# Hyperparameter optimization of hybrid quantum neural networks for car
classification

Asel Sagingalieva  Andrii Kurkin  Artem Melnikov  Daniil Kuhmistrov  Michael
Perelshtein  Alexey Melnikov  Terra Quantum AG, 9400 Rorschach, Switzerland
Andrea Skolik1,3 David Von Dollen2,3 1Volkswagen Data:Lab, 80805 Munich,
Germany  2Volkswagen Group of America, Auburn Hills, MI 48326, USA  3Leiden
University, 2333 CA Leiden, The Netherlands

###### Abstract

Image recognition is one of the primary applications of machine learning
algorithms. Nevertheless, machine learning models used in modern image
recognition systems consist of millions of parameters that usually require
significant computational time to be adjusted. Moreover, adjustment of model
hyperparameters leads to additional overhead. Because of this, new
developments in machine learning models and hyperparameter optimization
techniques are required. This paper presents a quantum-inspired hyperparameter
optimization technique and a hybrid quantum-classical machine learning model
for supervised learning. We benchmark our hyperparameter optimization method
over standard black-box objective functions and observe performance
improvements in the form of reduced expected run times and fitness in response
to the growth in the size of the search space. We test our approaches in a car
image classification task, and demonstrate a full-scale implementation of the
hybrid quantum neural network model with the tensor train hyperparameter
optimization. Our tests show a qualitative and quantitative advantage over the
corresponding standard classical tabular grid search approach used with a deep
neural network ResNet34. A classification accuracy of 0.97 was obtained by the
hybrid model after 18 iterations, whereas the classical model achieved an
accuracy of 0.92 after 75 iterations.

## INTRODUCTION

The field of quantum computing has seen large leaps in building usable quantum
hardware during the past decade. As one of the first vendors, D-Wave provided
access to a quantum device that can solve specific types of optimization
problems johnson2011quantum . Motivated by this, quantum computing has not
only received much attention in the research community, but was also started
to be perceived as a valuable technology in industry. Volkswagen published a
pioneering result on using the D-Wave quantum annealer to optimize traffic
flow in 2017 neukart2017traffic , which prompted a number of works by other
automotive companies mehta2019quantum ; ohzeki2019control ; yarkoni2021multi .
Since then, quantum annealing has been applied in a number of industry-related
problems like chemistry streif2019solving ; xia2017electronic , aviation
stollenwerk2019quantum , logistics feld2019hybrid  and finance
grant2021benchmarking . Aside from quantum annealing, gate-based quantum
devices have gained increased popularity, not least after the first
demonstration of a quantum device outperforming its classical counterparts
arute2019quantum . A number of industry-motivated works have since been
published in the three main application areas that are currently of interest
for gate-based quantum computing: optimization streif2021beating ;
streif2020training ; amaro2021case ; dalyac2021qualifying ; luckow2021quantum
, quantum chemistry and simulation arute2020hartree ; malone2021towards , and
machine learning rudolph2020generation ; skolik2021layerwise ;
skolik2021quantum ; peters2021machine ; alcazar2020classical . Research in the
industrial context has been largely motivated by noisy intermediate-scale
quantum (NISQ) devices – early quantum devices with a small number of qubits
and no error correction. In this regime, variational quantum algorithms (VQAs)
have been identified as the most promising candidate for near-term advantage
due to their robustness to noise cerezo2021variational . In a VQA, a
parametrized quantum circuit (PQC) is optimized by a classical outer loop to
solve a specific task like finding the ground state of a given Hamiltonian or
classifying data based on given input features. As qubit numbers are expected
to stay relatively low within the next years, hybrid alternatives to models
realized purely by PQCs have been explored zhang2021variational ;
mari2020transfer ; zhao2019qdnn ; dou2021unsupervised ;
sebastianelli2021circuit ; pramanik2021quantum . In these works, a quantum
model is combined with a classical model and optimized end-to-end to solve a
specific task. In the context of machine learning, this means that a PQC and
neural network (NN) are trained together as one model, where the NN can be
placed either before or after the PQC in the chain of execution. When the NN
comes first, it can act as a dimensionality reduction technique for the
quantum model, which can then be implemented with relatively few qubits.

