pickup rates. Finally, the beneficiaries also show a
preference for the first and last time slots of the day. Addressing the
correct time slots to place calls, driving engagement for calls that do
connect despite lower pick up rates, and increasing the number of attempts to
counter the effect of technical failures can potentially boost the performance
of these beneficiaries.

Figure 6. Listenership patterns of beneficiaries with low pick up rates and
low listenership

###  4.3. Technical Failures and pick up rates

Certain beneficiaries are more susceptible to technical failures than others,
which can affect their pickup rates and might eventually lead to them getting
dropped out of the program. Dropouts are crucial in Kilkari as they save
resources for the program by avoiding unnecessary attempts to phone numbers
that are no longer in service, or to people who were incorrectly registered in
the program. However, an unintended effect may be the possibility of dropouts
in poor network coverage areas. While such issues can be addressed in due
course with improvements in network coverage, it is critical to make multiple
attempts to ensure maximum beneficiaries can receive technically successful
calls.

Fig.7 shows the listening trajectory of a beneficiary who picks up all the
calls. The first graph shows the attempt day on which the call was picked
(’blue’ if attempt day = 1, ’green’ if attempt day = 2, ’maroon’ if attempt
day = 3, ’red’ if attempt day = 4). The second graph shows the engagement of
the beneficiary every week (green bar implies engaging and red bar implies
non-engaging). The red hashed bars in both the graphs indicate calls that
haven’t been picked up. Finally, the final graph plots Boolean values of
whether the beneficiary received technically successful (green) and failed
(red) calls. Weeks with only red dots indicate that no technically successful
calls were made that week.

We see that the beneficiary loses up to 5 weeks of messages due to a series of
technical failures. It is interesting to note that after 6 weeks of no
listenership, beneficiaries are dropped from the program.Kilkari has the
flexibility of increasing the number of attempt days that beneficiaries are
contacted over, hence identifying the segment of beneficiaries prone to
technical failures and reaching them over a greater number of attempts can
help further boost pickup rates in the program.

Figure 7.  A beneficiary who picks up all calls except when they are complete
technical failures - the beneficiary loses upto 5 weeks of messages due to
technical failures

###  4.4. Analysis of time slot on listener pickup and engagement rates

Currently, calls in Kilkari are made at random times during the day. For
analysis, we have constructed seven time slots in the interval from 8 AM to 10
PM, divided into two-hour intervals. Calling beneficiaries in their preferred
time slot can help in increasing pickup and engagement rates. As seen in
Fig.8, there is a clear preference in the total cohort for morning time slots,
which boasts of the highest pickup rates. However, for beneficiaries with low
pickup rates and low engagement, all slots perform equally poorly with the
exception of the first and last time slot (Fig.9). As discussed previously,
one possible reason is the issue of phone ownership, where the beneficiary has
access to the phone in only the earlier or later parts of the day.

Figure 8.  Pickup rates per time slot for the full cohort  Figure 9.  Pickup
rates per time slot for bucket 4

Currently, there are plans to update the IVR infrastructure to allow for
maximum outreach. While bandwidth will no longer be an issue in the newer
version of the IVR system, concurrency of calls still acts as a bottleneck.
Thus, it might not be possible to call all beneficiaries in the morning or
evening slots and leave the remaining slots with no bandwidth utilization. We
aim to build a predictive model to calculate the top-k preferred time slots of
beneficiaries, and optimize calling times in a way that can bring up the
overall pickup and engagement rates of the cohort while working under the
constraint of the number of concurrent calls that can be made at a particular
time.

