# Dynamic Temporal Reconciliation by Reinforcement learning

Himanshi Charotia
Mastercard, AI Garage
Gurugram
Abhishek Garg
Mastercard, AI Garage
Gurugram
Gaurav Dhama
Mastercard, AI Garage
Gurugram
Naman Maheshwari
Mastercard, AI Garage
Gurugram

###### Abstract

Planning based on long and short term time series forecasts is a common
practice across many industries. In this context, temporal aggregation and
reconciliation techniques have been useful in improving forecasts, reducing
model uncertainty, and providing a coherent forecast across different time
horizons. However, an underlying assumption spanning all these techniques is
the complete availability of data across all levels of the temporal hierarchy,
while this offers mathematical convenience but most of the time low frequency
data is partially completed and it is not available while forecasting. On the
other hand, high frequency data can significantly change in a scenario like
the COVID pandemic and this change can be used to improve forecasts that will
otherwise significantly diverge from long term actuals.

We propose a dynamic reconciliation method whereby we formulate the problem of
informing low frequency forecasts based on high frequency actuals as a Markov
Decision Process (MDP) allowing for the fact that we do not have complete
information about the dynamics of the process. This allows us to have the best
long term estimates based on the most recent data available even if the low
frequency cycles have only been partially completed. The MDP has been solved
using a Time Differenced Reinforcement learning (TDRL) approach with
customizable actions and improves the long terms forecasts dramatically as
compared to relying solely on historical low frequency data. The performance
of the proposed Reinforcement Learning system has been evaluated on stock
exchange NIFTY 50 dataset[1]. The result also underscores the fact that while
low frequency forecasts can improve the high frequency forecasts as mentioned
in the temporal reconciliation literature (based on the assumption that low
frequency forecasts have lower noise to signal ratio) the high frequency
forecasts can also be used to inform the low frequency forecasts. However,
this is not achievable without accepting the fact that the dynamics of the
system in practical scenarios can change significantly in a short amount of
time and the temporal reconciliation model has to account for these
uncertainties.

_K_ eywords Time Series Reconciliation $\cdot$ Reinforcement Learning $\cdot$
Temporal Difference Learning

##  1 Introduction

Many real-life decision problems involve multiple time horizons. For example,
in energy planning expected demand has to be forecasted for the upcoming
hours, the next day, the next week, and even longer horizons [2, 3]. Another
example is inventory management where short term(monthly) forecasts are needed
for next month’s stocking. Whereas it is often beneficial to take into account
long term forecasts for taking decisions related to establishing contracts of
purchasing a product [4, 5]. These decisions are made one-two year in advance
using long term forecasts (1-2 years). This is to make sure that decisions
made now have a positive impact on future possibilities.

In this context, temporal aggregation and reconciliation techniques have been
found to be useful in improving forecasts among multiple horizons [6, 7, 8].
Different temporal aggregations can reveal important information about the
underlying data-generating process. When temporal aggregation is applied to a
time series, it can strengthen or attenuate different features [9]. Temporal
hierarchies forecasts for different horizons can be made with generating
forecasts independently for time series at each level by different (simple)
methods. These forecasts produced by different approaches and based on
different information are most likely incoherent. Reconciliation is necessary
because optimal decision-making requires coherent forecasts.

Thus, the main focus while forecasting hierarchical time series is to utilize
the information available across all levels of a given hierarchy for producing
coherent forecasts. Existing work in hierarchical forecasting deploys
methodology in which first base forecasts are independently generated for each
time series in the hierarchy and then combination/revision step is done in a
post-processing step to ensure coherence. However , all the methods used in
the industry or literature rely on the availability of complete data across
all levels of the temporal hierarchy. While this offers mathematical
convenience it is often not available to the decision maker using the
forecasts. Due to the dynamic and stochastic nature of time series, it is very
difficult to precisely estimate future changes (significant trend or mean
shift) and to adjust different horizon forecasts based on these changes. An
example can be a significant trend or mean shift in high frequency patterns at
a daily or a weekly level such as those encountered in the COVID pandemic.
There are high chances that the long term actuals will significantly diverge
from the forecasts. All the existing methods lead to static temporal
reconciliation and thus will perform poorly under dynamic scenarios.

Reinforcement learning especially deep reinforcement learning (DRL) has been
successfully applied in various fields including AlphaGo [10], ATARI games
[11] and robotics [12] . Reinforcement learning is a task-independent learning
scheme. It is suitable for problems where there is no supervised information
but only feedbacks from an external environment. The problem of informing
forecasts among hierarchies for reconciliation without prior/complete
knowledge of data distribution can be categorized as the same. Furthermore,
reinforcement learning is a data-driven approach that is able to capture
complex changing dynamics in the data and is well equipped to overcome the
deficiencies of the existing methods.

We propose a dynamic reconciliation method whereby we formulate the problem of
informing low frequency forecasts based on high frequency actuals as a Markov
Decision Process (MDP) allowing for the fact that we do not have complete
information about the dynamics of the process. This allows us to have the best
long term estimates based on the most recent data available even if the low
frequency cycles have only been partially completed. The MDP has been solved
using a Time Differenced Reinforcement learning (TDRL) approach and improves
the long terms forecasts dramatically as compared to relying solely on
historical low frequency data. The temporal difference is an agent learning
from an environment through episodes with no prior knowledge of the
environment. We have designed our own action and reward function. Based on the
factors described above, the contributions of this paper can be summarized as
below:

  * •

To the best of our knowledge, our work presents the first RL approach for the
hierarchical reconciliation problem.

  * •

Dynamic reconciliation framework which works on partial data also.

  * •

A tunable action design function that tries to incorporate forecast change
with tolerance while reconciling the forecast which ensures that produced
reconciled forecast do not deviate from expected values.

The rest of this paper is organized as follows. Section 2 introduces the
reinforcement learning basics. In Section 3, we describe the detailed
methodology of the proposed TDRL approach towards the hierarchical time series
reconciliation and give a detailed analysis on how to design actions and
$\epsilon$-greedy policy. We report and discuss experimental results in
Section 4. We talk about future directions and conclude in Section 5.

##  2 Background

###  2.1 Hierarchical Reconciliation

Temporal hierarchies for forecasting are constructed for any time series by
means of non-overlapping temporal aggregation. Such aggregation typically
leads to tree structure but need not necessarily. For example, grouped [13],
temporal [14], and cross-temporal aggregations [15] can be alternative
aggregation paths.

Consider a multi-level hierarchy $\mathbf{Y}_{t}\in\mathbb{R}^{n}$ at time t
having $t=1,...,T$. Here $y_{t,i}\in\mathbb{R}$ is the value of the i-th (out
of n) univariate time series. The index i denotes level of hierarchy. Level 0
(i=0) denotes the completely aggregated series, level 1 the first level of
disaggregation, down to level K containing the most disaggregated time series.

We refer to the time series at the leaf nodes of the hierarchy as bottom-level
series and rest of the series can be termed as aggregated series. We also call
the forecasts for all time series in the hierarchy generated without any
reconciliation approach as base forecasts denoted by
$\mathbf{\hat{Y}}_{t}$(not to be confused with bottom-level). We can split the
vector of all series $\mathbf{Y}_{t}$ into m bottom entries and r aggregated
entries where n = r + m. And $\mathbf{Y}_{t}=[u_{\textit{t}}\hskip
2.84544ptv_{\textit{t}}]^{\top}$ with $u_{\textit{t}}\in\mathbb{R}^{r}$ and
$v_{\textit{t}}\in\mathbb{R}^{m}$. Now, the aggregation matrix
$S\in\\{0,1\\}^{n\times m}$ is defined and the $\mathbf{Y}_{t},v_{\textit{t}}$
and $S$ satisfy

$\mathbf{Y}_{t}=Sv_{\textit{t}}$ (1)

Existing approaches for generating a coherent forecasts for a hierarchical
time series follow a two-step procedure: (i)generate h-step-ahead forecast for
each time series independently to obtain base forecasts $\hat{Y}_{T+h}$ and
(ii) produce revised h-step-ahead forecasts $\tilde{Y}_{T+h}$ through
reconciliation given by equation below:

$\tilde{Y}_{T+h}=SP\hat{Y}_{T+h}$ (2)

for some appropriately chosen matrix P of order $m\times n$. P matrix is used
to extract and combine the relevant elements of the base forecasts
$\hat{Y}_{T+h}$.

###  2.2 Temporal difference Reinforcement learning

Reinforcement learning (RL) is a machine learning approach inspired by
behaviorist psychology. In RL, an agent interacts with its environment by
sequentially taking actions, observing consequences, and altering its
behaviors to maximize a cumulative reward. RL is usually modeled as a MDP
which consists of a state space $\mathcal{S}=\\{s\\}$, an action space
$\mathcal{A}=\\{a\\}$, state transition dynamics
$\mathcal{T:}{\mathcal{S}\times\mathcal{A}}\rightarrow\mathcal{P(\ S)\ }$
where $\mathcal{P(\ S)\ }$ is the set of probability measures on S, an
immediate reward function
$r{:\mathcal{S}\times\mathcal{A}}\rightarrow\mathbb{R}$, and a discount factor
$\gamma\in[\ 0,1]\ $ . A policy, denoted by
$\mathcal{\pi:}\mathcal{S}\rightarrow\mathcal{P(\ A)\ }$ where $\mathcal{P(\
A)\ }$ is the set of probability measures on $\mathcal{A}$, fully defines the
behavior of an agent. The agent uses its policy to interact with the
environment and gives a trajectory of states, actions, and rewards
${s_{1}},a_{1},r_{1},...,s_{T},a_{T},r_{T}$ (T = $\infty$ indicates a infinite
horizon MDP and otherwise an episodic one) over
${:\mathcal{S}\times\mathcal{A}\times\mathbb{R}}$. The cumulative discounted
reward constitutes the return $\mathcal{R}=\sum_{t=1}^{T}{\gamma^{t-1}}r_{t}$
. The agent’s goal is to learn an optimal policy $\pi^{\ast}$ that maximizes
the expected return from the start state.
$\pi^{\ast}=arg{max_{\pi}}\mathbb{E}[\ {R|\pi}]\ $. A common method for
learning optimal policy or optimal state-value function is temporal difference
(TD) learning, which estimates the value of a state by bootstrapping from the
value estimates of successor states using Bellman-style equations. TD methods
work by updating the state-value estimates to reduce the TD error, which
describes the difference between the current estimate of the state-value and a
new sample obtained from interacting with the environment.

