# Nuclear mass predictions using machine learning models

Esra Yüksel<EMAIL_ADDRESS>Department of Physics, University of
Surrey, Guildford, Surrey GU2 7XH, United Kingdom  Derya Soydaner
<EMAIL_ADDRESS>Department of Brain and Cognition, University of
Leuven (KU Leuven), Leuven, Belgium  Hüseyin Bahtiyar
<EMAIL_ADDRESS>Y. Tahsinbey St. 8/4, 34841, Maltepe,Istanbul

(August 28, 2024)

###### Abstract

The exploration of nuclear mass or binding energy, a fundamental property of
atomic nuclei, remains at the forefront of nuclear physics research due to
limitations in experimental studies and uncertainties in model calculations,
particularly when moving away from the stability line. In this work, we employ
two machine learning (ML) models, Gaussian Process Regression (GPR) and
Support Vector Regression (SVR), to assess their performance in predicting
nuclear mass excesses using available experimental data and a physics-based
feature space. We also examine the extrapolation capabilities of these models
using newly measured nuclei from AME2020 and by extending our calculations
beyond the training and test set regions. Our results indicate that both GPR
and SVR models perform quite well within the training and test regions when
informed with a physics-based feature space. Furthermore, these ML models
demonstrate the ability to make reasonable predictions away from the available
experimental data, offering results comparable to the model calculations.
Through further refinement, these models can be used as reliable and efficient
ML tools for studying nuclear properties in the future.

Keywords: Nuclear mass, machine learning, regression, explainable AI

##  I Introduction

The atomic nucleus, a strongly correlated many-body system, is characterized
by its proton (Z) and neutron (N) numbers. Mass is a fundamental property of
atomic nuclei, playing a crucial role in our understanding of strong nuclear
interactions and a vital role in nuclear astrophysical calculations, such as
r-process simulations, where they serve as inputs [1, 2]. Experimentally, it
is possible to study nuclei near the stability lines, and accurate nuclear
data for the masses of nuclei are available [3, 4]. However, despite
significant advancements in nuclear facilities, measurements on the neutron-
rich side of the nuclear chart remain unfeasible in the near future, and the
exploration of the majority of nuclei involved in the r-process has yet to be
undertaken. For instance, the neutron drip line, which indicates the position
of the last bound nucleus, has been confirmed only up to Z = 10 [5], and the
boundaries of the nuclear landscape are not known experimentally. Therefore,
our understanding of nuclear properties away from the stability line and the
limits of the nuclear landscape relies heavily on theoretical calculations.

Up to now, several theoretical models have been used to investigate nuclear
properties and determine the location of drip lines. Within this framework,
macroscopic-microscopic (mic-mac) global nuclear mass models, such as
Weizsäcker-Skyrme Nuclear Mass Tables (WS4) [6] and Finite-Range Droplet Model
(FRDM-2012) [7], have demonstrated considerable success and been extensively
used in $r-$process calculations over the years. However, despite the success
of the mic-mac models in fitting experimental data on the measured masses of
nuclei, the root mean square (rms) deviations with respect to the known
experimental data are not at the desired level. The rms deviation, in relation
to the available mass data, was found to be 0.298 MeV when using the WS4 model
[6] and 0.662 MeV using the FRDM(2012) model [7]. Furthermore, these models
have been fitted using the available experimental data, which makes the
behavior of nuclei on the neutron-rich side of the nuclear chart still
somewhat uncertain.

More sophisticated methods, such as self-consistent mean-field (SCMF) theories
based on the Hartree-Fock-Bogolyubov (HFB) approach with nuclear energy
density functionals (EDF), have also long been employed to investigate the
properties of nuclei. Although using microscopic tools in calculations is a
computationally demanding task, large-scale computations of the nuclear chart
are nonetheless available. In recent years, both relativistic and non-
relativistic calculations have been performed using different EDFs to probe
the properties of nuclei and to define the boundaries of the nuclear landscape
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. While these models perform well around
the stability line with respect to the available experimental data, they
reveal local variations and significant discrepancies that increase with
neutron number, ultimately impacting the location of the drip lines. The major
source of these differences is the missing incomplete correlations on the
purely mean-field level of the HFB description, the usage of different
interactions that are optimized using different strategies, pairing
correlations, and the impact of the continuum. Typically, the rms deviations
of these mass tables compared to the available experimental data are quite
high and range between 2.0 and 5.0 MeV, depending on the interaction used in
the calculations. One of the most recent functionals, which are optimized
using the experimental data of all available nuclei, has reached an rms
deviation between 0.5 and 0.6 MeV [18, 19, 20, 21, 14]. Considering all of
these factors, there is a need to find fast and reliable methods for
determining nuclear properties, especially away from the stability line.

In recent years, machine learning (ML) models have gained considerable
attention within the scientific community, demonstrating notable success,
including also the field of nuclear physics (see Ref. [22] and references
therein). These models have proven capable of directly predicting nuclear
properties using experimental data [23, 24, 25, 26, 27, 28, 29, 30, 31, 28].
Recently, the ML models have also been used to improve mic-mac and microscopic
model predictions. Within this framework, the most popular tool is Bayesian
neural networks (BNNs), which have been used to improve the results of
microscopic calculations by training on the residuals. These residuals
represent the differences between experimental data and microscopic
calculations, and the BNNs have gained considerable attention and success in
that respect [32, 33, 34, 35]. In this context, ML models can be used as
reliable and efficient tools to probe nuclear properties; however, more
studies are necessary to better understand their predictive capability.

In this study, our goal is to assess the performance of the two ML models in
predicting the nuclear mass excess ($M$) of nuclei, rather than correcting
existing mic-mac or microscopic model predictions. We use the Gaussian Process
Regression (GPR) and Support Vector Regression (SVR) ML models to calculate
the mass excess of nuclei. These models are trained using available
experimental data along with the relevant physics-based feature space. Then,
we evaluate the performance of these ML models in predicting the mass excess
of nuclei, examining also their extrapolation capabilities far beyond the
training and test regions.

##  II Machine Learning models

In this section, we present an overview of the ML models employed in our
calculations: SVR and GPR. Additionally, we describe the experimental data
used to train these models and provide details about the physics-based feature
space involved.

###  II.1 Support Vector Regression

The SVR [36] is an ML model specifically designed for tackling regression
tasks, offering a unique approach to predict continuous outcomes by leveraging
the principles of support vector machines (SVMs) [37]. In contrast to
classification-focused methods, SVR seeks a hyperplane that optimally fits the
training data with minimized error. Central to SVR is the concept of _support
vectors_ , which are critical data points closest to the hyperplane’s
boundaries. The model aims to align as many data points as possible within the
optimal hyperplane, fitting within a specified tolerance margin. It
simultaneously controls margin violations, addressing instances where data
points exceed the boundaries. This brings a hyperparameter $\epsilon$, which
controls the width of the hyperplane [38].

The strength of SVR is particularly notable in addressing non-linear
regression problems, often yielding enhanced results [39]. At the core of
SVR’s approach to these problems is the effective _kernel trick_. This
technique is crucial when dealing with input data that is not linearly
separable in its original feature space. By employing the kernel trick, SVR
can implicitly project the data into a higher-dimensional space, achieving
linear separability. This projection is facilitated by a kernel function,
which efficiently calculates the dot product of data point pairs in this
higher-dimensional space without the need for explicit calculation of
transformed features. Thus, SVR involves mapping input data into a higher-
dimensional space using kernel functions, allowing for the capture of
nonlinear relationships. Among various kernel functions, the Radial Basis
Function (RBF) kernel is a widely used choice in SVR applications defined as:

$\displaystyle K_{G}(x,x^{\prime})$
$\displaystyle=\exp\left(-\frac{||x-x^{\prime}||^{2}}{2\sigma^{2}}\right)$ (1)
$\displaystyle=\exp\left(-\gamma||x-x^{\prime}||^{2}\right)$

In this context, $x$ and $x^{\prime}$ represent two data points, and their
Euclidean distance is denoted as $||x-x^{\prime}||$, while $\gamma$ is the
kernel coefficient. Eq. 1 quantifies the similarity or dissimilarity between
these data points, based on their distance in the input feature space. This
results in higher similarity for closer data points, and conversely, lower
similarity for those more distant [38, 40]. The tuning of hyperparameters
plays a crucial role in SVR. Proper parameter adjustment is important to
prevent overfitting or underfitting, ensuring the model generalizes well to
unseen data. The regularization hyperparameter, denoted as $C$, is essential
in striking the balance between maximizing the margin and minimizing the
training error. Additionally, the $\epsilon$ hyperparameter is important in
determining the tolerance margin, within which the epsilon-insensitive loss
function does not penalize errors. Data points within this margin do not
contribute to the loss function, enhancing the model’s robustness against
minor prediction errors and improving its resilience to outliers. In our
experiments, we set the $\epsilon$ to 0.002, the $C$ to 1000, and the $\gamma$
to 0.03. Another hyperparameter, the _tolerance_ , which indicates the desired
precision for convergence, is set to 1e-5.

SVR offers a versatile framework for regression tasks, utilizing kernels to
capture diverse relationships and incorporating a margin of tolerance to
enhance robustness. Practical hyperparameter tuning and understanding the role
of kernels are fundamental for optimizing the model performance across various
datasets and maximizing its efficiency in real-world applications.