In this work, we use a hybrid quantum-classical model to perform image
classification on a subset of the Stanford Cars data set KrauseStarkDengFei-
Fei_3DRR2013 . Image classification is an ubiquitous problem in the automotive
industry, and can be used for tasks like sorting out parts with defects.
Supervised learning algorithms for classification have also been extensively
studied in quantum literature havlivcek2019supervised ; schuld2019quantum ;
schuld2020circuit ; rebentrost2014quantum , and it has been proven that there
exist specific learning tasks based on the discrete logarithm problem where a
separation between quantum and classical learners exists for classification
liu2021rigorous . While the separation in liu2021rigorous  is based on Shor’s
algorithm and therefore not expected to transfer to realistic learning tasks
as the car classification mentioned above, it motivates further experimental
study of quantum-enhanced models for classification on real-world data sets.

In combining PQCs and classical NNs into hybrid quantum-classical models, we
encounter a challenge in searching hyperparameter configurations that produce
performance gains in terms of model accuracy and training. Hyperparameters can
be considered values that are set for the model and do not change during the
training regime, and may include variables such as learning rate, decay rates,
choice of optimizer for the model, number of qubits or layer sizes. Often in
practice, these parameters are selected by experts based upon some a priori
knowledge and trial-and-error. This limits the search space, but in turn can
lead to producing a suboptimal model configuration.

Hyperparameter optimization is the process of automating the search for the
best set of hyperparameters, reducing the need for expert knowledge in
hyperparameter configurations for models, with an increase in computation
required to evaluate configurations of models in search of an optimum. In the
1990s, researchers reported performance gains leveraging a wrapper method,
which tuned parameters for specific models and data sets using best-first
search and cross validation KOHAVI1995304 . In more recent years, researchers
have proposed search algorithms using bandits Li2017 , which leverage early
stopping methods. Successive Halving algorithms such as the one introduced in
pmlr-v28-karnin13  and the parallelized version introduced in Li2018  allocate
more resources to more promising configurations. Sequential model-based
optimization leverages Bayesian optimization with an aggressive dual racing
mechanism, and also has shown performance improvements for hyperparameter
optimization Hutter2011 ; Lindauer_Hutter_2018 . Evolutionary and population-
based heuristics for black-box optimization have also achieved state-of-the-
art results when applied to hyperparameter optimization in numerous
competitions for black-box optimization vermetten2020sequential ; back1996 ;
awad2020squirrel . In recent years, a whole field has formed around automating
the process of finding optimal hyperparameters for machine learning models,
with some prime examples being neural architecture search elsken2019neural
and automated machine learning (AutoML) hutter2019automated . Automating the
search of hyperparameters in a quantum machine learning (QML) context has also
started to attract attention, and the authors of berganza2022towards  have
explored the first version of AutoQML.

Our contribution in this paper is not only to examine the performance gains of
hybrid quantum-classical models vs. purely classical, but also to investigate
whether quantum-enhanced or quantum-inspired methods may offer an advantage in
automating the search over the configuration space of the models. We show a
reduction in computational complexity in regard to expected run times and
evaluations for various configurations of models, the high cost of which
motivate this investigation. We investigate using the tensor train
decomposition for searching the hyperparameter space of the HQNN framed as a
global optimization problem as in globalOptimizationTensorTrains . This method
has been successful in optimizing models of social networks in
sergey2019tensor , and as a method of compression for deep neural networks
WANG2021320 .

## RESULTS

###  .1 Hyperparameter Optimization

[width=1]Fig1-HPO.pdf

Figure 1: The hyperparameter optimization problem description (a). The tabular
methods for hyperparameter optimization: the grid search algorithm (b) and the
tensor train algorithm (c-d).

The problem of hyperparameter optimization (HPO) is described schematically in
Fig. 1(a). Given a certain data set and a machine learning (ML) model, the
learning model demonstrates an accuracy $A(\bar{h})$ which depends on the
hyperparameters $\bar{h}$. To achieve the best possible model accuracy, one
has to optimize the hyperparameters. To perform the HPO, an unknown black-box
function $A(\bar{h})$ has to be explored. The exploration is an iterative
process, where at each iteration the HPO algorithm provides a set of
hyperparameters $\bar{h}$ and receives the corresponding model accuracy
$A(\bar{h})$. As a result of this iterative process, the HPO algorithm outputs
the best achieved performance $A(\bar{h}_{\mathrm{opt}})$ with the
corresponding hyperparameters $\bar{h}_{\mathrm{opt}}$.