##  3 Related Work

Top-down and bottom-up approaches have traditionally been used to produce
coherent forecasts for a hierarchy. In Top-down, forecasts are generated at
the top level for the time series and then dis-aggregated down all the way to
the bottom level. Whereas for Bottom-up, forecasts are generated at the most
granular level and then aggregated up [16, 17]. In both of these methods, the
generation of forecasts for the entire hierarchy is dominated by a single
level of aggregation where the forecasts are produced, ignoring information at
all other levels. To optimally combine forecasts from all the series of the
hierarchy, [18] proposed the use of ordinary least-squares (OLS) estimator
after formulating the forecast reconciliation problem for a structural
hierarchy as a linear regression model. In the regression, the independent
base forecasts are modeled as the sum of the expected values of the future
series and coherency error. [13] suggested using weighted least squares (WLS),
taking account of the variances on the diagonal of the covariance matrix of
aggregation error but ignoring the off diagonal covariances. Later, [19]
considered the generalized least-squares (GLS) estimator using
$P={(S^{\top}{W_{h}}^{-1}S)}^{-1}{(S^{\top}{W_{h}}^{-1})}$, where $W_{h}$ is
the covariance matrix of the h-period-ahead forecast errors
$\hat{\varepsilon}_{T+h}={Y}_{T+h}-\hat{Y}_{T+h}$. MinT found that the
incorporation of correlation information into the reconciliation procedure to
be beneficial for forecast accuracy, when combined with a simple shrinkage
estimator. The advantages of the MinT approach are that its revised forecasts
are coherent by construction and it uses information from all levels of
hierarchy simultaneously. Disadvantages are the strong assumption of base
forecasts to be unbiased. [14] showed that it is possible to use the
reconciliation framework proposed by [18] to produce coherent forecasts by
representing temporally aggregated series as hierarchical time series. The
issue of getting coherent forecasts along both cross-sectional and temporal
dimensions (i.e., cross-temporal coherency) has been dealt with by [20, 15,
21] and [22].

##  4 Method

###  4.1 Problem Definition

For this paper, we will use only 2-level of hierarchy. Level-0 denotes monthly
aggregated data and Level-1 denotes its disaggregation into daily data. The
framework will take as input aggregated (monthly) forecasts, bottom level
(daily) forecasts $\hat{Y}$ which are provided to any reconciliation approach.
In addition to this, the framework will have access to ongoing high frequency
actual. High frequency actual can help the agent to learn the daily shares and
will help the agent to model ongoing dynamics in the data.

We consider an episodic MDP with a discount factor $\gamma=1$ where an
episode(typically one month) starts with a low frequency forecast. From the
external model, the agent also gets a share of each day in a month. The agent
is given high frequency forecast $\hat{y}_{t}$ at timestamp t, the agent takes
an action of increasing or decreasing forecast per day to capture the daily
variation/fluctuation. The main goal of the agent is to update monthly
forecasts based on lower level forecasts and actual. This is being done using
reward function R. The goal of the agent is to generate reconciled low
frequency forecasts based on the high frequency actual share seen per day.
More specifically, the core elements of the MDP are further explained as
follows:

  * •

$\mathcal{S}$: States in the environment. The state here represents the number
of days present in a month. This will help the agent to keep track of how many
days have passed in a month and the corresponding daily share. State $s_{t}$
is represented as : 1) t : the current day of the month, 2) State value will
be remaining monthly total to be adjusted.

  * •

$\mathcal{A}$: It constitutes a set of actions present to the agent in a state
S. We have designed our own set of actions which will be discussed in the
later section.

  * •

$\mathcal{T}$ : Transition function of MDP. We have used $\epsilon-$ greedy
policy which mixes two behavior of switching between random and Q policy using
the probability hyperparameter $\epsilon$.

  * •

$\mathcal{R}$: A reward $r_{t}$ is a scalar which measures the goodness of the
action $a_{t}$ taken by the agent in the state $s_{t}$. Reward $r_{t}$ at each
time step t will be high frequency actual encountered for that day.

  * •

$\gamma$ : We set reward discount factor $\gamma$ = 1

###  4.2 Action Design

Actions help the agent to interact with the environment and get feedback. As
the main goal is the adjustment of the forecast, we have defined actions
accordingly. The Agent has to perform an action of increase or decrease in
forecast per day to capture the daily variation/fluctuation. But how much to
increase or decrease from daily level forecast can have infinite values. To
make this action space discrete, we have introduced the tolerance level
$\varepsilon$ parameter which helps in bucketing the action space. In any
state $s_{t}$, agent can increase or decrease the high frequency forecast
based on tolerance level $\varepsilon$. Lets take an example of 3 actions,
then for any time step t the Agent can make updated daily level forecast to be

  1. 1.

$\geq\hat{y_{t}}+\varepsilon$

  2. 2.

$=\hat{y_{t}}\pm\varepsilon$

  3. 3.

$\leq\hat{y_{t}}-\varepsilon$

The tolerance level $\varepsilon$ parameter can be tuned according to the
problem at hand and helps in deciding how much fluctuation from the daily
level forecast is permitted to the agent. To illustrate the concept, lets
consider $\varepsilon$ =5$ and daily level forecast value at time t to be 30$.
Then the agent can take one of the 3 actions of increasing/decreasing
forecasts defined as:

  1. 1.

$\geq 30+5$

  2. 2.

$=30\pm 5$

  3. 3.

$\leq 30-5$

###  4.3 Policy

Epsilon-Greedy is a method for selecting actions that balance both exploration
and exploitation by choosing between exploration and exploitation randomly.
The epsilon-greedy, where $\epsilon$ refers to the probability of choosing to
explore. A policy is $\epsilon$ -greedy with respect to an action-value
function estimate Q if for every state,

  * •

with probability $1-\epsilon$ , the Agent selects the greedy action, and

  * •

with probability $\epsilon$ , the Agent selects an action uniformly at random
from the set of available (non-greedy and greedy) actions.

The probability of selecting non-greedy actions increases with larger value of
$\epsilon$. To construct a fixed policy $\pi$ that is $\epsilon$ -greedy with
respect to the current action-value function estimate Q , we have set the
policy as

$1-\epsilon+\frac{\epsilon}{|A(S)|}$ (3)

if action a maximizes Q ( s , a ). Else

$\frac{\epsilon}{|A(S)|}$ (4)

for each $\textit{s}\in S$ and $\textit{a}\in A(s)$.

Figure 1:  Environment design for agent training

###  4.4 State-Value Update

For finding the optimal state-value function, we have used SARSA variation of
Temporal Difference-(0) Algorithm. TD(0) improves upon the drawbacks of the
Monte Carlo (MC) method in which the agent has to wait until the end of an
episode to obtain the actual return (experienced return) before it can update
and make any improvements to the value function estimate. TD(0) updates the
state value function at every time step t without waiting for the end of the
episode. For finite-state MDP under a policy $\pi$, the update rule is given
by:

$V(S_{t})\leftarrow V(S_{t})+\alpha(R_{t+1}+\gamma V(S_{t+1}-V(S_{t}))$ (5)

where $V(S_{t})$ denotes is state value function for state $S_{t}$, $\alpha$
is step-size parameter, $\gamma$ is discount factor and $R_{t+1}$ is immediate
reward recieved at time step t+1.

Figure 2:  An example of state-action initialization for a time series.
$\hat{y_{1}}$ to $\hat{y_{6}}$ are forecasted values of a time series observed
at t $\in\\{{1,....,6}\\}$ respectively. Considering time step t=2, $S_{2}$
will be initialized as 150-(10+20)=120. Three actions for $S_{2}$ are defined
based on $\epsilon$=5 and $\hat{y_{2}}$=20.  Figure 3: High frequency dataset
starting from January 2019 to March 2020. X-axis denotes dates and Y-axis
denotes open price value of Nifty 50 index. Test period is highlighted in
green.

###  4.5 Reinforcement learning to Reconciliation

Putting them together, we present our Dynamic Temporal Reconciliation (DTR)
framework as given in Fig 1. The framework is built using TD(0) which takes as
input high frequency forecasts, where the state values are first initialised
as difference between forecasted low frequency total and cumulative high
frequency forecasts seen till time step t for every t $\in\\{{1,....,n}\\}$.
Action value function are initialised using the three actions defined in
Section 3.2 based on forecast for every t $\in\\{{1,....,n}\\}$ for the
construction of corresponding $\epsilon$-Greedy policy. See Fig. 2 for an
illustration. Based on the $\epsilon$-greedy policy, the agent takes an action
$a_{t}\in A$ under state $s_{t}\in S$. Here our reward function $r_{t}$ is not
directly beneficial for the agent at each step but combined with the value
update function, it helps in achieving the overall goal of the agent.

Table 1: Performance comparison for reconciled task for different settings for
tolerance level $\varepsilon$ with fixed $\epsilon$=0.05 (RMF: Reconciled
Monthly Forecast)
Note : the columns represent three values ( RMF / $MAPE$ / $\%_{f}$ ) Date | Actual | Forecasted |  | RMF
---
$\varepsilon$ = 0.1
| RMF
---
$\varepsilon$ = 0.2
| RMF
---
$\varepsilon$ = 0.3
01/03/20 | 11386 | 11354 | 367706 / 25% / 0% | 367706 / 25% / 0% | 364148 / 24% / 1%
02/03/20 | 11387 | 11472 | 368892 / 25% / 0% | 367706 / 25% / 0% | 367706 / 25% / 0%
03/03/20 | 11218 | 11513 | 367706 / 25% / 0% | 367706 / 25% / 0% | 367706 / 25% / 0%
04/03/20 | 11351 | 11527 | 367706 / 25% / 0% | 367706 / 25% / 0% | 364148 / 24% / 1%
05/03/20 | 11306 | 11536 | 367706 / 25% / 0% | 367706 / 25% / 0% | 367706 / 25% / 0%
06/03/20 | 10943 | 11668 | 368892 / 25% / 0% | 370079 / 26% / 1% | 371265 / 26% / 1%
07/03/20 | 10876 | 11668 | 367706 / 25% / 0% | 367706 / 25% / 0% | 367706 / 25% / 0%
08/03/20 | 10809 | 11786 | 367706 / 25% / 0% | 367706 / 25% / 0% | 367706 / 25% / 0%
09/03/20 | 10742 | 11783 | 368892 / 25% / 0% | 367706 / 25% / 0% | 367706 / 25% / 0%
10/03/20 | 10538 | 11783 | 333308 / 13% / 9% | 370079 / 26% / 1% | 364148 / 24% / 1%
11/03/20 | 10334 | 11783 | 335680 / 14% / 9% | 367706 / 25% / 0% | 371265 / 26% / 1%
12/03/20 | 10040 | 11877 | 334494 / 14% / 9% | 365334 / 24% / 1% | 367706 / 25% / 0%
13/03/20 | 9108 | 11877 | 332122 / 13% / 10% | 298910 / 2% / 19% | 367706 / 25% / 0%
14/03/20 | 9268 | 11877 | 330936 / 12% / 10% | 306027 / 4% / 17% | 374823 / 27% / 2%
15/03/20 | 9428 | 11877 | 334494 / 14% / 9% | 296537 / 1% / 19% | 367706 / 25% / 0%
16/03/20 | 9588 | 11877 | 335680 / 14% / 9% | 372451 / 26% / 1% | 371265 / 26% / 1%
17/03/20 | 9285 | 11927 | 333308 / 13% / 9% | 303654 / 3% / 17% | 367706 / 25% / 0%
18/03/20 | 9088 | 11968 | 333308 / 13% / 9% | 298910 / 2% / 19% | 371265 / 26% / 1%
19/03/20 | 8063 | 12001 | 332122 / 13% / 10% | 296537 / 1% / 19% | 257394 / 13% / 30%
20/03/20 | 8284 | 12001 | 330936 / 12% / 10% | 298910 / 2% / 19% | 271628 / 8% / 26%
21/03/20 | 8172 | 12023 | 330936 / 12% / 10% | 298910 / 2% / 19% | 257394 / 13% / 30%
22/03/20 | 8059 | 12023 | 333308 / 13% / 9% | 306027 / 4% / 17% | 264511 / 10% / 28%
23/03/20 | 7946 | 12006 | 333308 / 13% / 9% | 294165 / 0% / 20% | 257394 / 13% / 30%
24/03/20 | 7848 | 12000 | 336866 / 14% / 8% | 301282 / 2% / 18% | 260953 / 11% / 29%
25/03/20 | 7735 | 11998 | 333308 / 13% / 9% | 296537 / 1% / 19% | 257394 / 13% / 30%
26/03/20 | 8451 | 11998 | 333308 / 13% / 9% | 301282 / 2% / 18% | 371265 / 26% / 1%
27/03/20 | 8949 | 12111 | 332122 / 13% / 10% | 298910 / 2% / 19% | 381940 / 30% / 4%
28/03/20 | 8761 | 12111 | 335680 / 14% / 9% | 296537 / 1% / 19% | 371265 / 26% / 1%
29/03/20 | 8574 | 12111 | 334494 / 14% / 9% | 301282 / 2% / 18% | 367706 / 25% / 0%
30/03/20 | 8386 | 12077 | 330936 / 12% / 10% | 294165 / 0% / 20% | 257394 / 13% / 30%
31/03/20 | 8529 | 12091 | 333308 / 13% / 9% | 303654 / 3% / 17% | 257394 / 13% / 30%
Total | 294452 | 367706 |  |  |

##  5 Experimental Results

In this section, we present the empirical study of DTR along with the
experimental setup and implementation details of DTR.