###  II.2 Gaussian Process Regression

When we consider a linear model expressed as $y=w^{T}x$, this model describes
a linear relationship for every different value of $w$. If we introduce a
prior distribution for $w$, denoted as $p(w)$, the distribution of possible
$y$ values at any given $x$, $y(x|w)$, emerges from sampling $w$ from $p(w)$.
This is the main idea of a _Gaussian process_. When $p(w)$ follows a Gaussian
distribution, each resulting $y$ is also Gaussian, being a linear combination
of Gaussians. Specifically, our interest lies in the joint Gaussian
distribution of $y$ values computed at $N$ input data points $x^{t}$, where
$t=1,\dots,N$ [41, 42]. We typically assume a Gaussian prior with zero-mean
for $w$, as shown in Eq.2:

$p(w)\sim\mathcal{N}(0,(1/\alpha)I)$ (2)

GPR [41, 43] operates by leveraging Gaussian processes to model distributions
over functions. Initially, the algorithm establishes a prior distribution over
functions, assuming Gaussian-distributed function values at input points. This
prior distribution forms the foundation, characterized by a mean function and
a kernel function. As training data is observed, this prior is updated to a
posterior distribution using Bayes’ theorem. This update incorporates the
observed data, refining the model’s beliefs about the underlying function. The
resulting posterior distribution enables predictions at new data points,
providing not only a mean prediction but also an associated measure of
uncertainty, which is crucial for decision-making in uncertain scenarios.

The choice of kernels is important in GPR. In our work, we utilize a
combination of the RBF kernel (Eq.1) and the White kernel (Eq.3). The RBF
kernel is particularly effective at capturing intricate data patterns,
adapting to various scales, and ensuring smooth connections between data
points. Meanwhile, the White kernel models noise within the dataset. Adjusting
the noise level hyperparameter within the White Kernel is essential, striking
a balance between capturing the underlying signal and accommodating inherent
noise. This delicate interplay between kernels enables our GPR model to
provide robust predictions, while acknowledging and quantifying uncertainties.
To optimize the model, we adjust the kernel parameters by spanning a range of
values to obtain the optimum values for our calculations. In our experiments,
the length scale of the RBF kernel is set to 1.0, with its lower and upper
bounds on the length scale being (1e-4, 1e5). For the White Kernel, the noise
level is set at 1, with the noise level’s lower and upper bounds set at
(1e-10, 1e1).

$K_{W}(x,x^{\prime})=\begin{cases}\text{noise level}&\text{if
}x_{i}==x^{\prime}_{j}\\\ 0&\text{otherwise}\end{cases}$ (3)

GPR’s strength lies not only in its predictive accuracy but also in its
ability to provide nuanced insights into the reliability of those predictions.
This is achieved through its uncertainty quantification, which offers a
probabilistic measure of confidence in its predictions. By effectively
quantifying the uncertainty associated with each prediction, GPR enhances
decision-making processes in various domains. This dual capability of
delivering precise predictions while simultaneously assessing their
reliability makes GPR a valuable tool across a wide range of applications.

Figure 1: The training set (gray circles), test set (red circles), and
extrapolation set (blue circles) used in the ML models. Both the training and
test sets include nuclei from AME2020, with the exception of the newly
measured 71 nuclei from AME2020, which are exclusively designated for the
extrapolation set [4].

###  II.3 The experimental data and feature space

In this study, our objective is to develop a ML model that predicts the mass
excess of atomic nuclei using both experimental data on the mass excess of
nuclei and a physics-based feature space. The experimental mass excess values
are taken from the atomic mass evaluation 2020 (AME2020) [4] for nuclei with
$Z,N\geq 8$ (2386 nuclei). Then, the experimental data is randomly divided
into two subsets: $75.0\%$ (1789 nuclei) for training and $25.0\%$ (597
nuclei) for testing. The nuclei in the training and test sets remain the same
for all calculations. The performance of the ML models is also assessed beyond
the training and test data sets. The AME2020 data includes new experimental
information for 71 nuclei compared to the previous AME2016 [3]. These nuclei
have been utilized to test the extrapolation capabilities of the models. The
estimated mass excess values, derived from the trends in the mass surface
(TMS) of nuclei, are also utilized to compare our findings in the
extrapolation region. The selection of the training and test sets, as well as
the new data from AME2020 (extrapolation set), is shown in Figure 1.
Additionally, we evaluate the extrapolation performance of the models by
extending calculations to the neutron-rich region beyond the reach of current
experimental facilities, probing the limits of their predictive capabilities.

Figure 2: The absolute value of mass excess differences between the GPR and SVR predictions for training and test set using different features (see Table 1) and the AME2020 data [4]. Table 1: The ML models with different features. Model | Feature Space
---|---
SVR/GPR-5 |  $Z$, $N$, $A$, $A^{2/3}$, $(N-Z)/A$
SVR/GPR-8 |  $Z$, $N$, $A$, $A^{2/3}$, $(N-Z)/A$,
|  $\nu_{Z}$, $\nu_{N}$, $PF$
SVR/GPR-10 |  $Z$, $N$, $A$, $A^{2/3}$, $(N-Z)/A$,
|  $\nu_{Z}$, $\nu_{N}$, $PF$, $Z_{eo}$, $N_{eo}$
SVR/GPR-12 |  $Z$, $N$, $A$, $A^{2/3}$, $(N-Z)/A$,
|  $\nu_{Z}$, $\nu_{N}$, $PF$, $Z_{eo}$, $N_{eo}$, $Z_{shell}$, $N_{shell}$

As it is well known, the use of appropriate inputs during training can
significantly impact the performance of ML models [28, 30, 31, 26, 27].
Therefore, in our models, we incorporate relevant features of nuclei that can
influence mass predictions. Our feature space consists of 12 inputs: $Z$, $N$,
$A$, $A^{2/3}$, $(N-Z)/A$, $Z_{eo}$, $N_{eo}$, $\nu_{Z}$, $\nu_{N}$, $PF$,
$Z_{shell}$, and $N_{shell}$. Here, the bulk properties are defined as the
proton (neutron) number $Z$ ($N$), the mass number ($A$), and $A^{2/3}$ for
volume and surface terms. The term $\frac{N-Z}{A}$ is a measure of isospin
asymmetry. The odd-even nature of protons ($Z_{eo}$) and neutrons ($N_{eo}$)
is defined as follows: $Z_{eo}$ ($N_{eo}$) equals zero when $Z$ ($N$) is even
and one when $Z$ ($N$) is odd. We also provide information about the nuclear
magic numbers: $\nu_{Z}$ and $\nu_{N}$ represent the valence number of protons
and neutrons measured from the nearest closed shell. The nuclear magic numbers
for protons and neutrons are taken as $Z(N)=8,20,28,50,82,126,184$. The
promiscuity factor ($PF$) is represented by the formula
$PF=\frac{\nu_{Z}\cdot\nu_{N}}{\nu_{Z}+\nu_{N}}$, and serves as a measure of
valence proton-neutron $(p-n)$ interactions [44]. Lastly, the system is
informed about the nuclear shells with $Z_{\text{shell}}$ and
$N_{\text{shell}}$; they represent the shell model orbitals of the last proton
and neutron. The values of $Z_{\text{shell}}$ and $N_{\text{shell}}$ are
defined as 0, 1, 2, 3, or 4, depending on whether the proton or neutron number
falls within the specified ranges: 1–28, 29–50, 51–82, 83–126, and above 127,
respectively [45]. In order to assess the importance of the feature space in
model calculations, we implement ML models with different features. The inputs
used in our ML models are given in Table 1.

##  III Results

Figure 2 displays the absolute differences between the results of GPR (upper
panels) and SVR (lower panels) with different inputs and the experimental data
taken from AME2020 [4]. The feature space of the ML models is provided in
Table 1. The rms deviations for the training and test sets of each selected
model are also presented in Figure 2. Using only the bulk properties of nuclei
to construct the model, GPR-5 yields reasonable results, with rms errors of
0.91 and 1.08 MeV for the training and test sets, respectively, better than
most of the microscopic model calculations. On the other hand, the performance
of SVR-5 is lower compared to GPR-5, with rms errors of 2.40 and 2.55 MeV for
the training and test sets, respectively.

In Figure 2, it is evident that increasing the physics-based feature space
significantly improves the performance of the models. The importance of the
physics-based feature space has also been discussed in previous studies, with
similar results obtained using different ML models [26, 31, 30, 28]. It has
been noticed that the inclusion of the odd-even nature of protons and neutrons
($Z_{eo}$, $N_{eo}$) leads to a significant improvement in ML predictions.
Subsequently, the results improve further with the inclusion of information on
the nuclear shells, $Z_{\text{shell}}$ and $N_{\text{shell}}$. Utilizing 12
inputs (GPR-12) in the calculations, we achieved an rms deviation of 0.14 and
0.26 MeV for the training and test sets, respectively. These rms deviation
values are even better to well-known mic-mac mass models, suggesting that the
GPR model effectively captures the given information of nuclei and makes
reasonable predictions. Additionally, we observe that GPR performs better for
medium-heavy and heavy nuclei, while deviations are slightly higher for light
nuclei. The poorer performance in light nuclei is attributed to the lower
number of available experimental data in this region. Similar results are also
obtained using the SVR model. However, we find that SVR requires more data and
information to make reasonable predictions for training and test set nuclei,
and GPR outperforms SVR in that respect.