Model | Features | Balanced Accuracy | Precision@5 | AUC
---|---|---|---|---
Logistic Regression | duration, attempt, status, date | 0.756635 | 0.871566 | 0.828824
Logistic Regression | duration, attempt, status | 0.756471 | 0.869925 | 0.828644
Feedforward NN | duration, attempt, status | 0.756136 | 0.872381 | 0.828275
Feedforward NN | duration, attempt | 0.756120 | 0.872837 | 0.828449
LSTM | duration, attempt | 0.756097 | 0.876322 | 0.828866
Feedforward NN | duration, attempt, status, date | 0.755964 | 0.871730 | 0.828090
Logistic Regression | duration, attempt | 0.755885 | 0.874805 | 0.828331
LSTM | duration, attempt, status, date | 0.755800 | 0.866317 | 0.827406
LSTM | duration, attempt, status | 0.754932 | 0.865422 | 0.826992
Random | duration, attempt, status, date | 0.501689 | 0.410071 | 0.501652
Random | duration, attempt | 0.501505 | 0.409128 | 0.501464
Random | duration, attempt, status | 0.500297 | 0.408431 | 0.498771
Table 2. Overall Results in predictive modeling for low-engagement prediction Model | Features | Balanced Accuracy | Precision@5 | AUC
---|---|---|---|---
Feedforward NN | duration, attempt, status, date | 0.797638 | 0.430001 | 0.872797
Logistic Regression | duration, attempt, status, date | 0.796492 | 0.424793 | 0.870911
Feedforward NN | duration, attempt, status | 0.796021 | 0.425285 | 0.870253
Logistic Regression | duration, attempt, status | 0.795685 | 0.422127 | 0.870115
Feedforward NN | duration, attempt | 0.795241 | 0.420856 | 0.869499
Logistic Regression | duration, attempt | 0.794780 | 0.416838 | 0.868858
LSTM | duration, attempt | 0.794487 | 0.421922 | 0.869370
Random | duration, attempt | 0.503113 | 0.055565 | 0.503159
Random | duration, attempt, status, date | 0.502418 | 0.053555 | 0.500930
Random | duration, attempt, status | 0.501334 | 0.055196 | 0.499087
Table 3. Overall Results in predictive modeling for low-call pickup prediction

###  4.5. Predicting low-listeners

To segment beneficiaries into different clusters and to help NGOs plan for
timely interventions that can increase program retention, we make use of
beneficiaries’ historical trajectories in a time-series model to predict low
pickup rates and low engagement rates. Our model does not make use of any
beneficiary demographic features, and is thus especially relevant for mHealth
programs, where beneficary information is often sensitive and limited.

We use the dataset described in Section 3.3 to predict whether beneficiaries
would showcase low-listenership behaviour in the future. Specifically, we
consider two definitions of low-listenership, low-pickup and low engagement.
We thus learn separate binary classification models for these two target
variables. As input, we consider $N_{train}=6$ weeks of listenership
indicators. This includes time series of duration of calls listened to every
week, number of call attempts every week and number of calls with every status
codes every week. We consider the following status codes that is logged in the
Kilkari system: call picked up, phone busy, phone switched off, phone out of
network, any other reason of not reaching. This results in a total of 7
features every week. Crucially, we develop predictive models which only use
listenership behaviour data and no beneficiary specific information.

We consider the following models in our experiments:

  * •

Random: This is the baseline model which predicts low listenership by sampling
from uniform random distribution

  * •

Logistic Regression: This uses a logistic regression model to predict the
target variable. Time series features are flattened and given as input to the
model

  * •

Feedforward NN: This uses a dense feedforward neural network. We use 3 layers
with 128 hidden units each and finally a sigmoid acitvation for output. Binary
crossentropy loss is used to optimise the weights of the neural network. This
model also uses flattened time series as input.

  * •

LSTM: We use Keras implementation of Long Short Term Memory model for encoding
sequential information from beneficiaires listenership. Specifically, we use
LSTM cell with 128 hidden units followed by three layers of 128 hidden units
each. Finally, we use sigmoid activation for output. and Binary crossentropy
loss as optimization objective.

We evaluate all models on three metrics:

  * •

Precision@K%: This is the precision in predicting low-listenership flag if we
set threshold to $(100-K)^{\text{th}}$ percentile of values. This measures the
fraction of low-listeners that we will find if top $K\%$ of beneficiaires are
chosen according to model predictions. Since health resources are limited, we
use ($K=5\%$) in our experiments.

  * •

Balanced Accuracy: Since we have imbalanced target distribution, we extend the
notion of accuracy to imbalanced classes. Balanced accuracy is the arithmetic
mean of sensitivity and specificity.

  * •

AUC: We also report the area under the ROC curve (AUC) metric to compare all
the models.

In table 2 and table 3, we compare different models on multiple evaluation
metrics for low-engagement and low-pickup targets respectively. We notice for
both the targets, ML models can perform much better than random. This
showcases that input features are predictive of low listenership. However, we
do not see any added value from time series model like LSTM, as opposed to
logistic regression or feedforward neural network. Lastly, we find that adding
additional features, such as status of calls and attempts results in slight
increase in predictive performance.

##  5\. Ethics and Data Usage

Acknowledging the responsibility associated with real-world AI systems for
undeserved communities, we have closely coordinated with domain experts from
the NGO throughout our analysis. This study falls into the category of
secondary analysis of the aforementioned dataset. We use the previously
collected engagement trajectories of different beneficiaries participating in
the service call program to train the predictive model and evaluate the
performance. All the data collected through the program is owned by the NGO
and only the NGO is allowed to share data while the research group only
accesses an anonymized version of the data.