###  5.1 Dataset

We perform experimental validation of our approach by using NIFTY 50 data
set[1]. The NIFTY 50 index is the National Stock Exchange of India’s benchmark
broad based stock market index for the Indian equity market. NIFTY 50 stands
for National Index Fifty, and represents the weighted average of the top 50
Indian company stocks in 17 sectors. This data set contains a time series
starting from January 2000 till October 2021 containing indexes depicting the
performance of the stock in form of Open price, Close price, High value per
day, Low value per day, and Total volume traded per day. For most of the
markets across the globe, the impact of COVID started after February 2020
which is why there is a steep decrease in the value of the time series in
March 2020. The series starts recovering from April 2020. To capture this
dynamic decrease, we have used the Open price index with the data starting
from January 2019 till February 2020 as a training set and tested on March
2020. Given National Stock Exchange is only open on the weekdays from Monday
to Friday, we interpolated the missing weekend dates. Fig. 3 shows high
frequency data aggregated at the weekly level.

###  5.2 Performance Evaluation

To evaluate the model performance, we have used the mean absolute percentage
error (MAPE). This metric is calculated as:

$MAPE=\frac{1}{n}{\sum_{t=1}^{n}\frac{|Y_{t}-\hat{Y_{t}}|}{|Y_{t}|}\ast
100(\%)},$ (6)

where n is number of observations in time series, $Y_{t}$ and $\hat{Y_{t}}$ is
observed and forecastes values at time step t.

We report the performance of different settings of proposed algorithms in
Table 1. For this experimentation, we have used fixed $\epsilon$-greedy policy
along with 3 actions defined based on Section 3.2. The results in all
experiments are given by “ Reconciled Monthly forecast/ MAPE of reconciled
forecast wrt monthly actual($MAPE_{rec}$ )/ Percentage improvement over
monthly forecast without reconciliation ($\%_{f})$ ”. As can be seen, the
proposed DTR clearly outperform the both high frequency aggregated forecast as
well as low frequency without reconciliation forecast. The framework starts
with aggregated high frequency forecast and based on the actual seen per day
changes its estimate for low frequency forecast. The results reveal how the
DTR framework adapts to high frequency actual based on different parameter
initialisation. If there is unexpected increase/decrease in daily values
between the days, the framework increases/decreases its estimated low
frequency forecast which is shown in Table 1. On Day 13, there is a huge
decrease in high frequency actual going from ( 10040 -> 9108 ) which makes the
framework to adjust estimated low frequency forecast going from ( 365334 ->
298910 ) based on 5% tolerance and $\varepsilon$ as 20 % . The reconciled
forecast are highly dependent on tolerance level. Tolerance level parameter is
problem specific and should be adjusted based on the business needs. We have
tested our framework on values of $\varepsilon$ at ( 10%, 20%, 30% ) of
average high frequency forecast value which is ( 36736, 73473, 110209 )
respectively.

###  5.3 Hyperparameter Analysis

The proposed framework has two hyper-parameters $\epsilon$ and $\varepsilon$.
$\epsilon$ is used to balance between exploration and exploitation which helps
in selecting the action with the highest estimated reward most of the time.
$\varepsilon$ is a parameter to adjust the strictness of the agent. Table 3
contains the effects of $\epsilon$ and $\varepsilon$ on the agent’s
performance. This provides a user the flexibility to change the agent’s
behavior according to their requirement. Smaller $\epsilon$ will rely on
greedy action that agent believes has the best long-term effect. A higher
$\epsilon$ will help agent to explore more. Similarly smaller $\varepsilon$
will lead to adjustment in forecast near the mean value of forecast only
whereas a higher $\varepsilon$ will help agent to model high variance changes
also.

Table 2: Effect on agent by varying $\varepsilon$ and $\epsilon$ for NIFTY 50 dataset  tolerance | epsilon | $MAPE_{rec}$ | $\%_{f}$
---|---|---|---
10% | 0.05 | 13% | 9%
20% | 0.05 | 3% | 17%
30% | 0.05 | 13% | 30%
10% | 0.1 | 15% | 8%
20% | 0.1 | 5% | 16%
30% | 0.1 | 5% | 24%
10% | 0.2 | 16% | 7%
20% | 0.2 | 6% | 15%
30% | 0.2 | 4% | 23%

##  6 Conclusion

In this paper we present a Dynamic Temporal Reconciliation (DTR) framework by
incorporating a dynamic adjustment layer into traditional Reconciliation
algorithms. The integration allows us to better model the complex changes in
high frequency forecast and adjust long term estimates accordingly. Therefore,
DTR provides a powerful tool to handle incomplete low frequency data for
reconciliation tasks based on high frequency actual and is well equipped to
overcome the deficiencies of the existing reconciliation methods. For
reconciliation, we propose a Reinforcement learning framework which is
formulated as a MDP. This has been solved using a Time Differenced
Reinforcement learning (TDRL) approach with custom action design and reward
function. Proposed framework is data-driven and can be customised based on
business needs. We examine the framework on NIFTY 50 dataset[1], and DTR
framework has demonstrated superior performance over baseline forecasts.
Although we introduce DTR in an environment with only 3 actions and fixed
$\epsilon$-greedy policy, the framework can be customised by changing numbers
of actions designed and by learning a dynamic policy function.

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# Entangling quantum logic gates in neutral atoms via the microwave-driven
spin-flip blockade

Vikas Buchemmavari<EMAIL_ADDRESS>Center for Quantum Information and
Control, University of New Mexico, Albuquerque, NM, USA  Department of Physics
and Astronomy, University of New Mexico, Albuquerque, NM, USA  Sivaprasad
Omanakuttan  Center for Quantum Information and Control, University of New
Mexico, Albuquerque, NM, USA  Department of Physics and Astronomy, University
of New Mexico, Albuquerque, NM, USA  Yuan-Yu Jau  Center for Quantum
Information and Control, University of New Mexico, Albuquerque, NM, USA
Sandia National Laboratories, Albuquerque, NM, USA  Ivan Deutsch
<EMAIL_ADDRESS>Center for Quantum Information and Control, University of New
Mexico, Albuquerque, NM, USA  Department of Physics and Astronomy, University
of New Mexico, Albuquerque, NM, USA

(2023/5/1)

###### Abstract

The Rydberg dipole-blockade has emerged as the standard mechanism to induce
entanglement between neutral atom qubits. In these protocols, laser fields
that couple qubit states to Rydberg states are modulated to implement
entangling gates. Here we present an alternative protocol to implement
entangling gates via Rydberg dressing and a microwave-field-driven spin-flip
blockade [1]. We consider the specific example of qubits encoded in the clock
states states of cesium. An auxiliary hyperfine state is optically dressed so
that it acquires partial Rydberg character. It thus acts as a proxy Rydberg
state, with a nonlinear light-shift that plays the role of blockade strength.
A microwave-frequency field coupling a qubit state to this dressed auxiliary
state can be modulated to implement entangling gates. Logic gate protocols
designed for the optical regime can be imported to this microwave regime, for
which experimental control methods are more robust. We show that unlike the
strong dipole-blockade regime usually employed in Rydberg experiments, going
to a moderate-spin-flip-blockade regime results in faster gates and smaller
Rydberg decay. We study various regimes of operations that can yield high-
fidelity two-qubit entangling gates and characterize their analytical
behavior. In addition to the inherent robustness of microwave control, we can
design these gates to be more robust to thermal fluctuations in atomic motion
as well to laser amplitude, and other noise sources such as stray background
fields.

##  I Introduction

Arrays of optically trapped atoms have emerged as a powerful platform for
quantum information processing (QIP) [2, 3, 4, 5]. This architecture has a
number of unique capabilities including the potential to scale to arrays with
hundreds of atoms [6, 7], the ability to reconfigure the geometry through atom
transport [8, 9, 10], and multiple atomic species and mechanisms with which to
encode, manipulate, and measure quantum information [11, 12, 13, 14, 15, 16,
17, 18]. Applications include quantum metrology [19, 20, 21, 22, 23],
simulation of many-body physics [7, 6, 21, 24, 21], optimization of
combinatoric problems [25, 26, 27], and universal quantum computing with
potential paths to fault-tolerant error correction [28, 29, 10]. The
development of neutral atom quantum computing has paralleled the development
of atomic clocks, both in the traditional alkali atoms (cesium and rubidium)
that define microwave frequency standards and in alkaline earth-like atoms
(strontium and ytterbium) that define optical frequency standards [30, 31,
32]. While the latter provides a path to ultra-precise clocks, manipulating
qubits at microwave frequencies offers potential advantages for coherent
control required for QIP, and will be the focus of this work.

The standard entangling mechanism for neutral-atom QIP is the Rydberg dipole-
blockade, arising from the strong Van der Waals interaction $V_{rr}$ [3].
Typically protocols for entangling gates are achieved through the phases that
are accumulated as Rabi oscillations between ground and Rydberg states lead to
a closed loop trajectories on the Bloch sphere [12]. Progress has been made in
the recent years to improve and optimize these gates, especially using quantum
optimal control algorithms [33, 34], which have been demonstrated in
experiments [35, 15]. While the fidelity is fundamentally limited by the
lifetime of Rydberg states, the current gate infidelities are dominated by
other noise sources including inhomogeneities due to atomic thermal motion,
laser noise, and imperfect shielding of stray electric fields. In principle,
some of these effects can be mitigated using dynamical decoupling or other
composite pulse schemes and robust control techniques [36, 37].