We also compare our findings with previous ML studies in which different ML
models have been used to predict nuclear mass excess. A recent application of
the probabilistic ML algorithm, the Mixture Density Network (MDN), has yielded
rms deviations of around 0.5-0.6 MeV with respect to the AME2016 [30] when
supplemented with physics-based feature space. Additionally, one of the first
applications of ML models in nuclear physics was performed using SVMs [24],
predicting nuclear mass excess long ago. It yielded rms deviations of 0.35,
0.5, and 0.71 MeV for the training, validation, and test sets, respectively.
Our findings, using two different ML models, not only align with these
previous studies but also demonstrate an improved performance, establishing
GPR and SVR as alternative and reliable tools for use in ML studies of nuclear
physics.

Extrapolation performance of ML models -  One of the most important issues in
ML studies is the low performance of the ML models when it comes to
extrapolation, namely, outside the training and test set regions. It is
essential to develop models that not only predict well-known experimental data
(training and test data) effectively but can also make accurate predictions
for parts of the nuclear chart that are challenging to measure experimentally.
Therefore, in this subsection, we assess the extrapolation capabilities of the
ML models by extending beyond the experimentally known region. Initially, we
test the performance of the ML models on the newly measured 71 nuclei from the
AME2020 data [4] (see Fig. 1). We present the rms deviations of each model in
Table LABEL:extra. Clearly, the accuracy of model predictions improves with
the use of appropriate features. Specifically, increasing the number of inputs
from 5 to 12 improves the performance of the GPR and SVR models in the
extrapolation region by $54.73\%$ and $67.96\%$, respectively. Furthermore,
the low rms errors of these ML models, which are comparable to those of modern
nuclear mass models, indicate that ML models are able to make reasonable
predictions even outside the training region.

Figure 3: The GPR-5 and SVR-5 mass excess predictions are shown for the selected isotopic chains as a function of the neutron number. The blue shaded region represents the $95.0\%$ confidence interval, and the gray region indicates the training and test set area, while the white region is used as extrapolation region. The estimated values for the mass excess predictions, away from the training and test set region, are derived from the trends in the mass surface (TMS) and are taken from [4]. Predictions of other mass models: mic-mac model WS4 [6], non-relativistic Skyrme-type BSk24 interaction [20], and relativistic point-coupling interaction DD-PCX [46], are also provided for comparison. Figure 4: The same as in Fig. 3 but using GPR-12 and SVR-12 ML models. Table 2: The root mean square deviations (given in MeV) for the extrapolation set (71 nuclei from AME2020). The calculations are performed using different inputs. Feature | 5 | 8 | 10 | 12
---|---|---|---|---
Model | $\sigma_{rms}^{extrap.}$ | $\sigma_{rms}^{extrap.}$ | $\sigma_{rms}^{extrap.}$ | $\sigma_{rms}^{extrap.}$
GPR | 1.48 | 1.10 | 0.75 | 0.67
SVR | 2.31 | 1.17 | 0.70 | 0.74
Figure 5: Upper panels: one-neutron separation energies for Kr (a) and Nd (b)
isotopic chains using GPR-12 and SVR-12 models. Lower panels: two-neutron
separation energies for Kr (c) and Nd (d) isotopic chains. The blue shaded
region represents the $95.0\%$ confidence interval. Theoretical model
calculations (WS4, BSk24, DD-PCX) and experimental data are also provided when
available [4].

How far can we go from the experimentally known region and get reasonable
results using ML models? In order to assess the extrapolation performance of
the ML models, we extend our calculations through the proton-rich and neutron-
rich regions. The results are presented for both the training and test regions
(gray region) and the extrapolation region (white regions), where no
experimental data currently exists. In Figure 3, we depict the predictions for
the mass excess of nuclei using GPR-5 and SVR-5 for selected isotopic chains
from various parts of the nuclear chart. The estimated values for the mass
excess predictions from the trends in the mass surface (TMS) are also used to
assess the performance of the models in the extrapolation region [4].
Additionally, we compare these predictions with results from well-known mass
tables: the mic-mac model WS4+RBF [6] and the non-relativistic (BSk24) [20]
calculations. The relativistic calculations with the point-coupling
interaction DD-PCX [46] are performed for even-even nuclei using the axially-
deformed Hartree-Bogoliubov (RHB) model with separable pairing [47], employing
20 harmonic oscillator shells for convergence in the calculations [15].

In GPR, the uncertainty is represented by the blue shaded region. It
represents the probability distribution over the possible functions. This
distribution is updated as more data or features are observed, which leads to
a more precise estimate of the function. Therefore, it is expected that the
uncertainty increases away from the training data, which is a direct
consequence of the roots of Gaussian Process in probability and Bayesian
inference. As can be seen from the upper panels of Figure 3, the GPR with only
5 features performs poorly when we move away from the training-test region,
and the uncertainty is quite high in the extrapolation region. Apart from the
Mg chain, the GPR can make reasonable predictions for the isotopic chains up
to an increase in neutron number around 4 or 5. Then, the results start to
deviate and do not follow the trends obtained in different mass models.
Although the rms deviations are higher for the training and test sets using
the SVR-5 model, it is seen that the SVR-5 model captures the trends better in
the extrapolation region.

By increasing the number of features in the GPR model, we observe a
significant improvement in the model’s performance in the extrapolation region
(Figure 4). Firstly, we note a considerable reduction in the uncertainties of
the predictions. Secondly, the predictions of the GPR-12 model align with a
trend that is similar and comparable to those obtained in different mass
models, albeit slightly higher nearby the drip line. Increasing the number of
features in the GPR model unequivocally enhances its generalizability and
improves uncertainty estimation. As mentioned above, SVR-12 demonstrates
improved predictions in both the training and test regions when the number of
features is increased. However, an increase in the number of features in the
SVR model does not lead to better results for the extrapolation region. The
predictions of the SVR models start to deviate from other mass models and
underestimate the mass excess values compared to them near the drip lines.

Finally, we explore the one- and two-neutron separation energies calculated
using the mass excess $M$ values obtained from our ML models and compare them
with those from other models and available experimental data. The one and two
neutron separation energies are calculated by

$\begin{split}S_{n}&=-M(A,Z)+M(A-1,Z)+m_{n}\\\
S_{2n}&=-M(A,Z)+M(A-2,Z)+m_{2n},\end{split}$ (4)

where $m_{n}$ represents the mass of the neutron. In the upper panels of
Figure 5, the results are displayed for the one-neutron separation energies of
Kr (a) and Nd (b) isotopic chains. It is evident that the ML models provide
reasonable predictions and are in agreement with the experimental data,
exhibiting the well-known odd-even staggering (OES) in binding energies. As
the neutron number increases, the results also show comparability with other
theoretical model calculations. However, near the drip lines, the ML models
start to deviate from other model calculations.

In the lower panels of Fig. 5, the two-neutron separation energies are
displayed for the Kr (c) and Nd (d) isotopic chains. It can be observed that
the ML models make reasonable predictions for the Kr chain. In comparison to
the SVR-12 model, the GPR-12 model’s predictions are more reasonable near the
drip lines and follow a smooth decreasing behaviour with increasing neutron
number. Additionally, the predictions of the GPR-12 model are comparable to
the WS4 model, while the SVR-12 model results align with the BSK4 model as
neutron number increases. When it comes to nuclei near the drip lines, the
predictions of the SVR-12 model become inaccurate and exhibit an increasing
pattern. The ML model predictions deviate from other mass models, particularly
for heavier Nd nuclei. It is also seen that the uncertainty in the GPR-12
predictions is higher for this chain in the extrapolation region. This
discrepancy is a natural consequence of both the limited number of available
experimental data points and the absence of information in the physics-based
feature space in this particular region.

Explainable AI - The implementation of ML models often faces the challenge of
their perceived ‘black box’ nature. To counter this issue, Explainable AI
(XAI) techniques have become increasingly popular for their role in
demystifying these models and enhancing understanding. Among a range of XAI
techniques, SHapley Additive exPlanations (SHAP) [48] has emerged as a
prominent technique that has achieved widespread recognition.

The SHAP technique utilizes the concept of SHAP values, derived from Game
Theory, which illustrates the individual contributions of players in a
cooperative coalition. This concept, originally known as Shapley values [49],
has been extensively studied in game theory literature [50]. Recently adapted
to AI research, specifically in XAI, this approach treats model features as
‘players’ and the prediction as the ‘game’. SHAP values assigned to these
features indicate their relative importance compared to a baseline reference.
Thus, this technique effectively highlights the features most influential in
the model’s decision-making process.

We apply the SHAP technique to interpret the results of the GPR-12 model more
in depth. For the test dataset, we compute the SHAP values, where each value
indicates the contribution of a specific feature to the model’s prediction.
These SHAP values are visually summarized in the Figure 6. The SHAP summary
plot offers an insightful illustration of how each feature influences the
predictions by the GPR-12 model. In this plot, features are ordered on the
y-axis based on their impact, with the most impactful feature positioned at
the top and the least impactful at the bottom. To manage the extensive
computational demands of calculating SHAP values for the GPR-12 model, we
adhered to the guidelines suggested in the official SHAP documentation
111https://shap-lrjball.readthedocs.io/en/latest/examples.html, utilizing
k-means clustering on the training data. We condensed the training data into
three clusters using k-means, assigning weights to each cluster proportionate
to the number of data points it encompasses. Experiments with varying numbers
of clusters, including more than three, consistently yielded comparable
results.

Figure 6: SHAP summary plot for the GPR-12 model. Each input is represented by
a horizontal bar on the plot, where the length of the bar reflects the SHAP
values’ magnitude. The color of each bar indicates the direction of the
feature’s influence on the prediction: blue for a decrease with lower feature
values and red for an increase with higher feature values, with the intensity
of the color denoting the magnitude of the feature’s value.