##  6\. Conclusion

In this preliminary work, we perform secondary data analysis for Kilkari, the
largest maternal mobile health program in the world. We characterize
beneficiaries’ engagement and pickup behaviour with the program through their
temporal patterns in voice message listenership. We showcase that apart from
beneficiaries’ call pickup rates, which has been the primary focus of several
previous works, call engagement rates, preferred time slots and good call
technical success ratios are also critical for successful outreach of the
program. Lastly, we demonstrate that historical data can be used to predict
low-listenership of beneficiaries to provide tailored approaches for
beneficiary clusters and help NGOs perform timely intervention to increase
beneficiary retention. Our proposed modelling approach relies only on past
listenership trajectories, thus removing dependence on any sensitive or
limited beneficiary information. These results also open up new line of ML
research in areas such as feature engineering, intervention optimization and
measuring behaviour change with better access to information such as voice
messages. Our analysis and techniques can be extended to other large-scale
mHealth programs.

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# Robust phase-controlled gates for scalable atomic quantum processors using
optical standing waves

Shannon Whitlock  European Center for Quantum Sciences and aQCess - Atom
Quantum Computing as a Service,
Institut de Science et d’Ingénierie Supramoléculaire (UMR 7006), University of
Strasbourg and CNRS<EMAIL_ADDRESS>
###### Abstract

A simple scheme is presented for realizing robust optically controlled quantum
gates for scalable atomic quantum processors by driving the qubits with
optical standing waves. Atoms localized close to the antinodes of the standing
wave can realize phase-controlled quantum operations that are potentially more
than an order of magnitude less sensitive to the local optical phase and
atomic motion than corresponding travelling wave configurations. The scheme is
compatible with robust optimal control techniques and spatial qubit addressing
in atomic arrays to realize phase controlled operations without the need for
tight focusing and precise positioning of the control lasers. This will be
particularly beneficial for quantum gates involving Doppler sensitive optical
frequency transitions and provides an all optical route to scaling up atomic
quantum processors.

##  1 Context and motivation

Trapped atoms and atom-like systems are a leading candidate for numerous
quantum technologies, including scalable quantum computing [1, 2, 3], optical
atomic clocks [4] and quantum sensors [5]. A key requirement in these
applications is to reliably control the full quantum state of the system,
typically by applying a sequence of phase-controlled optical pulses to the
atoms. However, a major issue arises when scaling up to many atoms and large
numbers of quantum operations, as errors associated to optical phase noise,
atom position fluctuations and motion lead to rapidly accumulating errors [6,
7, 8, 9].

A common way to realize phase-controlled quantum operations is by utilizing
atomic transitions with frequency differences in the radio frequency or
microwave regimes. These transitions can then be driven by stable microwave
fields or a pair of Raman lasers with a small effective wavevector in a
Doppler insensitive configuration [10]. On the other hand, protocols involving
optical frequency transitions [11, 12, 13, 14] can have significant advantages
when it comes to isolating the states and controlling differential light
shifts, performing high fidelity state-resolved readout and mediating strong
interactions exploiting highly-excited Rydberg states [3]. However optical
frequency transitions are generally much more sensitive to atom position
fluctuations and motional dephasing making it difficult to realize high-
fidelity phase-controlled operations.

Here a general method is proposed for realising robust and high fidelity
quantum operations (gates) that is also compatible with spatial addressing of
atomic qubits in large atomic arrays. By using two driving lasers in standing
wave configuration it is possible to strongly suppress the sensitivity to
local optical phase noise, including atom position variations and motion which
are dominant errors in current experiments [7, 15, 8]. Robust phase-
compensated gates compatible with standing wave or travelling wave
configurations can also be applied to individual qubits in large atomic arrays
using targeted light-shifts [16, 17, 18], where unwanted rotations on non-
targeted qubits are cancelled by exploiting a time-reversal symmetry and
geometric constraints. The proposed gates are robust, technically simple and
parallelizable which will enable scaling up atomic quantum processors and high
fidelity qubit control to large atomic registers.

Figure 1: Setup for realizing robust phase controlled gates using optical
standing waves. (a) Exemplary level diagram showing an optical transition for
encoding an optical frequency qubit and laser fields for trapping and qubit
addressing. (b) An interfering pair of counter propagating lasers produce an
optical standing wave characterised by a periodically varying coupling
strength $|\Omega(x)|$ and a phase $\varphi(x)$ which is spatially uniform
between adjacent nodes. By controlling both the amplitude and phase of the
lasers it is possible to realize phase-controlled quantum operations. (c)
Scheme for single atom addressing involving a global standing wave drive field
(red stripes) and an auxiliary far off-resonant beam (purple shaded region)
targeting the central qubit.