Alternative protocols for implementing entangling gates employ “Rydberg
dressing” whereby one of the qubit states is optically dressed with some
Rydberg character [38]. Due to the strong interaction between the atoms when
both are in the Rydberg state, the resulting light shift of two dressed atoms
is different from two independently dressed atoms. The difference in the light
shift energies, $J$, provides an entangling mechanism [39, 40, 19]. Of
particular interest is the phenomenon of the spin-flip blockade for qubits
encoded in the clock states of alkali atoms [1]. In the presence of Rydberg
dressing, a microwave photon can flip the qubit spin from the lower to upper
hyperfine state, but the spin of two qubits is blockaded when $J$ is
sufficiently large. Like its optical Rydberg-blockade counterpart[13], the
microwave spin-flip blockade provides a direct route to creating Bell states
[1].

In this paper, we extend the use of the spin-flip blockade from Bell-state
preparation to full two-qubit logic gates. A new ingredient is the
introduction of an auxiliary dressed hyperfine level that plays the role of
the Rydberg state but for microwave-driven transitions. Importantly, this
allows us to map the dipole-blockade physics implemented in the optical regime
to the spin-flip blockade in the microwave regime. Hence, any gate protocols
that can be implemented in the optical regime, using an optical/UV field
coupling to the Rydberg state, can be implemented in the microwave regime.
This can potentially lead to more robust implementations as we can employ the
mature tools of microwave control, and reduce other inhomogeneities such as
noise from residual Doppler shifts.

The remainder of this paper is organized as follows. In Sec. (II), we show how
to map the physics of the Rydberg blockade to the spin-flip blockade using
dressed states, which forms the basis of our gate protocols. In Sec. (III) we
consider specific gate protocols and the use of quantum control to optimize
performance. In Sec. (IV) we consider the most important sources of noise due
to thermal fluctuations in the atomic motion and show how robust optimal
control can be used to safeguard against this. Sec. (V) summarizes the results
and presents an outlook for future work.

##  II Mapping from optical to microwave control

Figure 1: Dressing scheme to map the optical Rydberg blockade to the microwave
spin-flip blockade. (a) Single atom level structure. Electronic ground states
$\ket{a}$ and $\ket{1}$ (e.g., hyperfine clock states) can be coupled by a
microwave/Raman field with Rabi frequency $\Omega_{\mu w}$ and detuning
$\Delta_{\mu w}$. The “auxiliary” state $\ket{a}$ is dressed through optical
coupling to Rydberg state $\ket{r}$ with Rabi frequency $\Omega_{\mathrm{L}}$
and detuning $\Delta_{L}$, leading to a one-atom light shift $E^{(1)}_{LS}$.
The admixture of Rydberg character modifies the microwave coupling Rabi
frequency $\tilde{\Omega}_{\mu w}=\langle a|\tilde{a}\rangle{\Omega}_{\mu w}$
and detuning, $-\tilde{\Delta}_{\mu w}=-\Delta_{\mu w}+E^{(1)}_{LS}$. (b)
Level diagram in the two-atom symmetric subspace, including atom-atom
interactions. In the symmetric state
$\ket{1a}_{+}=(\ket{1a}+\ket{a1})/\sqrt{2}$, only one atom can be optically
dressed through coupling to $\ket{1r}_{+}$ with Rabi frequency
$\Omega_{\mathrm{L}}$, leading to dressed state $\ket{1\tilde{a}}$. The state
$\ket{aa}$, in which two atoms can be optical excited, is coupled to
$\ket{ar}_{+}$ with Rabi frequency $\sqrt{2}\Omega_{L}$, but this state is
blockaded from excitation to $\ket{rr}$ by the Van der Waals energy $V_{rr}$.
The result is a two-atom light shift $E_{LS}^{(2)}=2E_{LS}^{(1)}+J$, leading
to the dressed ground-state $\ket{\widetilde{aa}}$ shown on the right. The
state $\ket{11}$ is coupled to $\ket{1\tilde{a}}_{+}$ by microwave/Raman
photons, but the state $\ket{1\tilde{a}}_{+}$ is blockaded from excitation to
$\ket{\widetilde{aa}}$ due to the nonlinear light shift $J$ (the spin-flip
blockade). The level diagram for the optically-coupled triplet
$\\{\ket{aa},\ket{ar}_{+},\ket{rr}\\}$ maps directly to the microwave-coupled
dressed triplet $\\{\ket{11},\ket{1\tilde{a}}_{+},\ket{\widetilde{aa}}\\}$
level diagram.

To begin we describe how coherent control associated with the Rydberg-blockade
in the optical regime can be mapped to the microwave regime via Rydberg-
dressing and the spin-flip blockade. For concreteness, we consider
${}^{133}\mathrm{Cs}$ where qubits are stored in the hyperfine levels of the
electronic ground state $6S_{1/2}$, though this easily generalizes to other
species for both alkali and alkaline-earth-like atoms. We take the logical
states to be the clock states, $\ket{0}=\ket{F=4,m_{F}=0}$,
$\ket{1}=\ket{F=3,m_{F}=0}$.

In the presence of a bias magnetic field, when a two-qubit gate is to be
performed, the $\ket{0}$-state of each atom can be shelved in the
$\ket{F=3,m_{F}=1}$ state, where it is effectively removed from further
interactions. The state $\ket{F=4,m_{F}=0}\equiv\ket{a}$ is then an auxiliary
state, which is coupled to a Rydberg state $\ket{r}$ via a near-resonance
optical/UV laser field with detuning $\Delta_{\mathrm{L}}$ and Rabi frequency
$\Omega_{\mathrm{L}}$, as shown in Fig. 1a. For a single atom this results in
a dressed Autler-Townes doublet

$\displaystyle\ket{\tilde{a}}$
$\displaystyle=\cos\frac{\theta}{2}\ket{a}+\sin\frac{\theta}{2}\ket{r}$ (1)
$\displaystyle\ket{\tilde{r}}$
$\displaystyle=\cos\frac{\theta}{2}\ket{r}-\sin\frac{\theta}{2}\ket{a}$

where $\tan\theta=-\Omega_{\mathrm{L}}/\Delta_{\mathrm{L}}$. These states are
light-shifted by energies
$-\Delta_{\mathrm{L}}/2\pm\sqrt{\Omega_{\mathrm{L}}^{2}+\Delta_{\mathrm{L}}^{2}}/2$
(here and throughout $\hbar=1$). Due to the admixture, the dressed
$\ket{\tilde{a}}$ state then plays the role of the Rydberg-state for
entangling the spin qubits. The states $\ket{1}$ and $\ket{\tilde{a}}$ can be
coupled by a microwave/Raman field with an effective Rabi frequency
$\tilde{\Omega}_{\mu w}=\cos(\theta/2)\Omega_{\mu w}$, where $\Omega_{\mu w}$
is the Rabi frequency coupling $\ket{1}$ to the bare state $\ket{a}$. When
driving this transition resonantly, if
$\sqrt{\Omega_{\mathrm{L}}^{2}+\Delta_{\mathrm{L}}^{2}}\gg\tilde{\Omega}_{\mu
w}$, we can neglect the coupling from $\ket{1}$ to $|\tilde{r}\rangle$.

To entangle two qubits we consider symmetrically dressing the atoms by the
same uniform laser and microwave/Raman fields. Interactions between the qubits
only occur when both atoms are initially in the $\ket{1}$ state and then
excited by microwaves. To describe the entangling interaction, we need only to
consider the symmetric subspace, $\ket{11}$, $\ket{1a}_{+}$, and $\ket{aa}$;
here and throughout $\ket{xy}_{\pm}\equiv(\ket{xy}\pm\ket{yx})/\sqrt{2}$. The
$\ket{1a}_{+}$ state is coupled to $\ket{1r}_{+}$ with Rabi frequency
$\Omega_{\mathrm{L}}$, yielding the same single atom dressed states as Eq.
(1), denoted $\ket{1\tilde{a}}_{+}$ and $\ket{1\tilde{r}}_{+}$. The state
$\ket{aa}$ is coupled to $\ket{ar}_{+}$ with Rabi frequency
$\sqrt{2}\Omega_{\mathrm{L}}$. For conditions well-approximated by a perfect
Rydberg blockade, the state $\ket{rr}$ can be ignored and the resulting
dressed states are denoted $\ket{\widetilde{aa}}$, $\ket{\widetilde{ar}}_{+}$.
This doublet is split by
$\sqrt{2\Omega_{\mathrm{L}}^{2}+\Delta_{\mathrm{L}}^{2}}$. The nonlinear
behavior of the light shift, due to the Rydberg blockage, implies that the
energy of the two interacting atoms is not equal to the sum of the energies of
the noninteracting atoms. The difference is the “entangling energy,” defined
as
$J=E_{\widetilde{aa}}-2E_{1\tilde{a}_{+}}=\frac{1}{2}[\Delta_{\mathrm{L}}\pm(\sqrt{2\Omega_{\mathrm{L}}^{2}+\Delta_{\mathrm{L}}^{2}}-2\sqrt{\Omega_{\mathrm{L}}^{2}+\Delta_{\mathrm{L}}^{2}})]$.
The $\pm$ sign corresponds to two branches of the light shift, adiabatically
connected to the red or blue side of resonance[40].

Critically, in the presence of dressing, when driving the microwave transition
$\ket{11}\rightarrow\ket{1\tilde{a}}_{+}$ with Rabi frequency
$\tilde{\Omega}_{\mu w}$, the transition
$\ket{1\tilde{a}}_{+}\rightarrow\ket{\widetilde{aa}}$ will be blockaded when
$J\gg\tilde{\Omega}_{\mu w}$. This is the spin-flip blockade, as first
observed in [1]. As seen in Fig. 1, the level structure of the dressed states
mimics that of the optically driven ground-Rydberg system with the dressed
auxiliary state as a proxy for the Rydberg state, and the spin-flip-blockade
replacing the Rydberg blockade. This allows us to map all gate protocols from
the optical regime to the microwave regime.

While the essential physics is well understood assuming a perfect Rydberg
blockade, this may not be the optimal operating point given the fundamental
limit on the gate fidelity determined by the decay rate of Rydberg state
$\Gamma_{r}$. More generally when two interacting atoms participate in
dressing, including some admixture of doubly-excited Rydberg states, the
result is the dressed triplet,

$\displaystyle\ket{\widetilde{aa}}$
$\displaystyle=\alpha\ket{aa}+\beta\ket{ar}_{+}+\gamma\ket{rr},$ (2)
$\displaystyle\ket{\widetilde{ar}}_{+}$
$\displaystyle=\alpha_{2}\ket{ar}_{+}+\beta_{2}\ket{aa}+\gamma_{2}\ket{rr},$
$\displaystyle\ket{\widetilde{rr}}$
$\displaystyle=\alpha_{3}\ket{rr}+\beta_{3}\ket{aa}+\gamma_{3}\ket{ar}_{+},$

with dressed energies $E_{\widetilde{aa}}$, $E_{\widetilde{ar}_{+}}$ and
$E_{\widetilde{rr}}$ respectively. In this case the Rabi frequency for the
coupling $\ket{11}\leftrightarrow\ket{1\tilde{a}}_{+}$ is given by
$\sqrt{2}\tilde{\Omega}_{\mu w}=\sqrt{2}(\cos\frac{\theta}{2})\Omega_{\mu w}$
and for $\ket{1\tilde{a}}_{+}\leftrightarrow\ket{\widetilde{aa}}$,
$\sqrt{2}\tilde{\Omega}^{\prime}_{\mu
w}=\sqrt{2}(\alpha\cos\frac{\theta}{2}+\beta\sin\frac{\theta}{2}/\sqrt{2})\Omega_{\mu
w}$. Again, for appropriate microwave detuning we can neglect coupling of
$\ket{1\tilde{a}}_{+}$ to $|\widetilde{ar}\rangle_{+}$ and
$\ket{\widetilde{rr}}$. Note, since the states
$\ket{\tilde{a}},\ket{\widetilde{aa}}$ have only partial Rydberg character,
their effective decay rates will be proportional to the admixture of Rydberg
state in the dressing,

$\displaystyle\Gamma_{\tilde{a}}$
$\displaystyle=\sin^{2}\frac{\theta}{2}\Gamma_{r}$ (3)
$\displaystyle\Gamma_{\widetilde{aa}}$
$\displaystyle=(|\beta|^{2}+2|\gamma|^{2})\Gamma_{r}$ (4)

where we assumed that the state $\ket{rr}$ decays twice as fast as state
$\ket{r}$.