The HPO could be organized in different ways. One of the standard methods for
HPO is a tabular method of grid search (GS), also known as a parameter sweep
(Fig. 1(b)). To illustrate how a grid search works, we have chosen two
hyperparameters: the learning rate ($h_{1}$) and the multiplicative factor of
learning rate ($h_{2}$). They are plotted along the $x$-axis and the $y$-axis,
respectively. The color on the contour shows the accuracy of the model
$A(h_{1},h_{2})$ with two given hyperparameters changing from light pink (the
lowest accuracy) to dark green (the highest accuracy). In the GS method, the
hyperparameter values are discretized, which results in a grid of values shown
as big dots. The GS algorithm goes through all the values from this grid with
the goal of finding the maximum accuracy. As one can see in this figure, there
are only three points at which this method can find a high accuracy with 25
iterations (shown as 25 points in Fig. 1(b)). This example shows that there
could be a better tabular HPO in terms of the best achievable accuracy and the
number of iterations used.

###  .2 Tensor train approach to hyperparameter optimization

Here, we propose a quantum-inspired approach to hyperparameter optimization
based on the tensor train (TT) programming. The TT approach was initially
introduced in the context of quantum many-body system analysis, e.g., for
finding a ground state with minimal energy of multi-particle Hamiltonians via
Density Matrix Renormalization Groups (DMRG) DMRG . In this approach, the
ground state is represented in the TT format, often referred to as the Matrix
Product State in physics MPS . We employ the TT representation (shown in Fig.
1(c)) in another way here, and use it for the hyperparameter optimization. As
one can see in Fig. 1(c), the TT is represented as a multiplication of
tensors, where an individual tensor is shown as a circle with the number of
“legs” that corresponds to the rank of the tensor. $h_{1}$ and $h_{d}$ circles
are the matrices of $n\times r$ dimension, and $\\{h_{i}\\}_{i={2}}^{i={d-1}}$
is a rank 3 tensor of dimensions $n\times r^{2}$. The two arrows in the Fig.
1(c) illustrate sweeps right and left along with the TT. This refers to the
algorithm described below. Leveraging the locality of the problem, i.e., a
small correlation between hyperparameters, we perform the black-box
optimization based on the cross-approximation technique applied for tensors
tt_cross ; tt_min .

Similar to the previously discussed GS method, we discretize the
hyperparameter space with TT optimization (TTO) and then consider a tensor
composed of scores that can be estimated by running an ML model with a
corresponding set of hyperparameters. However, compared to GS, the TT method
is dynamic, which means that the next set of evaluating points in the
hyperparameter space is chosen based on the knowledge accumulated during all
previous evaluations. With TTO we will not estimate all the scores
$A(\bar{h})$ available to the model. Instead of this, we will approximate
$A(\bar{h})$ via TT, referring to a limited number of tensor elements using
the cross-approximation method tt_cross . During the process, new sets of
hyperparameters for which the model needs to be evaluated are determined using
the MaxVol routine maxvol . The MaxVol routine is an algorithm that finds an
$r\times r$ submatrix of maximum volume, i.e., a square matrix with a maximum
determinant module in an $n\times r$ matrix.

Hyperparameters are changed in an iterative process, in which one is likely to
find a better accuracy $A(\bar{h})$ after each iteration, and thus find a good
set of hyperparameters. Notably, the TTO algorithm requires an estimate of
$\mathcal{O}(dnr^{2})$ elements and $\mathcal{O}(dnr^{3})$ of calculations,
where $d$ is the number of hyperparameters, $n$ is a number of discretization
points, and $r$ is a fixed rank. If one compares it with the GS algorithm,
which requires estimation of $\mathcal{O}(n^{d})$ elements, one is expected to
observe practical advantages, especially with a large number of
hyperparameters.

Algorithm 1 Tensor Train Optimization

1:Accuracy $A(\bar{h}_{\mathrm{opt}})=0$

2:$i_{\mathrm{swp}}=1$

3:$\mathrm{Core}=1$

4:$j_{\mathrm{swp}}=1$

5:Discretize each of $d$ hyperparameters with $n$ points

6:Randomly choose $r$ combinations of $(h_{2},h_{3},\ldots,h_{d})$

7:while $i_{\mathrm{swp}}\leq n_{\mathrm{swp}}$ do

8: while $j_{\mathrm{swp}}\leq 2$ do

9: while $\mathrm{Core}<d$ do

10: if $\mathrm{Core}==1$ then

11: Estimate $A(\overline{h})$ of $nr$ elements with all $n$ values of $h_{1}$

12: if $A(\bar{h}_{\mathrm{opt}})<A(\bar{h})$ then

13: $A(\bar{h}_{\mathrm{opt}})=A(\bar{h})$

14: end if

15: MaxVol

16: Fix corresponding $r$ values of $h_{1}$

17: else

18: Estimate $A(\bar{h})$ of $nr^{2}$ elements for fixed
$(h_{1},\ldots,h_{\mathrm{Core}-1})$ with all $n$ values of
$h_{\mathrm{Core}}$