##  2 Phase-controlled gates using optical standing wave (OSW) fields

Our goal is to realize phase-controlled qubit manipulations, which for a
single qubit can be described by the following transformation acting on two
atomic states

$\displaystyle|0\rangle$
$\displaystyle\rightarrow\cos(\theta/2)|0\rangle-i\sin(\theta/2)e^{i\varphi}|1\rangle$
$\displaystyle|1\rangle$
$\displaystyle\rightarrow\cos(\theta/2)|1\rangle-i\sin(\theta/2)e^{-i\varphi}|0\rangle$
(1)

Generally the rotation angle $\theta$ and qubit phase $\varphi$ is determined
by the specific protocol used to drive the qubit. In the case of resonant
driving $\theta$ is determined by the strength and duration of the atom-light
coupling, while $\varphi$ is determined by, and very sensitive to the relative
phase between the atomic dipole and the optical field at the position of the
atom.

To realize the transformation (2) one can resonantly drive the atoms by two
counter propagating optical fields with the same amplitude and phase forming
optical standing wave (OSW). The drive Hamiltonian in the rotating wave
approximation can be written

$\hat{H}_{d}=\frac{\hbar}{2}\left(\Omega_{1}(t)e^{ikx}+\Omega_{2}(t)e^{-ikx}\right)|1\rangle\langle
0|+\textrm{h.c}$ (2)

where $\Omega_{\alpha}(t)=|\Omega_{\alpha}(t)|e^{i\varphi_{\alpha}(t)}$ are
the complex valued Rabi frequencies of the two drive lasers with wavevectors
$\pm k$, at the position of the atom $x$. For OSW gates
$\Omega_{1}(t)=\Omega_{2}(t)=\Omega(t)$ is assumed.

This Hamiltonian allows to realize arbitrary phase controlled single qubit
gates which are mostly insensitive to local optical phase noise and atomic
position fluctuations. While an optical travelling wave (OTW) has a phase that
is a linearly varying function of position $\propto kx$, an OSW on the other
hand interferes to produce a field with a spatially uniform phase which steps
between $\varphi$ and $\varphi+\pi$ between adjacent nodes (Fig. 1b). The
atoms are assumed to be localized close to the antinodes of the OSW, either by
making the atom array commensurate with the standing wave (Fig. 1c) or by
shifting the relative phase of the drive fields between gates to address
specific atoms. In this way it is possible to realize phase-controlled quantum
operations that are mostly insensitive to the precise positions or velocities
of the atoms. Instead the atom experiences a spatially varying intensity
$\propto\cos^{2}(kx)$. But by tailoring the time-dependence of $|\Omega(t)|$
and $\varphi(t)$ (e.g., using electro-optic or acousto-optic modulators which
can act on the timescale of several nanoseconds) it is possible to realize
different quantum gates that also correct for associated intensity noise (or
Rabi frequency) errors.

To assess the advantage of OSW fields over OTW fields the achievable gate
fidelity for three different gate protocols will be compared. The OTW-1 gate
consists of a drive pulse $\Omega_{1}(t)=\nobreak A\sin(\pi t/T)$ and
$\Omega_{2}(t)=0$ with $A=\pi^{2}/(4T)$ (Fig. 2a). The corresponding OSW-1
gate has $\Omega_{1}(t)=\Omega_{2}(t)=(A/2)\sin(\pi t/T)$ (note that the total
required intensity for the OSW-1 gate is half that of OTW-1 for the same gate
duration due to constructive interference). These simple parametric pulse
shapes are favorable for experimental implementations, particularly for fast
gates where bandwidth of the control systems might be limited. Finally the
OTW-1 and OSW-1 gates are compared with a gate based on a four pulse BB1
sequence [19] (OSW-BB1, see Fig. 2b) which additionally suppresses sensitivity
to Rabi frequency errors. In the absence of noise all three gates perfectly
realize a $\textrm{R}_{x}(\theta=\pi/2)$ gate, equivalent to a $\sqrt{X}$ gate
within a global phase factor, which is a basic operation for realising more
complex multiqubit gates and quantum circuits. See Appendix A for
generalizations to $U_{xy}(\theta,\phi)$ rotation gates according to Eq. (2).
Several $U_{xy}(\theta,\phi)$ gates can be concatenated to realize arbitrary
single qubit control using OSW fields.