In summary, through Rydberg dressing the optical physics of the dipole
blockade is mapped to the microwave spin-flip blockade, with $\ket{\tilde{a}}$
playing the role of $\ket{r}$ according to mapping:

$\displaystyle\Omega_{\mathrm{L}}$
$\displaystyle\rightarrow\tilde{\Omega}_{\mu w},\tilde{\Omega}^{\prime}_{\mu
w}$ (5) $\displaystyle V_{rr}$ $\displaystyle\rightarrow J$ (6)
$\displaystyle\Gamma_{r}$ $\displaystyle\rightarrow\Gamma_{\tilde{a}}$ (7)
$\displaystyle\Gamma_{rr}$ $\displaystyle\rightarrow\Gamma_{\widetilde{aa}}$
(8)

One key difference is the ability to control the entangling energy, $J$, by
controlling the laser parameters $\Omega_{\mathrm{L}},\Delta_{\mathrm{L}}$ and
the well defined nature of the state $\ket{\widetilde{aa}}$. This is in
contrast to the optical regime where the doubly-excited Rydberg spectrum can
be complex and $V_{rr}$ is strongly dependent on the interatomic distance. We
will use this to our advantage below by operating in the moderate-blockade
regime where $J\approx\Omega_{\mu w}$.

##  III Entangling gate protocol

Figure 2: Level diagram for implementing entangling gates in the microwave
regime. (a) The hyperfine states of the cesium electronic ground state,
labeled $\ket{F,M_{F}}$, are shown in the presence of a bias magnetic field,
near the clock states with $M_{F}=0$ in which a qubit is encoded. Before
implementing an entangling gate, the logical state $\ket{0}$ is shelved in
$\ket{3,1}$ through a microwave resonance. (b) $\ket{4,0}$ is now an auxiliary
state $|a\rangle$, dressed with a Rydberg laser (Rabi frequency
$\Omega_{\mathrm{L}}$, detuning $\Delta_{\mathrm{L}}$). (c) The phase of a
microwave/Raman field of correct frequency and polarization couples
$|1\rangle$ and dressed state $|\tilde{a}\rangle$, leaving $\ket{0}$
untouched, is modulated to implement the gate. The microwave here is shown
tuned to resonance with the dressed state and the Rabi frequency,
$\tilde{\Omega}_{\mu w}(t)$, is modified from its bare value due to the
admixture of $\ket{r}$ in $|\tilde{a}\rangle$. (d) Two-atom symmetric basis
showing the coupling of the logical basis to the Rydberg-dressed states. The
state $|01\rangle$($\ket{10}$) couples to the dressed state
$|0\tilde{a}\rangle$($\ket{\tilde{a}0}$) with modified Rabi frequency
$\tilde{\Omega}_{\mu w}$ . The two-atom state $|11\rangle$ couples to the
bright state $|\tilde{a}1\rangle_{+}$ and it to $|\widetilde{aa}\rangle$ with
modified Rabi frequencies $\sqrt{2}\tilde{\Omega}_{\mu
w},\sqrt{2}\tilde{\Omega}^{\prime}_{\mu w}$. The state
$|\widetilde{aa}\rangle$ has a nonlinear light shift that we call the
entangling energy, $J$, which determines the strength of the atom-atom
coupling.

In this section, we use the optical-to-microwave mapping described above as a
vehicle for implementing two-qubit entangling gates. Previous work was based
on adiabatic dressing [41, 40, 39], which is intrinsically robust to certain
forms of noise. Here we demonstrate the versatility of Rydberg dressing and
microwave control in its application of the most successful gate to date, the
Levine-Pichler (LP) gate [12]. The LP-gate is also well suited to optimal
control, as was theoretically studied in [33, 34] and recently demonstrated in
[35, 15] based on optical excitations. Microwave control will open the door to
additional robust protocols that can leverage well-developed technologies.

In the original LP-gate [12], a sequence of optical pulses with carefully
chosen Rabi frequency, detuning, and relative phase are used to implement a
$\mathrm{CZ}$ gate based on the phases accumulated by the computational basis
states as rotations occur on the Bloch sphere. We focus here on its
generalization based on quantum optimal control [33, 34]. Not only are these
protocols faster and yield higher fidelity, but they can be made robust to
some inhomogeneities. In addition, we can design waveforms beyond the perfect
blockade regime, which can lead to further optimization.

In the case of cesium considered here, the qubit is encoded in the clock
states $\ket{0}\equiv\ket{F=4,m_{F}=0}$ and $\ket{1}\equiv\ket{F=3,m_{F}=0}$,
and a pulse maps $\ket{F=4,m_{F}=0}\leftrightarrow\ket{F=3,m_{F}=1}$ to shelve
all of the qubit computational states in the $F=3$ manifold similar to [42].
The unpopulated state $\ket{F=4,m_{F}=0}$ is then designated as the auxiliary
state $\ket{a}$, which can be dressed using a Rydberg laser without coupling
any population in the computational states. This results in the creation of a
level diagram as shown in Fig. 2. A microwave/Raman laser field is used to
couple the qubit state $\ket{1}$ to the dressed state $\ket{\tilde{a}}$, with
effective Rabi frequency $\tilde{\Omega}_{\mu w}$, detuning
$\tilde{\Delta}_{\mu w}$ and phase $\xi_{\mu w}$. The dressed auxiliary state
acts as a proxy to the Rydberg state. We set $\Omega_{\mu w}$ to be a constant
and detuning $\tilde{\Delta}_{\mu w}=0$ with respect to the dressed state
$\ket{\tilde{a}}$ and modulate the phase $\xi_{\mu w}(t)$ of the microwave
field over a period of time $\tau$ to implement a $\mathrm{CZ}$ gate.

As shown in Fig. 2d, the computational basis states
$\ket{k};k\equiv\\{00,01,10,11\\}$ do not couple to each other during the
evolution. Thus we consider only diagonal gates, a process in which all the
basis states evolve as $\ket{k}\rightarrow\ket{\psi_{k}(t)}$ and the
populations return to these initial states at the end of the pulse, $t=\tau$.
The net result is an accumulation of phase,
$\ket{k}\rightarrow\ket{\psi_{k}(\tau)}=e^{i\phi_{k}}\ket{k}$. If these phases
satisfy the condition $\phi_{11}-2\phi_{01}=\pm\pi$, the gate that is
implemented is equivalent to a CZ-gate up to local
$e^{-i\phi_{01}\sigma_{z}/2}$ gates on both qubits. We will refer to this
family of gates as “$\mathrm{CZ}$ gate” hereon for simplicity. Hence, for a
unitary two-qubit gate that we implement, $U$, we define the $\mathrm{CZ}$
gate fidelity as

$\mathcal{F}=|1+e^{-i\phi_{01}}\bra{01}U\ket{01}+e^{-i\phi_{10}}\bra{10}U\ket{10}-e^{-i\phi_{11}}\bra{11}U\ket{11}|^{2}/16.$
(9)

Note that $\phi_{00}=0$ and $\phi_{01}=\phi_{10}$. Also note that because all
the population begins and ends in the $F=3$ hyperfine manifold, the dressing
lasers can be turned on and off rapidly, without the need for adiabatic
ramping.

To implement a $\mathrm{CZ}$ gate, we design a control waveform to modulate
the microwave phase $\xi_{\mu w}(t)$. As a proof-of-principle, we choose the
waveform to be $N$ piece-wise constant pulses such that $\xi_{\mu
w}(t)=\xi_{i}$ if $t\in[i\frac{\tau}{N},(i+1)\frac{\tau}{N})$. This waveform
implements a unitary $U(\vec{\xi})=\prod_{i}U(\xi_{i})$. Our objective is to
find a $\vec{\xi}$ such that $U[\vec{\xi}]$ is locally equivalent CZ-gate
unitary. We cast this problem as a minimization problem where we are trying to
minimize the cost function $-\mathcal{F}[\vec{\xi}]$. The “Gradient Ascent
Pulse Engineering” (GRAPE) algorithm helps us efficiently calculate the
gradient $-\nabla_{\vec{\xi}}\mathcal{F}$ which greatly speeds up minimization
algorithms [43]. Using GRAPE, we can find a waveform that implements a CZ-gate
with arbitrarily close to unit fidelity as long as the gate time $\tau$ is
larger than the so-called “quantum speed limit” (QSL), denoted by $\tau_{*}$.
We seek to choose parameters such that $\tau_{*}$ is as small as possible,
given physical constraints. We will show that this is achieved by going beyond
the regime of a perfect spin-flip blockade, and choosing $J\approx\Omega_{\mu
w}$.

###  III.1 Optimal dressing strength

Figure 3:  The effects of the Rydberg-laser detuning on the entangling gate
implemented for $\Omega_{\mathrm{L}}/2\pi=10\,\mathrm{MHz}$, $\Omega_{\mu
w}/2\pi=1\,\mathrm{MHz}$. (a) The behavior of $J$ as a function of Rydberg
detuning $\Delta_{\mathrm{L}}$. $J_{\mathrm{max}}=0.29\Omega_{\mathrm{L}}$ at
resonant dressing and falls rapidly with the magnitude of the detuning. Around
$\Delta_{\mathrm{L}}/2\pi=-5.9\,\mathrm{MHz}$, we satisfy the condition
$J\approx\Omega_{\mu w}$. (b) As the magnitude of laser detuning is increased,
the effective Rabi frequencies, $\tilde{\Omega}_{\mu w}$ and
$\tilde{\Omega}^{\prime}_{\mu w}$ increase and the effective decay rates,
$\Gamma_{\tilde{a}}$ and $\Gamma_{\widetilde{aa}}$, decrease due to decreased
admixture of the Rydberg state in the dressed states
$\ket{\tilde{a}},\ket{\widetilde{aa}}.$ (c) Using optimal control, a CZ gate
waveform can be found at all laser detunings, as shown by the unit theoretical
gate fidelity orange dots. The Rydberg decay limited fidelity is shown in blue
dots, with the highest occurring around the point where $J\approx\Omega_{\mu
w}$, at $\mathcal{F}_{r}=0.9992$ for $1/\Gamma_{r}=150\,\mu s$. (d) The gate
time is also shortest around $J\approx\Omega_{\mu w}$ where $\tau\approx
1.26\,\mu\mathrm{s}$. This speedup is due to a combination of larger effective
Rabi frequencies compared to resonant dressing and the moderate blockade
speedup effect shown in A

In the absence of errors, one can reach arbitrary fidelity using optimal
control. The fundamental source of error is decoherence due to the finite
lifetime on the Rydberg state, $1/\Gamma_{r}$, which is of the order of
$100\mu s$ for typical principal quantum numbers, $n\sim 50$, including decay
stimulated by blackbody photons. Because the branching ratio for the decay of
population back to the computational subspace is small, we can approximate the
master equation simply by a non-Hermitian effective Hamiltonian
$H_{\mathrm{eff}}=H-i\frac{\Gamma_{r}}{2}\sum\ket{r}\bra{r}$, giving a
decoherence limited fidelity $\mathcal{F}_{r}$. Furthermore, we define the
time spent in the Rydberg state as $T_{r}=(T_{01}+T_{10}+T_{11})/4$, where
$T_{k}=\int_{0}^{\tau}\sum_{i=1,2}|\bra{r_{i}}\ket{\psi_{k}(t)}|^{2}dt$ and
$\ket{r_{1}},\ket{r_{2}}$ are the Rydberg states of the two atoms. We shall
call $T_{r}$ Rydberg time hereon for brevity. These are related as
$\mathcal{F}_{r}\approx 1-\Gamma_{r}T_{r}$. $T_{r}$ and $\mathcal{F}_{r}$ thus
act as figures of merit with which to compare various protocols.