The analysis reveals that $A^{2/3}$ is the most impactful factor in predicting
the mass, as shown by the SHAP values. It is closely followed by $Z$, $A$ and
$N$, both making noteworthy contributions to the model’s predictions. In
contrast, $Z_{eo}$ and $N_{eo}$ demonstrate a limited impact on predicting the
mass, as indicated by their lower placement on the plot. Nonetheless, their
inclusion is important to improve model predictions as we explain below (see
Fig.7). The SHAP values depicted in the Figure 6 clearly show the extent of
each input’s contribution to individual predictions. The $x$-axis represents
the relative importance of each feature based on their SHAP values. Inputs
with larger absolute SHAP values indicate a more significant effect on the
model’s predictions, whereas those with smaller absolute values have a lesser
influence. It is worth noting that we also examined the SHAP summary plot for
the SVR-12 model, and the results are identical to those presented in Figure
6.

Figure 7: Selected SHAP interaction plots. In these plots, an intense red
color indicates higher positive SHAP values, while a deep blue color signifies
lower negative SHAP values.

In Figure 6, the contributions of features beyond the top five may appear
minimal. However, the GPR-12 model outperforms its versions with fewer
features. This enhanced performance of the GPR-12 model can be attributed to
the interactions between features, which can also be examined in detail
through SHAP analysis. SHAP interaction plots reveal the joint impact of two
features on the model’s predictions. A majority of red in the interaction plot
suggest a positive joint contribution of both features to the model’s
prediction. This means that higher values of these features together are
likely to elevate the model’s output. In contrast, a majority of blue suggests
that the combined features negatively influence the model’s prediction, with
lower values of both features together expected to decrease the model’s
output. Thus, we can pinpoint critical feature interactions and enhance our
understanding of the model’s decision-making based on feature combinations.

In Figure 7, we present selected interaction plots derived from the SHAP
values of the GPR-12 model. To simplify our discussion, we choose interactions
between $Z$ and other features to illustrate. For instance, the combined
effect of higher values of $Z$ and $A$ (see Fig. 7(a)) impacts the model’s
prediction positively, while lower values have a negative impact on the
output. A similar situation is also observed among $Z$, $\nu_{N}$ (c), and
$Z_{\text{shell}}$ (f). It is also seen that the interaction between $Z$ and
$PF$ (e) shows variations according to the region of interest. Nonetheless,
low values of $Z$ and $PF$ have a negative impact on the output. On the other
hand, there is no such interaction between $Z$ and $(N-Z)/A$ or $Z_{eo}$, as
shown in panels (b) and (d) of Fig. 7. It is worth noting that the combined
effect of $Z$ and $(N-Z)/A$ demonstrates a positive impact across all regions.

##  IV Conclusion

This study presents successful implementations of two ML models, SVR and GPR,
using the available experimental data and physics-based feature space to make
predictions for the mass excess of atomic nuclei. The ML models achieve good
results not only in accurately predicting nuclear mass excesses for training
and test sets but also in demonstrating robust extrapolation capabilities. Our
comprehensive analysis, which includes the extrapolation region using the
newly measured data from AME2020 and the region beyond, underscores the
models’ success in handling a diverse range of nuclear data. In addition to
demonstrating the effective application of ML models, our study incorporates
SHAP, an Explainable AI (XAI) technique, enhancing the interpretability of our
ML models.

It is evident that GPR and SVR can be effectively utilized as reliable and
efficient tools for predicting nuclear properties. This study highlights the
potential of these ML models as powerful tools in nuclear physics and opens up
new avenues for future research. These ML models can be further refined to
improve their performance, especially near the drip lines, and be employed to
explore other aspects of nuclear properties.

##  V Acknowledgements

E.Y. acknowledges support from the Science and Technology Facilities Council
(UK) through Grant No. ST/Y000013/1.

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Current affiliation: ]Département de Physique de l’École Normale Supérieure,
24 rue Lhomond, 75005 Paris, France

# Erasure tolerant quantum memory and the quantum null energy condition in
holographic systems

Avik Banerjee<EMAIL_ADDRESS>Center for Quantum Information Theory of
Matter and Spacetime, and Center for Strings, Gravitation and Cosmology,
Department of Physics, Indian Institute of Technology Madras, Chennai 600036,
India  Tanay Kibe<EMAIL_ADDRESS>Center for Quantum Information
Theory of Matter and Spacetime, and Center for Strings, Gravitation and
Cosmology, Department of Physics, Indian Institute of Technology Madras,
Chennai 600036, India  Nehal Mittal<EMAIL_ADDRESS>[  Center for
Quantum Information Theory of Matter and Spacetime, and Center for Strings,
Gravitation and Cosmology, Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  Ayan Mukhopadhyay
<EMAIL_ADDRESS>Center for Quantum Information Theory of Matter and
Spacetime, and Center for Strings, Gravitation and Cosmology, Department of
Physics, Indian Institute of Technology Madras, Chennai 600036, India  Pratik
Roy<EMAIL_ADDRESS>Center for Quantum Information Theory of Matter
and Spacetime, and Center for Strings, Gravitation and Cosmology, Department
of Physics, Indian Institute of Technology Madras, Chennai 600036, India

###### Abstract

Investigating principles for storage of quantum information at finite
temperature with minimal need for active error correction is an active area of
research. We bear upon this question in two-dimensional holographic conformal
field theories via the quantum null energy condition (QNEC) that we have shown
earlier to implement the restrictions imposed by quantum thermodynamics on
such many-body systems. We study an explicit encoding of a logical qubit into
two similar chirally propagating excitations of finite von-Neumann entropy on
a finite temperature background whose erasure can be implemented by an
appropriate inhomogeneous and instantaneous energy-momentum inflow from an
infinite energy memoryless bath due to which the system transits to a thermal
state. Holographically, these fast erasure processes can be depicted by
generalized AdS-Vaidya geometries described previously in which no assumption
of specific form of bulk matter is needed. We show that the quantum null
energy condition gives analytic results for the minimal finite temperature
needed for the deletion which is larger than the initial background
temperature in consistency with Landauer’s principle. In particular, we find a
simple expression for the minimum final temperature needed for the erasure of
a large number of encoding qubits. We also find that if the encoding qubits
are localized over an interval shorter than a specific localization length,
then the fast erasure process is impossible, and furthermore this localization
length is the largest for an optimal amount of encoding qubits determined by
the central charge. We estimate the optimal encoding qubits for realistic
protection against fast erasure. We discuss possible generalizations of our
study for novel constructions of fault-tolerant quantum gates operating at
finite temperature.

##  I Introduction

Quantum thermodynamics is an emerging framework for the study of feasibility
of quantum channels and resources like energy and entanglement needed for
their function Brandão and Gour (2015); Guryanova  _et al._ (2016); Yunger
Halpern and Renes (2016); Yunger Halpern  _et al._ (2016); Goold  _et al._
(2016); Gour  _et al._ (2018); Chitambar and Gour (2019). Most studies have
been in the domain of finite-dimensional quantum systems where many
thermodynamic notions have been generalized via information-theoretic
quantities such as hypothesis-testing relative entropy between a state and a
suitable Gibbs ensemble Yunger Halpern and Renes (2016). Extensions of these
studies to quantum field theories are of interest for finding general
principles of construction of efficient quantum engines and fault-tolerant
quantum gates using many-body systems.

In the context of topologically ordered many-body systems, issues regarding
fault tolerant quantum computation at finite temperature have been studied
extensively Kitaev (2003); Nayak  _et al._ (2008); Brennen and Pachos (2008);
Pachos (2012). Especially with regard to construction of self-correcting
quantum memories for storing quantum information without the need of active
error correction Brown  _et al._ (2016), there are no-go results for one and
two (spatial) dimensional models which utilize stabilizer codes (e.g. the
toric code model), following simply from the result that the energy barrier
between degenerate ground states does not scale with the system size Bravyi
and Terhal (2009); Landon-Cardinal and Poulin (2013). The lack of finite
temperature stability of encoded information is supported by the rapid decay
of topological order parameters at finite temperature Alicki  _et al._ (2009);
Chesi  _et al._ (2010). Quantum thermodynamics could be applied to design
systems that are ideal for storing quantum memory.

Recently, it has been demonstrated in Kibe  _et al._ (2021) that the quantum
null energy condition (QNEC) Bousso  _et al._ (2016) can be a powerful tool to
examine the restrictions, beyond classical thermodynamics, which physical
processes should satisfy in two-dimensional conformal systems. For two-
dimensional conformal field theories ($2D$ CFTs), QNEC states that the non-
vanishing components of the energy-momentum tensor, namely $t_{\pm\pm}$ (with
$\pm$ denoting the left and right moving future-directed null directions
$x^{\pm}=t\pm x$), should be bounded from below by null derivatives of the
entanglement entropy $S$ of any spatial interval ending at the point of
observation as follows Wall (2012); Koeller and Leichenauer (2016);
Balakrishnan  _et al._ (2019):

$\mathcal{Q}_{\pm}\coloneqq 2\pi\langle
t_{\pm\pm}\rangle-S^{\prime\prime}-\frac{6}{c}{S^{\prime}}^{2}\geq 0.$ (1)

Here, primes denote derivatives obtained from infinitesimal displacements of
the endpoint coinciding with the point of observation along $\pm$ directions,
and $c$ is the central charge of the CFT. Generic transitions between thermal
states carrying momentum in holographic $2D$ CFTs driven by instantaneous
energy-momentum inflow from an infinite memoryless bath were examined in Kibe
_et al._ (2021). It was found that non-violation of QNEC not only implies that
the entropy and temperature should increase as implied by classical
thermodynamics, but also for a fixed increase in temperature (entropy) there
are both lower and upper bounds on the increase in entropy (temperature).
Furthermore, upper and lower bounds on the early time quadratic and late time
ballistic growths of entanglement were obtained analytically. The strictest
bounds implied by the inequality (1) are always produced by the semi-infinite
entangling interval. These bounds thus could apply to any system whose
infrared behavior is described by a CFT with a sufficiently large central
charge and a sparse spectrum.