Figure 2: Infidelity of the $\text{R}_{x}(\pi/2)$ optical travelling wave gate
(OTW-1) and optical standing wave (OSW-1 and OSW-BB1) gate sequences. (a)
Amplitude and phase of the control field for a half-sine shaped control pulse
with a combined pulse area of $\pi/2$ (for OTW-1 the control amplitude is
scaled by a factor of 2). (b) BB1 pulse sequence which is optimized to cancel
amplitude noise. (c) Comparison of gate infidelities as a function of the
local optical phase $kx$ showing superior robustness of the OSW-1 and OSW-BB1
gate pulses.

Figure 2c shows the gate infidelity $\epsilon=1-F$ as a function of the local
optical phase $kx$ assuming the atom is at rest during the gate time. $F$ is
estimated by simulating the time-evolution operator $\hat{U}$ as a sequence of
400 piecewise constant segments and calculating
$F=|\textrm{Tr}(\hat{U}_{\textrm{target}}^{\dagger}\hat{U})|^{2}/4$, where
$\hat{U}_{\textrm{target}}=((1,-i),(i,1))/\sqrt{2}$. The simulations show the
OTW-1 gate is most sensitive to the local optical phase with quadratic
dependence on $kx$ with $\epsilon>0.1$ for $|kx|=0.5$. In contrast the OSW-1
gate exhibits a less sensitive quartic dependence on $|kx|$ with
$\epsilon=0.01$ for $|kx|=0.5$. The residual infidelity in the OSW
configuration can be attributed to the spatial varying intensity, which can be
corrected using more sophisticated pulse sequences. For example, the OSW-BB1
composite pulse sequence suppresses errors to 12th order in $|kx|$ with
$\epsilon<10^{-5}$ for $|kx|=0.5$.

While the OSW configurations appear to be a vast improvement over the OTW
configuration in the static case, it is important to know whether this
improvement holds when the atoms move. To check this the Hamiltonian
$\hat{H}_{d}$ is simulated with a time-dependent phase factor $kx(t)$ where
$x(t)$ is obtained from the classical trajectory of an atom in a one-
dimensional harmonic trap, i.e. $\dot{x}(t)=v(t),\dot{v}(t)=F(t)/m$ with
$F(t)=m\omega^{2}x(t)$, with random initial position $x(0)$ and velocity
$v(0)$ sampled from a Maxwell-Boltzmann distribution, which is an
approximation to the full quantum mechanical problem. Calculations are
performed assuming realistic experimental parameters for 171Yb atoms,
including a qubit transition wavelength of 578 nm (standing wave period 289
nm), trap frequency of $\omega/2\pi=100\,$kHz (trap period $\tau=10\,\mu$s)
and a temperature of $5\,\mu$K based on current experiments [20, 14, 21, 22].
The average infidelity of each gate, for different gate times $T$, was
obtained by averaging the infidelity $1-F$ obtained from the unitary evolution
over 2000 classical trajectories. The Rabi frequencies of the OSW gates were
scaled by a factor 1.03 to correct for the lower intensity seen by the atoms
when averaging over their spatial distribution.

Figure 3: Average infidelity of $\text{R}_{x}(\pi/2)$ gates for a harmonically
trapped 171Yb atom as a function of the gate time in units of the trap period.
The position and velocity distribution is sampled from a thermal Boltzmann
distribution with a temperature of $5\,\mu$K and trap frequency of $100\,$kHz
which is then evolved according to the classical equations of motion. Each
point corresponds to the mean of 2000 trajectories. The distribution of
fidelities is highly non-Gaussian, with a median fidelity typically $3.7\pm
0.5$ and $6\pm 1$ times smaller than the mean and a standard deviation $3.0\pm
0.5$ and $2.6\pm 0.3$ times larger than the mean for the OSW-1 and OSW-BB1
gates respectively.

Fig. 3 shows the calculated gate infidelity for the OSW-1 gate is
approximately 30 times smaller than the OTW-1 gate for $T\lesssim 0.5\tau$,
and approximately a factor of 2 smaller for $T>1.8\tau$. The OSW-BB1 gate
exhibits a higher sensitivity to atomic motion, however for very short and
very long gate durations it can outperform the other two gates. Assuming a
maximum Rabi frequency of $2\pi\times 200\,$kHz it is possible to obtain fast
gates with $\epsilon_{\textrm{OTW-1}}\approx 0.032\leavevmode\nobreak\
(T_{\textrm{OTW-1}}=2.0\mu$s), $\epsilon_{\textrm{OSW-1}}\approx 0.0014$
($T_{\textrm{OSW-1}}=1.4\,\mu$s), and $\epsilon_{\textrm{OSW-BB1}}\approx
0.018$ ($T_{\textrm{OSW-BB1}}=8.8\,\mu$s). For the OSW gates the residual
infidelity is dominated by trajectories with a large initial displacement
which could be further improved using higher trap frequencies or lower
temperatures.