In the protocol above, the detuning of the Rydberg laser affects the strength
of the entangling energy, $J$, as well as the decay rates
$\Gamma_{\mathrm{\tilde{a}}},\Gamma_{\widetilde{aa}}$ and the effective Rabi
frequencies $\tilde{\Omega}_{\mu w},\tilde{\Omega}_{\mu w}^{\prime}$. In the
context of adiabatic dressing gates, we have shown that the optimal operating
point was to tune the Rydberg laser toward resonance. Far off resonance, the
rapid decrease in $|J|$ outweighed the decrease in decoherence [40]. For this
diabatic optimal control protocol based on the spin-flip blockade, there are
additional considerations: the cost of a smaller entangling energy for larger
detunings can be offset by larger $\tilde{\Omega}_{\mu w},\tilde{\Omega}_{\mu
w}^{\prime}$ and smaller $\Gamma_{\tilde{a}},\Gamma_{\widetilde{aa}}$. Below,
we will show that if $J_{\mathrm{max}}>\Omega_{\mu w}$, there is an optimal
point and when the dressing is such that $J\sim\Omega_{\mu w}$ we get the
minimum $T_{r}$. The waveforms that minimize $T_{r}$ also generally minimize
the total gate time $\tau_{*}$. If $J_{\mathrm{max}}<\Omega_{\mu w}$, which is
the experimentally less likely scenario, resonant dressing remains optimal.

We demonstrate this optimal dressing for an example parameter set:
$\Omega_{\mathrm{L}}/2\pi=10$ MHz, $\Omega_{\mu w}/2\pi=1$ MHz and a perfect
Rydberg blockade, $V_{rr}\approx\infty$. For resonant Rydberg dressing at
$\Delta_{\mathrm{L}}=0$, the value of $|J|$ peaks at
$J_{\mathrm{max}}=(2-\sqrt{2})\Omega_{\mathrm{L}}/2\approx
0.29\Omega_{\mathrm{L}}$ [1] and the dressed state
$\ket{\tilde{a}}=(\ket{a}+\ket{r})\sqrt{2}$ has the strongest Rydberg
character leading to the weakest effective microwave Rabi frequencies,
$\tilde{\Omega}_{\mu w}=\Omega_{\mu w}/\sqrt{2},\tilde{\Omega}^{\prime}_{\mu
w}=\frac{1}{2}(1+\frac{1}{\sqrt{2}})\Omega_{\mu w}$, and largest effective
decay rates $\Gamma_{\tilde{a}}=\Gamma_{\widetilde{aa}}=\Gamma_{r}/2$. As we
increase the laser detuning, reducing the dressing strength, we reduce the
value of $J$ along with the Rydberg character in the dressed state $|\langle
r|\tilde{a}\rangle|^{2}$, thus reducing the photon scattering rate and
increasing $\tilde{\Omega}_{\mu w}$. At various dressing strengths we plot the
effective Rabi frequencies $\tilde{\Omega}_{\mu
w},\tilde{\Omega}^{\prime}_{\mu w}$, and decay rates
$\Gamma_{\tilde{a}},\Gamma_{\widetilde{aa}}$ in Fig. 3a,b.

We use GRAPE to find the QSL for each of these scenarios and plot the QSL gate
time $\tau_{*}$ and Rydberg time $T_{r}$ in Fig. 3c,d. We can see the best
operating point at $\Delta_{\mathrm{L}}/2\pi=-5.9\,\mathrm{MHz}$, where
$J=\Omega_{\mu w}=2\pi\times 1\,\mathrm{MHz}$. This corresponds to a blockade
ratio of $J/\Omega_{\mu w}=1$. We find the gate time in this scenario to be
$\tau_{*}=1.3\mu s=1.3\times 2\pi/\Omega_{\mu w}$, which is approximately
$25\%$ faster than the resonant dressing scenario. The phase waveform
$\xi_{\mu w}(t)$ of this gate shown in Fig. 4a is similar to that found in
[33] for optical control. The gate properties are given in Fig. 4b,c, which
shows plots of the population in the various levels as a function of time. For
the $\ket{11}$ state, during the central part of the of waveform, a majority
of the population is held is the doubly-excited dressed state
$\ket{\widetilde{aa}}$. This leads to faster gates.

Figure 4: Simulation of a CZ-gate implemented through microwave phase
modulation driving Rydberg-dressed hyperfine states. Parameters:
$\Omega_{\mathrm{L}}/2\pi=10\,\mathrm{MHz},\Delta_{\mathrm{L}}/2\pi=-5.9\,\mathrm{MHz},\Omega_{\mu
w}/2\pi=1\,\mathrm{MHz}$. For these dressing parameters, $J=2\pi\times
0.999\,\mathrm{MHz}\approx\Omega_{\mu w}$, corresponding to the optimal point
in Fig. 3. (a) The phase waveform of the microwave field, $\xi_{\mu w}(t)$
found using GRAPE optimization. (b) The populations in various levels during
the evolution of state when initially in the logical $\ket{01}$ state (also
the same for $\ket{10}$) when only one atom can be excited by the microwave.
(c) The population occupations during the evolution of state when initially in
the logical $\ket{11}$ state and two atoms can be excited by the microwave. We
work here beyond the regime of the perfect spin-flip blockade. In the middle
portion of the gate, most of the population gets pumped into
$\ket{\widetilde{aa}}$ where it accumulates the entangling phase due to the
nonlinear light shift, $J$.

For these parameters, the dressed states are given by
$\ket{\tilde{a}}=0.85\ket{a}+0.52\ket{r}$ and
$\ket{\widetilde{aa}}=0.81\ket{aa}+0.59\ket{ar}_{+}$. This results in the
effective Rabi rates $\tilde{\Omega}_{\mu w}/2\pi\approx 0.85\,\mathrm{MHz}$,
$\tilde{\Omega}^{\prime}_{\mu w}/2\pi\approx 0.9\,\mathrm{MHz}$. The effective
decay rates in this case are $\Gamma_{\tilde{a}}=0.27\Gamma_{r}$,
$\Gamma_{\widetilde{aa}}=0.35\Gamma_{r}$. For a Rydberg lifetime of $150\,\mu
s$, we get a decay-limited-fidelity of $99.92\%$. This is the best
$\mathcal{F}_{r}$ achievable in this protocol with our chosen Rabi frequencies
as shown in Fig. 3d, due to a combination of fast gate time, moderately large
effective Rabi frequencies and moderately small dressed decay rates.

This particular speedup in the moderate blockade regime can be understood from
the foundations of Lie-algebraic control theory. Control in the Hilbert space
occurs due to the mixing of noncommuting Hamiltonians whose affect on the
dynamics is seen in the Magnus expansion through nested commutators [44]. Here
the two noncommuting Hamiltonian are the microwave-driven Rabi oscillations
and the nonlinear light shift. The speed limit is fastest when these effects
are on the same order, $J\approx\Omega_{\mu w}$. A similar effect is seen in
the optical regime, when $V_{rr}\approx\Omega_{\mathrm{L}}$ for the optical
gates (see Appendix A), but $V_{rr}$ is not easily controlled and the doubly
excited Rydberg state often leads to other inelastic processes. In contrast,
$J$ can be accurately controlled by tuning the Rydberg laser detuning, and has
a soft-core dependence on distance [38, 1].

More generally, Fig. 5a shows a plot of the QSL as a function of the ratio of
the laser and microwave Rabi frequencies $\Omega_{\mathrm{L}}/\Omega_{\mu w}$
and laser detuning, $\Delta_{\mathrm{L}}/\Omega_{\mathrm{L}}$. The valley
running across the plot corresponds to the optimal dressing strength that
minimizes the gate time. This valley reliably corresponds to dressing
strengths at which $J=\Omega_{\mu w}$ in the regime where
$J_{\mathrm{max}}>\Omega_{\mu w}$. If $\Omega_{\mathrm{L}}<3.5\Omega_{\mu
w}\implies J_{\mathrm{max}}<\Omega_{\mu w}$, we can never achieve
$J\sim\Omega_{\mu w}$. In this scenario, resonant dressing is the optimal,
similar to [40], as seen in Fig. 5a. We look at the properties of these
optimally dressed gates for different limits of
$\Omega_{\mathrm{L}}/\Omega_{\mu w}$.