States which saturate (1) are termed quantum equilibrium states Ecker  _et
al._ (2019). Essentially, these are states with Virasoro hair (chirally
propagating excitations) on top of the vacuum or a thermal state. These are
holographically dual to Bañados geometries Banados (1999). Methods developed
in Kibe  _et al._ (2021) allow the study of transitions between two such
quantum equilibrium states in holographic systems.

In this work, we utilize these methods to study a simple encoding of a logical
qubit in a quantum equilibrium state with two similar chirally propagating
excitations on a finite temperature background, and its fast erasure in which
the system transitions to a finite temperature state via instantaneous energy-
momentum inflow from the bath. The non-violation of QNEC sets a lower bound on
the final temperature in consistency with the Landauer principle Landauer
(1961); Sagawa (2017); Esposito  _et al._ (2010); Reeb and Wolf (2014),
according to which information erasure should imply flow of energy from the
information-carrying degrees of freedom to non-information-carrying degrees of
freedom Reeb and Wolf (2014). For a large number of encoding qubits, the
threshold final temperature needed for the erasure increases monotonically
with the number of qubits, and has a simple analytic expression. Crucially, we
show that the fast erasure is not allowed if the encoded information is
localized on a length scale shorter than a specific value $b_{c}$ which is
determined by the initial background temperature and the number of encoding
qubits. The length scale $b_{c}$ turns out to be maximal when the number of
encoding qubits are about $c/6$. We provide estimates for the number of
encoding qubits etc for realistic applications to erasure tolerant quantum
memories.

Figure 1: A schematic depiction of the fast erasure process in which the
energy density has been plotted as a function of the space and time
coordinates. Two spacelike intervals (blue) that are related by reflection
about $x=0$ have the same entanglement entropy.

##  II Holographic quantum memory
and fast erasure

A simple encoding of a logical qubit into the Hilbert subspace of a CFT
spanned by the Virasoro descendents of the vacuum, which can readily be
generalized to a finite temperature background, is as follows. Consider the
stereographic projection of the point on the Bloch sphere representing the
logical qubit state onto a point $z_{*}$ on the complex plane. The Lorentzian
continuation describes a point with coordinates $(t_{*},x_{*})$ or
equivalently $(x^{+}_{*},x^{-}_{*})$. The encoding is achieved by the state
with $\langle
t_{\pm\pm}\rangle=\mathcal{L}_{\pm}(x^{\pm})=\mathcal{L}(x^{\pm}-x^{\pm}_{*})$
comprising two similar left and right moving energy-momentum excitations on
vacuum or a finite temperature background. The logical qubit can be readily
decoded by measuring the energy density at any time and deducing the
coordinates $(t_{*},x_{*})$ where the centers of the two chirally propagating
excitations coincide. For simplicity, we consider $(t_{*},x_{*})$ to be the
origin.

More specifically, we will consider $\mathcal{L}(x^{\pm})$ of the form

$\displaystyle\mathcal{L}(x^{\pm})=-\frac{1}{2}\text{Sch}\left(\int^{x^{\pm}}w(y)dy,\,x^{\pm}\right)+w(x^{\pm})^{2}$
(2)

where Sch denotes the Schwarzian derivative, and $w(x)$ is a positive constant
plus any smooth function with finite support such that $w(x)>0$. As a specific
example, we choose

$w(x)=\frac{a}{b^{2}+x^{2}}+d,$ (3)

which gives

$\displaystyle\mathcal{L}(x^{\pm})=$
$\displaystyle\,\frac{3a^{2}{x^{\pm}}^{2}+\left(a+d\left(b^{2}+{x^{\pm}}^{2}\right)\right)^{4}}{\left(b^{2}+{x^{\pm}}^{2}\right)^{2}\left(a+d\left(b^{2}+{x^{\pm}}^{2}\right)\right)^{2}}$
(4) $\displaystyle+$
$\displaystyle\,\frac{a\left(b^{2}-3{x^{\pm}}^{2}\right)\left(a+d\left(b^{2}+{x^{\pm}}^{2}\right)\right)}{\left(b^{2}+{x^{\pm}}^{2}\right)^{2}\left(a+d\left(b^{2}+{x^{\pm}}^{2}\right)\right)^{2}},$

where $a$ is the amplitude and $b$ the width of the Lorentzian function $w$.
See Fig. 2 for a plot of $\mathcal{L}(x^{\pm})$. Since
$\lim_{x^{\pm}\to\pm\infty}\mathcal{L}_{\pm}(x^{\pm})=d^{2}$,
$\mathcal{L}(x^{\pm})$ is described by two similar chirally propagating
excitations on top of a background temperature $T=\frac{d}{\pi}$. In a
holographic CFT, with large central charge $c$ and sparse spectrum, this state
can be described by a Bañados geometry (see below).

The number of qubits used for the encoding can be estimated from the von-
Neumann entropy that is in excess of the background thermal contribution.
Using the holographic entanglement entropy prescription Ryu and Takayanagi
(2006); Hubeny  _et al._ (2007), one can readily obtain that the entanglement
entropy of an interval with endpoints $(-\frac{l}{2},t)$ and $(\frac{l}{2},t)$
in the large $l$ limit is

$\displaystyle S_{\rm ent}$ $\displaystyle=S_{\rm ent}^{\rm th}(d)+s_{\rm
info}+\mathcal{O}\left(\frac{1}{l}\right),$ $\displaystyle s_{\rm info}$
$\displaystyle=\frac{c}{6}\frac{2\pi a}{b},$ (5)

where $S_{\rm ent}^{\rm th}(d)=\frac{c}{3}\left(dl-\log(2d)\right)$ is simply
the thermal contribution corresponding to the temperature $T=\frac{d}{\pi}$.
Therefore, $s_{\rm info}$, which is independent of the entangling length $l$,
provides an estimate of the number of encoding qubits.

Figure 2: A plot of $\mathcal{L}(x^{\pm})$ (or equivalently energy density at
the moment of quench $t=0$) and $D(x_{0})$, the coefficient of quadratic
growth of entanglement entropy of the interval $x\geq x_{0}$. Here $a=0.01$,
$b=0.19$, $T^{i}=\frac{0.2}{\pi}$ and $T^{f}=\frac{1}{\pi}$.

We model information erasure as a quench that transforms the inhomogeneous
state with encoded information to a thermal state with
$\mathcal{L}_{\pm}=\mu^{2}$ (and with temperature $\mu/\pi$) as depicted in
Fig. 1. The quench involves inhomogeneous inflow of energy and momentum from
an infinite memoryless bath and can be realized physically via appropriate
time-dependent couplings between the bath and the system Balasubramanian  _et
al._ (2011); Hubeny and Rangamani (2010); Chesler and Yaffe (2014). In the
holographic setup, we can model this via an appropriate bulk energy-momentum
tensor which is infalling from the boundary of spacetime without assuming any
explicit realization of the bulk matter. We will consider the energy-momentum
inflow from the bath describing the quench to be instantaneous 111The time-
scale of the quench should be much larger than the ultraviolet cutoff of the
CFT.. In this case, the form of the bulk energy-momentum tensor is fixed
completely by initial and final states via the Israel junction conditions
Israel (1966). To simplify the discussion, we assume that the quench occurs at
$t=0$, when the propagating excitations coincide. However, our results remain
valid even if the quench occurs before or after.

The holographic description of the full process Brown and Henneaux (1986);
Maldacena (1998); Henningson and Skenderis (1998); Balasubramanian and Kraus
(1999) in the limit of large central charge is given by a generalized $2+1$
dimensional AdS-Vaidya geometry involving a transition between the Bañados
geometry depicting the initial encoding state to a BTZ (Bañados-Teitelboim-
Zanelli) black hole Banados  _et al._ (1992, 1993) describing the final
thermal state as described in Kibe  _et al._ (2021). The metric, which solves
Einstein’s equations with a negative cosmological constant $\Lambda=-1/L^{2}$,
takes the form

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$ $\displaystyle 2{\rm d}r{\rm
d}t-\left(\frac{r^{2}}{L^{2}}-2m(t,x)L^{2}\right){\rm
d}t^{2}+\frac{r^{2}}{L^{2}}{\rm d}x^{2},$ (6)

which is a special case of the more general transitions between arbitrary
Bañados geometries considered in Kibe  _et al._ (2021). The dual CFT shares
the coordinates $t,x$, and lives at the boundary $r\to\infty$ of this
spacetime. The metric is supported by a bulk energy momentum tensor
$T_{\mu\nu}$, which is conserved and traceless in the background metric (6),
and whose only non-vanishing component is

$\displaystyle T_{tt}$ $\displaystyle=$ $\displaystyle\frac{q(t,x)L^{2}}{r}.$
(7)

For our specific fast erasure process, we have

$\displaystyle m(t,x)$ $\displaystyle=$
$\displaystyle\theta(-t)(\mathcal{L}(x^{+})+\mathcal{L}(x^{-}))+2\theta(t)\mu^{2},$
(8)

with $\mathcal{L}(x)$ given by the even function (4) and

$\displaystyle 8\pi Gq(t,x)$ $\displaystyle=$ $\displaystyle
2\delta(t)\left(\mu^{2}-\mathcal{L}(x)\right),$ (9)

with $G$ being the three dimensional Newton’s constant. The bulk matter (7) is
thus an inhomogeneous infalling null shell. The non-violation of QNEC, as
described below, requires $q(t,x)>0$ due to which the bulk matter satisfies
the classical null energy condition .