##  3 Robust qubit addressing using targeted light shifts

Phase-controlled gates are also applicable to addressing in large atomic
arrays without the need to tightly focus and spatially position the counter-
propagating drive laser beams on individual atoms. The idea is to have all
atoms interact with a common standing wave drive field while one or more
target atoms are addressed by shifting their energy using an auxiliary focused
and far-detuned laser [16, 17, 18]. This decouples coherent qubit control and
spatial addressing and is applicable to 1D, 2D and 3D atomic arrays. Previous
light shift addressing protocols for atomic qubits used focused lasers to
induce targeted phase shifts [16, 18] or to shift target qubits into resonance
with a detuned drive field [17, 23]. However, in these examples one is
typically limited to weak drive fields compared to the light shift or
additional spin-echo sequences are required to correct unwanted rotations on
the non-target atoms, resulting in slow gates. Here another approach is
proposed for realizing fast and robust light-shift gates, which requires
relatively small light shifts and guarantees the non-target qubits perfectly
return to their original state at the end of the gate sequence, even in the
presence of realistic noise sources. The basic idea is to exploit time
reversal symmetry of the control Hamiltonian. For a resonant drive this can be
achieved by making the drive field an anti-symmetric function of time,

$\Omega(t+T/2)=-\Omega(-t+T/2),$ (3)

which ensures that the non-target qubits return to their original state at
$t=T$.

For target atoms a local ac Stark shift of the form $V(t)|1\rangle\langle 1|$
is added to $\hat{H}_{d}$, where $V(t)$ is proportional to the addressing
laser intensity and has a tailored time-dependence to realize a desired gate.
For simplicity $V(t)\approx 0$ is assumed for non-target atoms. The same drive
pulse $\Omega(t)$ can yield gates with different rotation angles $\theta$
enabling different gates to be applied to multiple qubits in parallel simply
by shaping the addressing field. Multiple layers of gates with different phase
angles $\varphi$ could be realized by concatenating pulses in the spirit of
the sinusoidally modulated, always rotating, and tailored SMART qubit protocol
recently demonstrated for spin qubits [24, 25], which can provide continuous
noise protection.

Figure 4a,b presents a new gate OSW-LS2 suitable for addressing atomic qubits
which has high robustness against both Rabi frequency and qubit frequency
errors, especially for the non-target qubits. While the OSW configuration is
analyzed, it should also be beneficial for travelling wave drives exploiting
Doppler free resonances. The gate was numerically optimized using a restricted
sum-of-sines basis with a cost function that provides second order
cancellation of qubit frequency errors using the geometric formalism [26]
(further details in Appendix B). Besides its robustness to noise, a remarkable
feature of the gate is that the required light shifts are comparable to, or
smaller than the drive Rabi frequency which will be beneficial for realizing
many gates in parallel. Fig. 4c,d shows the average infidelity (black lines)
as a function of relative Rabi frequency errors (uncorrelated for each beam
forming the OSW) and absolute qubit frequency errors (in units of $T^{-1}$)
where averaging is performed over different static values of the Rabi
frequency or qubit frequency sampled from normal distributions with standard
deviation defined on the horizontal axis. Nearly perfect correction of both
error types for non-target qubits are observed while target qubit errors are
$<10^{-4}$ (Rabi errors) and $<10^{-6}$ (qubit frequency errors) for noise
amplitudes as high as 2% even though the gate is not directly optimized
against target qubit errors. For comparison results are also shown for a non-
corrected gate OSW-LS1 (red lines, details in Appendix A) which is
significantly more sensitive to qubit frequency errors. For robustness with
respect to intensity imbalance of the standing wave beams see Appendix C.

Figure 4: Robust gate protocol with light shift addressing. (a) Time-dependent
controls for the OSW-LS2 gate. (b) Corresponding time-dependent populations
for a target qubit (solid lines) and non-target qubit (dashed line) starting
from the $|0\rangle$ state. (c) Average gate fidelity for target (solid black)
and non-target qubits (dashed black) with respect to static Rabi frequency
errors. (f) Similar comparison for static qubit frequency errors. The red
lines show a comparison for a gate which is not corrected for qubit frequency
errors (OSW-LS1, Appendix A).