In a realistic experimental scenario, we do not have access to unlimited field
strengths. For ${}^{131}\mathrm{Cs}$ under consideration, we are typically
limited by the microwave/Raman Rabi frequency, ($\Omega_{\mu
w}<J_{\mathrm{max}}\approx 0.3\Omega_{\mathrm{L}}$). For the limiting case
$\Omega_{\mathrm{L}}\gg\Omega_{\mu w}$, the optimal dressing strength is
achieved by detuning the laser such that $J\approx\Omega_{\mu w}$. In this
case, $\Delta_{\mathrm{L}}\gg\Omega_{\mathrm{L}}$, the weak dressing regime
(wdr) where[1]

$J^{wdr}\approx\frac{\Omega_{\mathrm{L}}^{4}}{8\Delta_{\mathrm{L}}^{3}}.$ (10)

Setting $J^{\mathrm{wdr}}=\Omega_{\mu w}$, for weak dressing we find the
optimal dressing detuning,

$\Delta^{\mathrm{wdr}}_{L}=\Omega_{\mathrm{L}}^{4/3}/2\Omega_{\mu w}^{1/3}$
(11)

Also in this regime, $\tilde{\Omega}_{\mu w},\tilde{\Omega}_{\mu
w}^{\prime}\rightarrow\Omega_{\mu w}$ (up to first order in
$\Omega_{\mathrm{L}}/\Delta_{\mathrm{L}}$) and
$\Gamma_{\tilde{a}}\rightarrow\frac{\Omega_{\mathrm{L}}^{2}}{4\Delta_{\mathrm{L}}^{2}}\Gamma_{r},\Gamma_{\widetilde{aa}}\rightarrow\frac{\Omega_{\mathrm{L}}^{2}}{2\Delta_{\mathrm{L}}^{2}}\Gamma_{r}$.
By finding the CZ gate waveform for this scenario, we find that

$T_{r}^{\mathrm{wdr}}=\frac{7\Gamma_{\tilde{a}}}{\Omega_{\mu
w}\Gamma_{r}}=\frac{3.5}{\Omega_{\mathrm{L}}^{2/3}\Omega_{\mu w}^{1/3}}.$ (12)

When we are not limited by microwave Rabi frequency ($\Omega_{\mu
w}>J_{\mathrm{max}}$), and $V_{rr}\gg\Omega_{\rm L}$, the best regime of
operation is on Rydberg resonance ($\Delta_{\mathrm{L}}=0$), the strong
dressing regime (sdr) similar to [40, 19]. The gate time $\tau$ is also
minimized for this scenario. For this resonant dressing, $\tilde{\Omega}_{\mu
w}=\Omega_{\mu w}/\sqrt{2},\tilde{\Omega}_{\mu w}^{\prime}=\Omega_{\mu
w}(1+1/\sqrt{2})$ and
$\Gamma_{\tilde{a}}=\Gamma_{\widetilde{aa}}=\frac{\Gamma_{r}}{2}$. We consider
the limit of very large $\Omega_{\mu w}$, i.e., $\Omega_{\mu
w}\gg\Omega_{\mathrm{L}}$, for which the gate time asymptotically reaches
$\pi/J_{\mathrm{max}}$. Hence, we find

$T_{r}^{\mathrm{sdr}}\sim\frac{\pi}{J_{\mathrm{max}}}\frac{(2\Gamma_{\tilde{a}}+\Gamma_{\widetilde{aa}})}{4\Gamma_{r}}\approx
1.66\times\frac{\pi}{\Omega_{\mathrm{L}}}.$ (13)

Figure 5: Behavior of gate-time $\tau_{*}$ and Rydberg time $T_{r}$ as a
function of dressing. (a) Gate time as a function of
$\Omega_{\mathrm{L}}/\Omega_{\mu w}$ and
$\Delta_{\mathrm{L}}/\Omega_{\mathrm{L}}$. The narrow valley in the surface
plot corresponds to the optimal dressing strengths for a given
$\Omega_{\mathrm{L}}/\Omega_{\mu w}$. For large
$\Omega_{\mathrm{L}}/\Omega_{\mu w}$, this valley closely tracks the curve for
$\Delta_{\mathrm{L}}/\Omega_{\mathrm{L}}$ that results in $J=\Omega_{\mu w}$.
In the regime where $.29\Omega_{L}<\Omega_{\mu w}$, the optimal dressing is on
laser resonance where $J_{\mathrm{max}}=0.29\Omega_{\mathrm{L}}$. (b) Gate
times $\tau_{*}$ as a function of $\Omega_{\mathrm{L}}/\Omega_{\mu w}$. The
red line corresponds to the weak-dressing-regime bound, $1.11\times
2\pi/\Omega_{\mu w}$, and orange line corresponds to strong-dressing regime
bound, $\pi/J_{\mathrm{max}}$ (c) Rydberg times $T_{r}$ as a function of
$\Omega_{\mathrm{L}}/\Omega_{\mu w}$. The red line corresponds to the weak-
dressing-regime bound, $3.5/\Omega_{\mathrm{L}}^{2/3}\Omega_{\mu w}^{1/3}$ Eq.
(12), and the orange line corresponds to the strong-dressing-regime bound,
$1.66\pi/\Omega_{L}$ Eq. (13).

In the wdr, the time spent in the Rydberg state
$\sim\Omega_{\mathrm{L}}^{-2/3}\Omega_{\mu w}^{-1/3}$ is generally longer than
that achievable by an optically driven gate at the same UV Rabi frequency
($\sim\Omega_{\mathrm{L}}^{-1}$). Currently, the state-of-the-art gate
fidelities are far from the limit set by the Rydberg lifetime, owing to
experimental noise. Hence, protecting against the inhomogeneous noise sources
is more important in the near future. In the following section, we explain how
the major sources of noise in our gate boil down to uncertainty in $J$ and how
this can be mitigated.

##  IV Robust control

The fidelity of entangling gates is fundamentally limited by the finite
lifetime of the Rydberg state. Another important source of error is thermal
fluctuations in atomic motion. This leads to inhomogeneities in the atomic
positions and momenta which affect their coupling to external fields and the
atom-atom interactions. Deleterious effects include Doppler shifts and
uncertainty in the Rabi frequencies due to position uncertainty of an atom
across a tightly focused beam profile. The microwave-driven gates have some
intrinsic robustness to such noise, and using the tools of robust optimal
control one can further reduce the residual effects of inhomogeneities.

To study this we include atomic motion with the following Hamiltonian
associated with two atoms that can be excited to the Rydberg state [39],

$\displaystyle H$ $\displaystyle=H_{0}+H_{\mathrm{rel}}+H_{\mathrm{com}}.$
(14) $\displaystyle H_{0}$ $\displaystyle=-\Delta\outerproduct{B}{B}-(2\Delta-
V_{rr})\outerproduct{rr}{rr}$
$\displaystyle+\frac{\Omega^{+}_{L}}{2}(\outerproduct{11}{B}+\outerproduct{B}{rr}+\mathrm{h.c.}).$
(15) $\displaystyle H_{\mathrm{rel}}$
$\displaystyle=-\Delta\outerproduct{D}{D}+\frac{\Omega^{-}_{L}}{2}(\outerproduct{11}{D}+\outerproduct{D}{rr}+\mathrm{h.c.})$
$\displaystyle-\frac{k_{L}p_{\mathrm{rel}}}{m}(\outerproduct{B}{D}+\outerproduct{D}{B}).$
$\displaystyle H_{\mathrm{com}}$
$\displaystyle=\frac{k_{L}P_{\mathrm{com}}}{2m}(\outerproduct{D}{D}+\outerproduct{B}{B}+2\outerproduct{rr}{rr}).$
(16)

Here $H_{0}$ is the ideal Hamiltonian, where the ground logical state
$\ket{11}$ is coupled to the singly-excited “bright state,”

$\ket{B}=(\ket{ra}+\ket{ar})/\sqrt{2}$ (17)

via Rabi frequency

$\Omega_{\mathrm{L}}^{+}\equiv(\Omega_{\mathrm{L}1}+\Omega_{\mathrm{L}2})/\sqrt{2}.$
(18)

where $\Omega_{\mathrm{L}1},\Omega_{\mathrm{L}2}$ are the Rydberg Rabi
frequencies of the two atoms.

Due to unequal Rabi frequencies, the so-called ”dark state” participates in
the dynamics

$\ket{D}=(\ket{ra}-\ket{ar})/\sqrt{2}.$ (19)

which couples to $\ket{11}$ with Rabi frequency

$\Omega_{\mathrm{L}}^{-}\equiv(\Omega_{\mathrm{L}1}-\Omega_{\mathrm{L}2})/\sqrt{2}.$
(20)

In addition, the bright and dark states are coupled to one another due to the
relative momentum ($p_{\mathrm{rel}}=(p_{1}-p_{2})/2$) as this affects the
relative phase of the laser seen by the two atoms. Here, $k_{L}$ is the laser
wavenumber and $m$ is the mass of an atom and $p_{1}$ and $p_{2}$ are the
momenta of the two atoms in the direction of laser propagation.

Finally, in $H_{\mathrm{com}}$, thermal fluctuations lead to a random Doppler
shift due to the center of mass momentum ($P_{\mathrm{com}}=p_{1}+p_{2}$) of
the two atoms. We note that for single-atom excitations driven from the
logical states $\ket{01}$ and $\ket{10}$, we will have a Doppler shifts and
potential uncertainty laser Rabi frequency. These effects are not included in
the Hamiltonian above as they play a minor role in gate infidelity.

Figure 6: Effects of Rydberg inhomogeneities and mitigating them via robust
control methods. (a) Level diagram for two-atom dressing in the presence of
atomic motion and imperfect Rabi frequencies. The asymmetric dressing of the
two atoms results in unequal Rydberg-laser Rabi frequencies,
$\Omega_{\mathrm{L}1},\Omega_{\mathrm{L}2}$, and coupling from $\ket{aa}$ to
both the bright state $|B\rangle=|ar\rangle_{+}$ and the dark state
$|D\rangle=|ar\rangle_{-}$, with Rabi frequencies
$\Omega_{\mathrm{L}}^{+}=\frac{\Omega_{\mathrm{L}1}+\Omega_{\mathrm{L}2}}{\sqrt{2}}$
and
$\Omega_{\mathrm{L}}^{-}=\frac{\Omega_{\mathrm{L}1}-\Omega_{\mathrm{L}2}}{\sqrt{2}}$
respectively. The center-of-mass motion results in a Doppler-shifted detuning
of $\frac{k_{L}P_{cm}}{2m}$ per atom in the Rydberg state. The relative
momentum results in a coupling between the bright and dark states with Rabi
frequency $\frac{k_{L}P_{rel}}{m}$. Because the dressed state is an eigenstate
of the total Hamiltonian in this subspace, the net effect of all motional
coupling is uncertainty in the light-shifts and the dressed microwave-Rabi
frequencies. The dominant effect is the uncertainty in the entangling energy,
$J$. (b) Sensitivity to uncertainty in $\Omega_{\mathrm{L}1}$ for the optimal
gate (orange) Fig. 4 and the robust gate (blue) found by using robust optimal
control. The dashed correspond to the ideal fidelity as function of
inhomogeneity, assuming unitary evolution, while the solid lines include
decoherence to to Rydberg decay. We choose
$\Omega_{\mathrm{L}1}/\Omega_{L}=\Omega_{\mathrm{L}2}/\Omega_{\mathrm{L}}$
($\Omega_{\mathrm{L}}$ is the ideal Rabi frequency). The robust waveform
achieves significant resilience to uncertainty in $\Omega_{\mathrm{L}}$ and
the price of longer gate times. This implies a trade-off with Rydberg decay.
The regime in which roust control is useful is seen for large inhomogeneities,
where the solid blue cure is above the organ curve. (c) Sensitivity to
uncertainty in $\Delta_{\mathrm{L}1}$, with $\Delta_{\mathrm{L}2}=0$. The
fidelity as a function of $\Delta_{\mathrm{L}1}/\Delta_{\mathrm{L}}$ using the
same waveform that optimizes robustness against inhomogeneity in
$\Omega_{\mathrm{L}1}$ shows significant robustness. Typically, a gate
optimized to be robust against one noise source has increased sensitivity
against other noises. In this case however, since both noises are effectively
the same uncertainity in the dressed state $\ket{\widetilde{aa}}$, we are able
to achieve robustness against both major noises.