The energy-momentum tensor of the dual CFT state can be be obtained via the
procedure of holographic renormalization Henningson and Skenderis (1998);
Balasubramanian and Kraus (1999). The non-vanishing components are

$\langle
t_{\pm\pm}\rangle=\frac{c}{12\pi}\left(\mathcal{L}(x^{\pm})\theta(-t)+\mu^{2}\,\theta(t)\right).$
(10)

where $c=3L/(2G)$ is the central charge Brown and Henneaux (1986); Henningson
and Skenderis (1998); Balasubramanian and Kraus (1999). The Ward identity
$\partial_{\mu}\langle t^{\mu\nu}\rangle=f^{\nu}$, where $f^{\nu}=L(q(t,x),0)$
is implied by (9), so that $q(t,x)$ describes the heat inflow from the bath to
the CFT. The condition $q(t,x)>0$, implied by the non-violation of QNEC as
discussed above, is thus equivalent to the Landauer principle which implies
that energy should flow irreversibly from information carrying degrees of
freedom to non-information carrying degrees of freedom during information
erasure.

The time-dependent entanglement entropy is given by the holographic
entanglement entropy prescription Ryu and Takayanagi (2006); Hubeny  _et al._
(2007), and therefore $\mathcal{Q}_{\pm}$ in (1) can be computed analytically
following the method developed in Kibe  _et al._ (2021) via the use of
uniformization maps. For a summary, see the Supplemental Material.

Let us discuss first how the entanglement entropy evolves. It is to be noted
that although the bulk metric becomes that of the BTZ black hole and the
energy-momentum tensor becomes thermal immediately after quench, the state
does not thermalize immediately as is evident from the time-dependence of the
entanglement entropy of an interval of length $l$. It has been shown in Kibe
_et al._ (2021) that the thermalization time for an arbitrary momentum
carrying thermal state is $l/2$ supporting earlier results Hubeny  _et al._
(2013); Liu and Suh (2014a, b).

As shown in Fig. 1, owing to reflection symmetry $x\rightarrow-x$ of our
geometry, the entanglement entropy of two spatial intervals related by
reflection about origin should be identical. Also $\mathcal{Q}_{+}$ of one of
these intervals would be same as $\mathcal{Q}_{-}$ of the other for
displacements of the endpoint closer to the origin.

Immediately after quench, i.e. at $t=0^{+}$, the entanglement entropy $S(t)$
of a semi-infinite interval $x\geq x_{0}$ grows quadratically, $S(t)\approx
S_{0}+Dt^{2}$, with

$D=\mu\left(4\mu-2d-\frac{w^{\prime}(x_{0})}{w(x_{0})}-2w(x_{0})\right).$ (11)

At larger times, the entanglement entropy of these semi-infinite intervals
grows linearly as $S(t)\approx 2(s^{f}-s^{i})t$ with $s^{f}=(c/3)d$ and
$s^{i}=(c/3)\mu$ being the initial and final thermal entropy densities as
observed before in Kibe  _et al._ (2021). This is consistent with the light-
cone like spreading of entanglement seen in Calabrese and Cardy (2009); Liu
and Suh (2014b, a); Calabrese and Cardy (2016); Cheneau  _et al._ (2012); Kim
and Huse (2013); Kaufman  _et al._ (2016) from both ends of an interval.

The quadratic growth coefficient $D$ in (11) is plotted in Fig. 2 as a
function of the endpoint $x_{0}$ of the semi-infinite interval. Note that the
reflection asymmetry originates from our choice of interval $x\geq x_{0}$. We
note that the coefficient $D$ is not positive definite and demanding $D\geq 0$
already bounds the final temperature $T^{f}$ from below. This bound can be
easily evaluated when the amplitude ($\propto a$) of the chiral excitations is
large for fixed width $b$ and initial background temperature $T^{i}=d/\pi$. In
this limit, the number of encoding qubits $s_{\rm info}$ given by (5) is
large, and we find that $D\geq 0$ implies

$T^{f}\geq T^{i}+\frac{a}{2\pi b^{2}}=T^{i}+\frac{3s_{\rm info}}{2c\pi^{2}b}.$
(12)

QNEC non-violation places a stronger lower bound than positivity of $D$ as
reported below.

##  III Erasure tolerance

The QNEC inequalities (1) produce the strongest bounds for the erasure process
when we consider the entangling region to be a semi-infinite interval as was
found previously Kibe  _et al._ (2021) in the context of transitions between
thermal states (see Supplementary Material for details). Owing to the
reflection symmetry of the setup, we consider half line intervals $x\geq
x_{0}$ in what follows unless explicitly mentioned otherwise. As detailed in
the Supplementary Material, QNEC prohibits the fast erasure process completely
when the initial background temperature ($d/\pi$) is zero and also if any
encoding is done via an $\mathcal{L}(x)$ which has infinite extent (and
entropy). The interesting question however is the minimal value of the final
temperature ($T_{c}=\mu_{c}/\pi$) required by the erasure if the encoding is
done by a finite number of qubits on a finite (initial) temperature
background. The results obtained from QNEC non-violation are consistent with
the Landauer principle which demands that information erasure must be
accompanied by increase in the thermodynamic entropy of the non-information-
bearing (thermal) degrees of freedom.

For a large number of encoding qubits, $s_{\rm info}$, a fixed background
temperature ($T^{i}=d/\pi$) and a fixed width ($b$) of the encoding
excitation, QNEC non-violation implies

$T^{f}\geq T^{i}+\frac{a}{\pi b^{2}}=T^{i}+\frac{3s_{\rm
info}}{c\pi^{2}b}=T^{i}+\frac{1}{\pi}\left(\max w(x)\right).$ (13)

Note that this is stronger than the bound (12) obtained by demanding the
increase of entanglement entropy just after quench. Interestingly, we also
find the same bound for large $a$ and $b$ but with fixed $s_{\rm info}$. The
bound stated in terms of $w(x)$ also holds for $w(x)=a\,\text{sech}(x/b)+d$ in
both these limits indicating that it could be valid for any encoding
excitation with a finite support.

Figure 3: The critical localization length $b_{c}$ is plotted against $s_{\rm
info}$ for a half line interval and fixed $d=0.2$. $b_{c}\to 0$ as $s_{\rm
info}\to 0$, since the excitation disappears in this limit.

Remarkably, if one squeezes the encoding qubits to a width $b<b_{c}$ for a
fixed background temperature $T^{i}=d/\pi$, the fast erasure process is
disallowed as the minimal final temperature needed for the erasure diverges.
For widths $b\geq b_{c}$, one can always satisfy QNEC inequalities if
$T^{f}\geq T^{c}$ making erasure possible. When $b<b_{c}$, $\mathcal{Q_{+}}<0$
for intervals $x\geq x_{0}$ (and $\mathcal{Q_{-}}<0$ for intervals $x\leq
x_{0}$) violating QNEC for any final temperature. In fact, in this case, for
large final temperature $\mu/\pi$, $\mathcal{Q}_{+}(t)/\mu\approx f(t)$, with
$f(t)$ independent of $\mu$ for any interval. This feature leads to a simple
algorithm for obtaining $b_{c}$ for a fixed $s_{\rm info}$ (see Supplementary
Material). We find that both $b_{c}$ and $T_{c}$ increase with the length of
the entangling interval monotonically, and reach finite values for the semi-
infinite interval which thus puts maximum restriction on the parameter space
where erasure is possible.

The dimensionless critical localization length $b_{c}T_{i}$ (for the semi-
infinite interval) depends on $s_{\rm info}$ as shown in Fig. 3. We note as
$s_{\rm info}\rightarrow\infty$, $b_{c}\sim 0.4/s_{\rm info}$. This is
consistent with (13) which states that in this limit erasure is possible for
sufficiently large $T^{f}$ for any width $b$. Similarly in the limit $s_{\rm
info}\rightarrow 0$ erasure should be possible for any width $b$ implying that
$b_{c}T^{i}$ should vanish in this limit also. As shown in Fig. 3, the
dependence of $b_{c}T_{i}$ on $s_{\rm info}$ is non-monotonic and $b_{c}T^{i}$
obtains its maximal value $\approx 0.04/T^{i}$ for an optimal $s_{\rm
info}\approx c/6$ allowing protection against fast erasure with reasonably
small density of encoding.

Figure 4: The minimum value of $(T^{f}-T^{i})/T^{i}$ necessary for the erasure
process ($\Delta$) is plotted against the number of encoding qubits $s_{\rm
info}$ for various fixed encoding widths $b$ when background temperature
$T^{i}=0.2/\pi$. For large $s_{\rm info}$, we obtain (13).