##  4 Discussion

In this paper several new gate protocols have been proposed for realizing high
fidelity and spatially addressable quantum operations that are insensitive to
qubit position fluctuations and motion, which is one of the major challenges
affecting scalable quantum processors based on atomic qubits. This includes a
new robust gate for addressing individual qubits that decouples control and
addressing lasers, requires modest light shifts and perfectly corrects for
errors on non-target qubits. Compared to travelling waves, standing wave
drives can provide typically an order of magnitude lower infidelities
approaching $10^{-3}$ for conditions achievable in current setups. The basic
idea can be applied also to more complex multiqubit gates, including Rydberg
mediated gates [27, 28], which combined with phase controlled single qubit
gates would form a universal gate set. It can also be applied to different
geometries including standing-waves produced by optical cavities (where the
spectral width of the gate pulses fit within the cavity bandwidth),
holographic techniques or integrated photonic devices which may bring
additional advantages including higher stability and faster quantum
operations.

The OSW approach is especially advantageous for situations involving optical
frequency transitions where traditional Doppler-insensitive excitation schemes
like two-photon Raman transitions are not available. This brings forward the
possibility of realising quantum computers based ultrastable optical clock
qubits or new control strategies for optical atomic clocks and sensors that
suffer less from the accumulation of errors associated with phase-control
operations.

## Acknowledgements

I thank S. Jandura, G. Pupillo, T. Bienaimé and Li Chang for valuable
discussions. This work has benefited from a state grant managed by the French
National Research Agency under the Investments of the Future Program with the
reference ANR-21-ESRE-0032 and the Horizon Europe programme HORIZON-
CL4-2021-DIGITAL-EMERGING-01-30 via the project “EuRyQa - European
infrastructure for Rydberg Quantum Computing” grant agreement number 10107014
and support from the Institut Universitare de France (IUF). It is also part of
the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm,
was supported by IdEx Unistra (ANR 10 IDEX 0002), and by SFRI STRAT’US project
(ANR 20 SFRI 0012) and EUR QMAT ANR-17-EURE-0024 under the framework of the
French Investments for the Future Program. Aspects of this work are described
in disclosure "CNRS 16199-01" submitted as part of the “Code Quantum”
maturation project supported by SATT Conectus.

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##  Appendix A Summary of gate protocols used in the paper

The gates reported in the main text can be used to realize general
$U_{xy}(\theta,\phi)$ rotation gates of the form Eq.(2) with the following
control amplitudes.

###  A.1 OTW

$\displaystyle\Omega_{1}(t)$
$\displaystyle=\frac{\theta\pi}{2T}\sin\left(\frac{\pi
t}{T}\right)e^{i\varphi},$ (4) $\displaystyle\Omega_{2}(t)$ $\displaystyle=0$
(5)

###  A.2 OSW

$\displaystyle\Omega_{1}(t)=\Omega_{2}(t)$
$\displaystyle=\frac{\theta\pi}{4T}\sin\left(\frac{\pi
t}{T}\right)e^{i\varphi},$ (6)

###  A.3 OSW-BB1

$\displaystyle\Omega_{1}(t)=\Omega_{2}(t)=\frac{\pi
p}{2T}\times\left\\{\begin{array}[]{ll}e^{i\phi_{a}}&\quad t(p/T)\leq 1\\\
e^{i\phi_{b}}&\quad 1<t(p/T)\leq 3\\\ e^{i\phi_{a}}&\quad 3<t(p/T)\leq 4\\\
e^{i\varphi}&\quad t/T\leq 1\end{array}\right.$ (11)

with $p=\theta/\pi+4$, $\phi_{a}=\arccos(-\theta/4\pi)$ and
$\phi_{b}=3\arccos(-\theta/4\pi)$[29].

###  A.4 OSW-LS1 ($\theta=\pi/2$)

$\displaystyle\Omega_{1}(t)$
$\displaystyle=\Omega_{2}(t)=\frac{3.1317}{T}\sin\left(\frac{2\pi
t}{T}\right)e^{i\varphi-i\pi/2},$ (12) $\displaystyle V(t)$
$\displaystyle=\frac{3.5859}{T}\sin^{2}\left(\frac{\pi t}{T}\right)$ (13)

###  A.5 OSW-LS2 ($\theta=\pi/2$) with $\sigma_{z}$ cancellation

$\displaystyle\Omega_{1}(t)$
$\displaystyle=\Omega_{2}(t)=\frac{7.5551}{T}\sin\left(\frac{2\pi
t}{T}\right)e^{i\varphi-i\pi/2},$ (14) $\displaystyle V(t)$
$\displaystyle=\frac{1}{T}\left(2.1366\sin\left(\frac{2\pi
t}{T}\right)+0.8875\sin\left(\frac{4\pi t}{T}\right)\right)^{2}$ (15)

##  Appendix B Robust OSW-LS2 gate

The light-shift addressing concept can be applied also to more sophisticated
OSW pulse sequences that can correct for different types of errors. For
example, uncontrolled variations of the qubit frequency caused by residual
light shifts from the tweezer traps or leakage of the addressing light on
neighbouring qubits could be especially problematic as it potentially affects
many qubits that interact with the drive field and not just the target qubits.