The net effect of thermal motion on dressing and the microwave-driven gates is
thus as follows. Firstly, Doppler shifts at microwave frequencies are
negligible due to a much smaller microwave wavenumber $k_{\mu
w}=2\pi/\lambda_{\mu w}$. The major source of Doppler noise is the dressing
laser itself, due to a wavenumber $k_{L}=2\pi/\lambda_{\mathrm{L}}$ which
$10^{5}$ times larger. This affects the system in two ways: (1) the dressed
states themselves are modified; (2) the light shifts and associated entangling
energies are modified. The dressed states $\ket{0\tilde{a}}$,
$\ket{\tilde{a}0}$, and $\ket{1\tilde{a}}_{+}$ receive a one-atom light shift,
only slightly modified due to the Doppler shift when the excitation to the
Rydberg state is sufficiently far from resonance, $\Delta_{\mathrm{L}}\gg
P_{\mathrm{com}}k_{L}/(2m)$. The dressed state $\ket{\widetilde{aa}}$ receives
a two-atom light shift. This arises as $\ket{aa}$ is coupled to the bright and
dark states, $\ket{B}$ and $\ket{D}$, which are themselves coupled together
through the relative motion as shown in Fig. 6a; here we assume a perfect
Rydberg blockade. This leads to a slightly different dressed state
$\ket{\widetilde{aa}}$ as compared to the ideal case, which is now an
admixture of $\ket{aa}$, $\ket{B}$, and $\ket{D}$ with amplitudes that depend
on the random motion of the particles. The strength of the entangling
interaction $J$ is modified as well. The changes in the dressed state changes
the dressed Rabi frequencies $\tilde{\Omega}_{\mu w},\tilde{\Omega}_{\mu
w}^{\prime}$ but this change is much weaker than the change in $J$, as seen in
Fig. 3. The dominant noise effect due to thermal motion is thus the
uncertainty in the value of $J$.

Given how the dressed states are modified by inhomogeneities, we can determine
its effect on gate fidelity. As discussed in Sec. II, there are two possible
regimes of operation. Under the condition of a strong spin-flip blockade, when
$J\gg\tilde{\Omega}^{\prime}_{\mu w}$, the actual value of $J$ is unimportant,
and its small uncertainty is negligible. The residual effect of
inhomogeneities is on the one-atom light shifts. This, again, is negligible
when the laser detuning is large compared to the Doppler width. In this
regime, the microwave-drive dressed gates are intrinsically robust to small
thermal fluctuations. Note, however, that this is not necessarily the regime
of the highest fidelity due to the finite Rydberg lifetime. In Sec. II we
found that the fastest gates with the smallest integrated Rydberg time was
when $J\approx\tilde{\Omega}^{\prime}_{\mu w}$. In that regime, we must
consider the uncertainty in $J$. To do so we can employ the methods of “robust
optimal control.”

We consider the method of inhomogeneous control originally introduced by
Khaneja [45, 46]. The key idea is to optimize the average fidelity
$\bar{\mathcal{F}}=-\int
d\vec{\lambda}P(\vec{\lambda})\mathcal{F}(\vec{\lambda})$, where
$\mathcal{F}(\vec{\lambda})$ is the fidelity at a fixed set of the uncertain
parameters, and $P(\vec{\lambda})$ is their probability distribution. We again
use GRAPE to optimize the modified cost function
$\bar{\mathcal{F}}=\\{2\mathcal{F}(\Omega_{1}=\Omega_{2}=\Omega_{\mathrm{L}})+\mathcal{F}(\Omega_{1}=\Omega_{2}=1.02\Omega_{\mathrm{L}})+\mathcal{F}(\Omega_{1}=\Omega_{2}=0.98\Omega_{\mathrm{L}})\\}/4$
where $\Omega_{\mathrm{L}}$ is the ideal Rydberg Rabi frequency. We are
effectively optimizing the fidelity in a $\pm 2\%$ window of the ideal Rabi
frequency, but with a greater weight at the ideal parameters. We expect to get
better robustness as we increase the gate time, but this generally worsens the
sensitivity against other noise. In optical control protocols, waveforms that
are robust control against both Rabi frequency uncertainty and Doppler noise
require even longer gate times, as shown in [36, 37]. In the dressed-state
case we find that pulses that are robust against Rabi frequency noise are also
robust against Doppler detuning noise, as shown in Fig. 6b,c. We attribute
this to the fact that effectively, both noises manifest as uncertainty in $J$
and we are actually engineering robustness against this one quantity which
manifests as reduced sensitivity to uncertainty in both
$\Omega_{\mathrm{L}},\Delta_{\mathrm{L}}$.

##  V Summary and Outlook

We proposed a new method to implement entangling gates between neutral atoms
based on the spin-flip blockade. This has the potential advantage that all
coherent control can be performed robustly with microwave fields. We showed
for the specific case of alkali atoms, such as cesium, by optically dressing
an auxiliary hyperfine state with Rydberg character, the level structure
associated with the optically-excited Rydberg blockade can be mapped to the
microwave-excited spin-flip blockade. In this way any protocol that could be
implemented with optical control can be mapped into the microwave regime.

We studied the specific example of the spin-flip version of the generalized
Levine-Pichler gate, implemented through modulation of the microwave phase,
whose waveform is found by quantum optimal control. In contrast to the optical
regime, where the doubly-excited Rydberg potential $V_{rr}$ is not well
defined at a short interatomic distances, in the microwave regime we consider
going beyond the perfect blockade regime. This is profitable as the doubly
spin-excited state is well defined, and the blockade strength is a much more
stable function of interatomic motion due to its soft-core nature. We find
that the optimal quantum speed limit occurs when the entangling energy $J$ is
approximately equal to the driving microwave Rabi frequency $\Omega_{\mu w}$.
As $J$ is controllable by laser detuning, we find that for a laser Rabi
frequency $\Omega_{L}\gg\Omega_{\mu w}$, we can achieve high fidelity in the
off-resonance regime. For $\Omega_{\mu w}/2\pi=1$ MHz, we can achieve a
fidelity $>99.9\%$, limited solely by the finite lifetime of the Rydberg
state. While higher fidelity is in principle possible with optically
controlled gates, in practice, the fidelity is far from the theoretical
maximum, limited otherwise by technical noise sources. The robustness of
microwave control may allow us to reach closer to the fundamental limits.

Of particular importance is noise arising from atomic thermal motion which
gives rise to Doppler shifts and breaking of addressing symmetry of the ideal
protocol. The laser detuning and amplitude are then uncertain. We show that
the dominant net effect of thermal motion boils down to uncertainty in one
parameter, the entangling energy, $J$. This allows us to use the method of
inhomogeneous control to engineer robustness against both laser amplitude and
phase noise simultaneously, with only a modest increase in gate time.

These conclusions extend to other forms of noise such as fluctuations in laser
intensity (Rabi frequency), phase (detuning) and uncontrolled electric fields
that lead to random Stark shifts (detuning). Such noise leads to inhomogeneous
broadening on the timescale $T_{2}^{*}$. When $T_{2}^{*}\ll 1/\Gamma_{r}$,
off-resonance dressing, $\Delta_{L}\gg 1/T_{2}^{*}$, enables higher fidelity
gates so long as there is sufficient laser power that we are not too far off
resonance, e.g., $\Delta_{L}\sim\Omega_{L}$, and $J$ remains large compared
the dressed-state decay rate.

While we have focused here in the microwave version of the generalized LP-
gate, in principle any protocol developed in the optical regime can be mapped
into the microwave regime. For example, the recently studied protocol
involving adiabatic rapid passage beyond the perfect blockade [47] can be
mapped to a microwave/Raman adiabatic passage. Such gates might achieve even
higher fidelity as they can take full advantage of the entangling energy $J$.
Overall, the scheme proposed here should enable exploration of different
tradeoffs in speed, noise, and robustness in order to obtain the highest
fidelity entangling gates.

##  VI Acknowledgements

We thank Matt Chow, Bethany Little, Anupam Mitra, Sven Jandura and Bharath
Hebbe Madhusudhana for helpful discussions. This work is supported by Sandia
Laboratory Directed Research and Development (LDRD) and Quantum Systems
through Entangled Science and Engineering (Q-SEnSE) funded by National Science
Foundation Quantum Leap Challenge Institute (NSF QLCI). Sandia National
Laboratories is a multimission laboratory managed and operated by National
Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary
of Honeywell International Inc., for the U.S. Department of Energy’s National
Nuclear Security Administration under contract DENA0003525. This paper
describes objective technical results and analysis. Any subjective views or
opinions that might be expressed in the paper do not necessarily represent the
views of the U.S. Department of Energy or the United States Government.

*

## Appendix A

Figure 7: Entangling gate time vs Rydberg blockade strength
$V_{rr}/\Omega_{\mathrm{L}}$ for optical control gates. The waveforms for the
gates are calculated using GRAPE. For strong blockade, the gate time
approaches $\tau\Omega_{\mathrm{L}}=7.6$. For a small blockade ratio, the gate
time approaches $\tau\Omega_{\mathrm{L}}=\pi/V_{rr}$. At the sweet spot of
moderate blockade strength $V_{rr}/\Omega_{\mathrm{L}}\approx 1$, the gate
time is $7/\Omega_{\mathrm{L}}$. The three colors correspond to three
different classes of waveforms, differentiated by their shapes[48].

In our studies of the microwave-driven gates on dressed states, we have seen
that optimal gates can be designed when $J/\Omega_{\mu w}\approx 1$. This maps
to $V_{rr}/\Omega_{r}\approx 1$ in the optical regime. While this moderate
blockade optical protocol is not experimentally common due to the difficulty
in controlling $V_{rr}$, analyzing it further provides valuable insight into
our gate protocol. In Fig. 7 we show the QSL gate time as a function of the
blockade ratio for the optical protocols at various
$V_{rr}/\Omega_{\mathrm{L}}$. We use GRAPE to find $\xi_{L}(t)$ laser-phase
waveforms that implement $\mathrm{CZ}$ gates similar to [33, 34] at various
$V_{rr}/\Omega_{\mathrm{L}}$. Perhaps surprisingly, we see that a blockade
ratio as low as $V_{rr}/\Omega_{\mathrm{L}}\approx 1$ doesn’t make the gate
much slower; In fact, the gates are slightly faster at this moderate blockade
ratio. At large blockade ratios, the state $\ket{rr}$ is essentially absent
from the dynamics which explains the saturation of the gate time at
$7.6/\Omega_{\mathrm{L}}$ and these gates were studied in [33, 34]. On the
other end, for very weak blockade ratios, the entangling energy is simply too
small, and a larger gate time is needed to compensate. At very small
$V_{rr}/\Omega_{\mathrm{L}}$, the gates approach a protocol similar to [39]
where the populations are driven from computational states $\ket{01},\ket{11}$
into the Rydberg states $\ket{0r},\ket{1r}_{+}$ and kept there until they
accumulate the right amount of dynamical phases,
$\phi_{11}-2\phi_{01}\approx(E_{rr}-2E_{r})\tau=\pi$. The gate time
asymptotically approaches $\pi/V_{rr}$ in this limit (similar to the weak-
blockade limit in [47]). Near the optimal point of
$V_{rr}\approx\Omega_{\mathrm{L}}$, the entirety of the nonlinear phase
accumulated, $\phi_{11}-2\phi_{01}$ is geometrical in nature, with no
contribution from dynamical phases. This explains the fast gate time, as these
geometrical gates tend to follow the geodesic paths in the Hilbert space,
resulting in optimal gate times[49]. This counter-intuitive result explains
why choosing a dressing regime where $J\approx\Omega_{\mu w}$ would give us
the optimal gate times, which is borne out by the results in Fig. 3c.

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