In Fig. 4, $\Delta$, the minimal value of $(T^{f}-T^{i})/T^{i}$ needed for the
erasure, has been plotted as a function of $s_{\rm info}$ for fixed widths $b$
and a background temperature $T^{i}$ of the encoding. For large $s_{\rm
info}$, the dependence is given by (13). However, as expected from the above
discussion, the dependence is prominently non-monotonic for intermediate
values of $s_{\rm info}$ especially for small values of $b$. In fact, as
expected from Fig. 3 and shown explicitly in Fig. 4, for $bT^{i}\lesssim
0.04$, there are two values of $s_{\rm info}$, which we denote as $s_{\rm
info}^{1}$ and $s_{\rm info}^{2}$, such that $\Delta$ diverges as $1/(s_{\rm
info}-s_{\rm info}^{1,2})$ as $s_{\rm info}$ approaches $s_{\rm info}^{1}$ and
$s_{\rm info}^{2}$ from below and above respectively, implying protection
against erasure for $s_{\rm info}^{1}\leq s_{\rm info}\leq s_{\rm info}^{2}$.

So far we have considered arbitrarily large final temperatures. However, the
final temperature in an experimental setup will be only as large as that of
the environment. This fixed value of maximal final temperature and a fixed
$s_{\rm info}$ determine a point in Fig. 4. Then there exists $b=b^{*}$ such
that the corresponding curve $\Delta$ vs $s_{\rm info}$ passes through this
point. In the experimental setup, erasure protection is then achieved for
$b\leq b_{*}$ since then one requires a value of $\Delta$ for erasure which is
higher than what the environment can provide. If the background temperature
$T^{i}$ of the quantum memory is $50K$, and the environment provides
$T^{f}=300K$, then for $c=100$ we find that to protect 50 encoding qubits we
need the excitations to be localized to $\lesssim 2\mu$m. For realistic
applications, $T^{f}$ needs to be smaller than the microscopic energy scale at
which the CFT ceases to be a valid description.

##  IV Discussion

It should be interesting to study whether squeezing the encoding qubits
protects against other errors, especially those corresponding to shifts of the
center of $\mathcal{L}(x)$, or distortions which produce multiple peaks and
therefore make the decoding erroneous. Furthermore, one should also
investigate slower erasure processes in which energy and momentum are pumped
in from the bath over a long time scale. We would also like to study
implementation of quantum gates via unitary conformal transformations on the
encoded qubits and study their fault tolerant properties. One would expect
that one can protect against different errors simultaneously by exploiting the
infinite dimensional Virasoro group suitably.

###### Acknowledgements.

It is a pleasure to thank Souvik Banerjee, Abhishek Chowdhury, Arnab Kundu,
Prabha Mandayam, Pramod Padmanabhan and Marios Petropoulos for helpful
discussions. The research of AB is supported by IFCPAR/CEFIPRA project 6304-3
and the research of TK is supported by the Prime Minister’s Research
Fellowship (PMRF). AM acknowledges the support of the Ramanujan Fellowship of
the Science and Engineering Board (SERB) of the Department of Science and
Technology (DST) of India, IFCPAR/CEFIPRA project 6304-3, the new faculty seed
grant of IIT Madras and the additional support from the Institute of Eminence
scheme of IIT Madras funded by the Ministry of Education of India. NM would
like to thank IIT Madras and his home institutions Paris-Saclay University and
École Normale Supérieure for their hospitality.

## Supplementary Material

### Cut and glue method for entanglement entropy and QNEC

A Bañados geometry is dual to a quantum equilibrium state Ecker  _et al._
(2019) for which

$\langle t_{\pm\pm}\rangle=\frac{c}{12\pi}\mathcal{L}_{\pm}(x^{\pm})$ (14)

and is described by the metric

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$ $\displaystyle 2{\rm d}r{\rm
d}t-\left(\frac{r^{2}}{L^{2}}-2(\mathcal{L}_{+}(x^{+})+\mathcal{L}_{-}(x^{-}))L^{2}\right){\rm
d}t^{2}$ (15)
$\displaystyle+2(\mathcal{L}_{+}(x^{+})-\mathcal{L}_{-}(x^{-}))L^{2}{\rm
d}t{\rm d}x+\frac{r^{2}}{L^{2}}{\rm d}x^{2}.$

Any Bañados geometry can be uniformized to the Poincaré patch, given by the
above metric with $\mathcal{L}_{\pm}(x^{\pm})=0$ and dual to the vacuum state,
using the following uniformization map Kibe  _et al._ (2021)

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{(}X_{b}^{+}(x^{+})+X_{b}^{-}(x^{-})$
$\displaystyle+\frac{{X_{b}^{+}}^{\prime}(x^{+})+{X_{b}^{-}}^{\prime}(x^{-})-2\sqrt{{X_{b}^{+}}^{\prime}(x^{+}){X_{b}^{-}}^{\prime}(x^{-})}}{\frac{r}{L^{2}}-\frac{{X_{b}^{+}}^{\prime\prime}(x^{+})}{2{X_{b}^{+}}^{\prime}(x^{+})}-\frac{{X_{b}^{-}}^{\prime\prime}(x^{-})}{2{X_{b}^{-}}^{\prime}(x^{-})}}\Bigg{)},$
$\displaystyle X$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Bigg{(}X_{b}^{+}(x^{+})-X_{b}^{-}(x^{-})$
$\displaystyle+\frac{{X_{b}^{+}}^{\prime}(x^{+})-{X_{b}^{-}}^{\prime}(x^{-})}{\frac{r}{L^{2}}-\frac{{X_{b}^{+}}^{\prime\prime}(x^{+})}{2{X_{b}^{+}}^{\prime}(x^{+})}-\frac{{X_{b}^{-}}^{\prime\prime}(x^{-})}{2{X_{b}^{-}}^{\prime}(x^{-})}}\Bigg{)},$
$\displaystyle R$ $\displaystyle=$ $\displaystyle
L^{2}\frac{{\frac{r}{L^{2}}-\frac{{X_{b}^{+}}^{\prime\prime}(x^{+})}{2{X_{b}^{+}}^{\prime}(x^{+})}-\frac{{X_{b}^{-}}^{\prime\prime}(x^{-})}{2{X_{b}^{-}}^{\prime}(x^{-})}}}{\sqrt{{X_{b}^{+}}^{\prime}(x^{+}){X_{b}^{-}}^{\prime}(x^{-})}},$
(16)

where $(T,R,X)$ denote the Poincaré patch coordinates. Clearly
$X_{b}^{\pm}(x^{\pm})$ are the boundary lightcone coordinates of the Poincaré
patch as evident from the limit $r\rightarrow\infty$. For the uniformization
map to be valid, $X_{b}^{\pm}(x^{\pm})$ should satisfy

${\rm Sch}(X_{b}^{\pm}(x^{\pm}),x^{\pm})=-2\mathcal{L}_{\pm}(x^{\pm}).$ (17)

This equation can be solved by first solving the Hill equation

$\Psi_{\pm}^{\prime\prime}(x^{\pm})-\mathcal{L}_{\pm}(x^{\pm})\Psi_{\pm}(x^{\pm})=0,$
(18)

which has two linearly independent solutions $\Psi^{1}_{\pm}$ and
$\Psi^{2}_{\pm}$. One can then readily verify that

$X_{b}^{\pm}(x^{\pm})=\frac{\Psi^{1}_{\pm}(x^{\pm})}{\Psi^{2}_{\pm}(x^{\pm})}$
(19)

is a solution to (17). Solutions for $X_{b}^{\pm}$ can be analytically
generated as follows. For $\mathcal{L}_{\pm}(x^{\pm})=1$ it is easy to see
that

$\Psi^{1}_{\pm}(x^{\pm})=\frac{1}{\sqrt{2}}\sinh
x^{\pm},\quad\Psi^{2}_{\pm}(x^{\pm})=\frac{1}{\sqrt{2}}\cosh x^{\pm}$ (20)

are two independent solutions to the Hill equation. The conformal covariance
of the Hill equation then implies that under the transformation
$x^{\pm}\to\int^{x^{\pm}}w(y)dy$ (for any positive definite function $w(y)$),

$\displaystyle\mathcal{L}_{\pm}(x^{\pm})$ $\displaystyle\to-\frac{1}{2}{\rm
Sch}\left(\int^{x^{\pm}}w(y)dy,x^{\pm}\right)+w(x^{\pm})^{2},$ (21)
$\displaystyle\Psi^{1}_{\pm}(x^{\pm})$
$\displaystyle\to\frac{1}{\sqrt{2w(x^{\pm})}}\sinh\left(\int^{x^{\pm}}w(y)dy\right),$
(22) $\displaystyle\Psi^{2}_{\pm}(x^{\pm})$
$\displaystyle\to\frac{1}{\sqrt{2w(x^{\pm})}}\cosh\left(\int^{x^{\pm}}w(y)dy\right).$
(23)

For the transformation of $\mathcal{L}_{\pm}(x^{\pm})$ above we have added the
anomalous term with appropriate central charge. We can thus obtain the
uniformization map corresponding to the Lorentzian $w(x)$ described in the
main text.

We note that the metric (6) is essentially two Bañados geometries separated by
an inhomogeneous null shell infalling from the boundary at the time of the
quench $t=0$ and satisfying Israel junction conditions Kibe  _et al._ (2021).
In order to compute the entropy and QNEC, we use two different uniformization
maps (Cut and glue method for entanglement entropy and QNEC) for the two
respective Bañados geometries before and after the quench. The hypersurface
$t=0$ has two separate images $\Sigma^{i,f}$ in the respective Poincaré
patches which are glued together by identifying points with the same physical
coordinates $x$ and $r$ as shown in Fig. 5.

Figure 5: Schematic diagram of the cut and glue method to compute the
entanglement entropy. The left and right halves correspond to the two Poincaré
patches before and after the quench at $t=0$. The two glueing hypersurfaces
$\Sigma^{i,f}$ are coloured. Points on these hypersurfaces with the same $x,r$
coordinates are identified. A geodesic ending at the boundary points $p_{1,2}$
is shown (black solid curve).