Finding a minimal basis and parameter sets which yield robust gates is a non-
trivial problem, especially since the solution landscape can be very rugged.
This problem was alleviated by restricting the optimization to a restricted
sum-of-sines basis constrained to satisfy Eq. (3) and by adding an additional
term to the cost function which favors solutions which are robust to
perturbations.

Assuming the full Hamiltonian for non-targeted qubits can be written
$\hat{H}(t)=\hat{H}_{d}(t)+\gamma\hat{\sigma}^{z}$, where
$\sigma^{z}=|0\rangle\langle 0|-|1\rangle\langle 1|$ and $\gamma$ represents
the qubit frequency difference with respect to the carrier frequency of the
control field, which is treated as a small perturbation. The Hamiltonian in
the interaction picture can be written as
$\hat{H}_{I}(t)=\gamma\hat{U}_{d}^{\dagger}(t)\hat{\sigma}^{z}\hat{U}_{d}(t)$.
The time-evolution induced by the perturbation can be determined from a Magnus
series expansion

$\hat{U}_{I}(T)=\mathds{1}+\hat{A}_{1}(t)+\hat{A}_{2}(t)+\ldots$

where $A_{1}$ and $A_{2}$ represent the first two orders of the expansion

$\displaystyle\hat{A}_{1}(T)$
$\displaystyle=\frac{1}{\gamma}\int_{0}^{T}dt_{1}\hat{H}_{I}(t_{1}),$ (16)
$\displaystyle\hat{A}_{2}(T)$
$\displaystyle=\frac{1}{2\gamma^{2}}\int_{0}^{T}dt_{1}\int_{0}^{t_{1}}dt_{2}[\hat{H}_{I}(t_{1}),\hat{H}_{I}(t_{2})],$
(17)

For perfect noise cancellation, we desire $U_{I}(T)\rightarrow\mathds{1}$. In
practice it is sufficient to cancel noise at first or second order, i.e.
$\hat{A}_{1}(t)=0$ or $\hat{A}_{1}(T)=\hat{A}_{2}(T)=0$ respectively. In the
geometric formalism [26, 30], this is equivalent to the Bloch vector tracing
out a closed curve (first order cancellation) with a projected area of zero in
the xy, xz and yz planes (second order cancellation).

A simple control pulse shape that satisfies $\hat{A}_{1}(T)=\hat{A}_{2}(T)=0$
and is subject to the time-reversal symmetry constraint is shown in Fig. 4. It
was found by minimizing

$\text{cost}=1-F+||\hat{A}_{1}(T)||$

using a restricted sum-of-sines basis for $\Omega(t)$ and $V(t)$, where
$||\hat{M}||=|\text{Tr}(\hat{M}\cdot\vec{\sigma})|$ with $\vec{\sigma}$ the
Pauli vector $\\{\sigma^{x},\sigma^{y},\sigma^{z}\\}$. In practice, for the
gates studied in this paper the symmetry constraint, together with
$\text{cost}=0$ also ensures second order noise cancellation
$\hat{A}_{2}(T)=0$.

##  Appendix C Sensitivity to standing wave intensity imbalance

In practice it might be difficult to guarantee exactly equal intensities for
the two beams which form the optical standing wave. This reduces the contrast
of the standing wave and introduced a small gradient of the local optical
phase at the antinode positions which can cause increased sensitivity to atom
position fluctuations. To quantify this effect, Fig. 5 shows the infidelity of
OSW gates as a function of position (in units of the wavevector $k$), assuming
a relatively large difference in intensities of 20% relative to the mean. One
sees that the effect on the OSW-1 gate is negligible, since the infidelity is
dominated by the intensity inhomogeneity effect. In comparison, the OSW-BB1
gate which corrects for intensity homogeneity, is more sensitive to the
intensity imbalance. Both OSW gates are an order of magnitude or more robust
than the OTW gate and are compatible with infidelities $\sim 10^{-4}$ even in
the presence of significant intensity imbalance.

Figure 5: Infidelity of the $\text{R}_{x}(\pi/2)$ optical travelling wave gate
(OTW-1) and optical standing wave (OSW-1 and OSW-BB1) gate sequences with (red
lines) and without (black lines, same as Fig. 2c on a logarithmic scale)
intensity imbalance, assuming a 20% difference in intensities relative to the
mean.