Geodesic lengths can be readily computed in the Poincaré patch in terms of the
endpoints $(T_{1,2},X_{1,2},R_{1,2})$ using the formula

$L_{\rm geo}=L\ln(\xi+\sqrt{\xi^{2}-1}),$ (24)

with
$\xi=(Z_{1}^{2}+Z_{2}^{2}-(T_{1}+Z_{1}-T_{2}-Z_{2})^{2}+(X_{1}-X_{2})^{2})/2Z_{1}Z_{2}$
and $Z_{1,2}=L^{2}/R_{1,2}$. For transitions between Bañados geometries, we
can thus use the uniformization map to analytically calculate geodesic lengths
in the two Poincaré patches and sum the contributions. Fig. 5 shows a geodesic
in such a geometry cut into three arcs by the gluing hypersurface. The points
$q_{1,2}$ where the geodesic intersects the gluing hypersurface can be
analytically obtained if the $t>0$ geometry is a non-rotating BTZ spacetime.
The lengths of each of these segments can therefore be readily computed using
the geodetic distance formula in the corresponding Poincaré patch. Similarly
the variation of the entanglement entropy under null deformations of one of
the end points $p_{1,2}$ can be computed and the QNEC can be obtained (see
Kibe  _et al._ (2021) for details).

### The $l\to\infty$ limit gives the strongest QNEC inequality

Plots for a representative fast erasure of a Lorentzian excitation are shown
in Fig. 6. It is clear from the plots that the $l\to\infty$ limit gives the
strictest QNEC inequality.

Figure 6: $\mathcal{Q}_{+}$ (top) is plotted for $0<t<l/2$ and
$\mathcal{Q}_{-}$ (bottom) is plotted for $0<t<{\rm min}(1.1,l/2)$ for various
lengths $l$ of the entangling interval and the fast quench that transforms a
Lorentzian excitation with $a=1$, $b=1$ and $d=0.1$ to a thermal state with
temperature $T^{f}=0.39$. The strictest inequality can be seen to be imposed
in the limit $l\to\infty$.

### Fast erasure is impossible if $T^{i}=0$

For $T^{i}=0$ and arbitrary final temperature, initial amplitude and width we
find that as $t\to\infty$

$\frac{6}{c}\mathcal{Q}_{+}=-4\frac{\mu}{t}.$ (25)

Thus QNEC is violated for any final temperature and therefore erasure is
impossible if $T^{i}=0$. This is illustrated in Fig. 7.

Figure 7: $\mathcal{Q}_{+}$ is plotted for $T^{i}=0$, $a=1$, $b=1$ and various
$T^{f}$. The violation becomes larger as we increase the final temperature.

### Method used to obtain the critical localization length $b_{c}$

As shown in figure 3 in the main text, for $b\geq b_{c}$, the fast erasure
process is possible with a large enough $T^{f}$. For $b<b_{c}$ we find that
QNEC is violated for a range of $s_{\rm info}$ and raising the final
temperature simply scales the QNEC but doesn’t cure the violation. This
scaling behaviour is illustrated in Fig. 8 and we use it to determine $b_{c}$
as described in the following algorithm.

Figure 8: A plot of the $\mathcal{Q}_{+}$ vs $t$ for $\frac{6}{c}s_{\rm
info}=2\pi$, $b=0.3$, $T^{i}=\frac{0.2}{\pi}$, $x_{0}=-1$. We have chosen the
entangling interval $x\geq x_{0}$ with $x_{0}=-1$. For two choices of
$\mu=10^{6}$ and $\mu=10^{7}$ (note $\mu=\pi T^{f}$), we see that the
$\mathcal{Q}_{+}(t)$ plot simply scales with $\mu$.

  1. 1.

For a fixed $s_{\rm info}$ pick a $\mu$ (recall that the final state has
temperature $T^{f}=\frac{\mu}{\pi}$).

  2. 2.

Vary the value of $b$ and find a $b_{c}$ such that QNEC is violated for all
$b<b_{c}$.

  3. 3.

Increase $\mu$ to $10\mu$.

  4. 4.

Repeat steps 2 and 3 to find a new $b_{c}$ that will be smaller than the older
one.

  5. 5.

Repeat steps 2-4 until scaling behaviour is reached and $b_{c}$ cannot be
decreased further by increasing $\mu$.

### Erasure of infinitely de-localized Mathieu excitations

Using the cut and glue method described in the main text, here we study fast
erasure of infinitely delocalized excitations to a thermal state with
$T^{f}=\frac{\mu}{\pi}$. We consider excitations of the Mathieu form with

$\displaystyle\mathcal{L}_{\pm}(x^{\pm})$
$\displaystyle=d^{2}\alpha^{\prime}(x^{\pm})-\frac{1}{2}\text{Sch}(\alpha(x^{\pm}),x^{\pm}),$
(26) $\displaystyle\alpha(x)$ $\displaystyle=x+q\sin(x),\qquad|q|\leq 1.$ (27)

This particular form is chosen so that we can analytically obtain the
uniformization map described in Kibe  _et al._ (2021). A representative plot
of these excitations is shown in Fig. 9. It is clear from the figure that
these excitations are simply oscillations on top of a background temperature
$T^{i}=\frac{d}{\pi}.$

Figure 9: A representative plot of $\mathcal{L}_{\pm}$ for $d=0.5$, $q=0.1$.

The expression for $\mathcal{Q}_{+}$ at $t=0$ for a half line $x\geq x_{0}$
interval is

$\displaystyle\mathcal{Q}_{+}$ $\displaystyle=\frac{c\mu^{3}q}{48\left(\mu
q\cos\left(\mu x_{0}\right)+1\right){}^{2}}$ (28)
$\displaystyle\times\big{(}-\mu q\left(2\sin\left(2\mu
x_{0}\right)+3\right)+\mu q\cos\left(2\mu x_{0}\right)$
$\displaystyle\quad\quad-4\sin\left(\mu x_{0}\right)-2\cos\left(\mu
x_{0}\right)\big{)}$

At $x_{0}=0$ we get

$\mathcal{Q}_{+}=\frac{c}{6}\left(-\frac{\mu^{3}q}{4\mu q+4}\right).$ (29)

Therefore for $\mathcal{Q}_{+}\geq 0$ we must have

$-\min\left(\frac{1}{\mu},1\right)\leq q\leq 0$ (30)

At $x_{0}=\frac{3\pi}{2\mu}$ we get

$\mathcal{Q}_{+}=\frac{c}{6}\left(-\frac{1}{2}\mu^{3}q(\mu q-1)\right).$ (31)

For $\mathcal{Q}_{+}\geq 0$ we must have in this case

$0\leq q\leq\min\left(\frac{1}{\mu},1\right).$ (32)

The two conditions (30) and (32) are inconsistent with each other and
therefore QNEC cannot be satisfied for all $x_{0}$. An erasure of such an
infinitely de-localized excitation is disallowed by QNEC.

### Creation of Lorentzian excitations is disallowed

In the main text we have analyzed the deletion of Lorentzian excitations. It
is natural to ask if such excitations can be created from the vacuum or a
thermal state via fast quenches of the form described in the main text. Fig.
10 shows a representative plot of $\mathcal{Q}_{+}$ for a quench that
transforms the vacuum state to a Lorentzian excitation with various background
temperatures. The QNEC in this case has to be computed numerically since the
intersection points of the geodesic with the glueing hypersurfaces cannot be
solved analytically. As we scale the length of the entangling interval we find
that the plots for the QNEC with entangling lengths $l=1000$ and $l=5000$
coincide and we are therefore able to conclude that $l=1000$ is numerically
the $l\to\infty$ limit. The plot in Fig. 10 shows that this transition is
disallowed by QNEC and scaling the final temperature doesn’t cure the QNEC
violation but makes it worse. We scan the background temperature of the final
state over the range $T^{f}=\frac{1}{\pi}$ to $T^{f}=\frac{10000}{\pi}$ and
always find that QNEC is violated.

Figure 10: $\mathcal{Q}_{+}$ for a quench transforming the vacuum state to a
Lorentzian excitation with $x_{0}=-1$, $a=1$, $b=1$ for various $T^{f}$. The
length of the entangling interval is taken to be $l=1000$, which is
numerically the $l\to\infty$ limit.

Similarly we can look at the creation of such excitations from an initial
thermal state. Without loss of generality we choose the initial temperature
$T^{i}=1/\pi$. We fix the parameters of the excitation to be $a=1$, $b=1$ and
scan over the background temperature. The numerical $l\to\infty$ limit is
essentially $l=1000$ as described before. The plot for $\mathcal{Q}_{+}$ with
$x_{0}=-1$ is shown in Fig. 11 which shows that the QNEC violation doesn’t
improve but simply scales as we raise the background temperature.

Figure 11: $\mathcal{Q}_{+}$ for a quench transforming a thermal state with
$T^{i}=\frac{1}{\pi}$ to a Lorentzian excitation with $x_{0}=-1$, $a=1$, $b=1$
and various $T^{f}$. The length of the entangling interval is taken to be
$l=1000$, which is numerically the $l\to\infty$ limit. We work with numeric
precision to 100 decimal places, therefore the $10^{-19}$ order of magnitude
is not numerical noise.

Thus we find that the creation of such Lorentzian excitations of the CFT from
the vacuum or a thermal state via an fast quench is disallowed by QNEC. It
would be interesting to understand what leads to a violation of QNEC in these
processes both geometrically and also from a CFT computation.

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