19: if $A(\bar{h}_{\mathrm{opt}})<A(\bar{h})$ then

20: $A(\bar{h}_{\mathrm{opt}})=A(\bar{h})$

21: end if

22: MaxVol

23: Fix corresponding $r$ values of $(h_{1},\ldots,h_{\mathrm{Core}})$

24: end if

25: $\mathrm{Core}=\mathrm{Core}+1$

26: end while

27: Change index order $(h_{d},\ldots,h_{1})$

28: Relabel $(h_{1},\ldots,h_{d})$

29: $\mathrm{Core}=1$

30: $j_{\mathrm{swp}}=j_{\mathrm{swp}}+1$

31: end while

32: $j_{\mathrm{swp}}=1$

33: $i_{\mathrm{swp}}=i_{\mathrm{swp}}+1$

34:end while

The TTO algorithm for the HPO is presented as the Algorithm 1 pseudocode that
also corresponds to Fig. 1(d). The TTO algorithm can be described with $9$
steps:

  1. 1.

Suppose each of $d$ hyperparameters is defined on some interval
$h_{i}\in[h_{i}^{\mathrm{min}},h_{i}^{\mathrm{max}}]$, where $i\in[1,d]$. One
first discretizes each of $d$ hyperparameters by defining $n$ points

${\\{h_{i}(1),h_{i}(2),\ldots,h_{i}(n)\\}}_{i=1}^{i=d}.$

  2. 2.

Then, we need to choose the rank $r$. This choice is a trade-off between
computational time and accuracy, which respectively require a small and a
large rank.

  3. 3.

$r$ combinations of

${\\{h_{2}^{1}(j),h_{3}^{1}(j),\ldots,h_{d}^{1}(j)\\}}_{j=1}^{j=r}.$ (1)

are chosen.

  4. 4.

In the next three steps, we implement an iterative process called the “sweep
right”. The first step of this iterative process is related to the first TT
core evaluation:

     * •

The accuracy of $nr$ elements is estimated with all $n$ values of the first
hyperparameter ${\\{h_{1}(i_{1})\\}}_{i_{1}=1}^{i_{1}=n}$ and for the $r$
combinations of
${\\{h_{2}^{1}(j),h_{3}^{1}(j),\ldots,h_{d}^{1}(j)\\}}_{j=1}^{j=r}$:

$\begin{split}{\\{A(h_{1}(i_{1}),h_{2}^{1}(j),h_{3}^{1}(j),\ldots,}\\\
{h_{d}^{1}(j))\\}}_{j=1,i_{1}=1}^{j=r,i_{1}=n}.\end{split}$ (2)

     * •

In this matrix of size $n\times r$ we search for a submatrix with maximum
determinant module:

$\quad\quad{\\{A(h_{1}^{1}(i_{1}),h_{2}^{1}(j),h_{3}^{1}(j),h_{d}^{1}(j))\\}}_{j=1,i_{1}=1}^{j=r,i_{1}=r}.$
(3)

The corresponding $r$ values of the first hyperparameter are fixed
$\\{h_{1}^{1}(i_{1})\\}_{i_{1}=1}^{i_{1}=r}$.

  5. 5.

The next step of this iterative process is related to the second TT core
evaluation:

     * •

We fix $r$ values $\\{h_{1}^{1}(i_{1})\\}_{i_{1}=1}^{i_{1}=r}$ of the previous
step as well as $r$ combinations
${\\{h_{3}^{1}(j),h_{4}^{1}(j),\ldots,h_{d}^{1}(j)\\}}_{j=1}^{j=r}$ of the
third step. We, then, estimate the accuracy of the $nr^{2}$ elements with all
$n$ values of the second hyperparameter
${\\{h_{2}(i_{2})\\}}_{i_{2}=1}^{i_{2}=n}$:

$\begin{split}{\\{A(h_{1}^{1}(i_{1}),h_{2}(i_{2}),h_{3}^{1}(j),\ldots,}\\\
{h_{d}^{1}(j))\\}}_{j=1,i_{1}=1,i_{2}=1}^{j=r,i_{1}=r,i_{2}=n}`\end{split}$
(4)

     * •

Again, in this matrix of size $nr\times r$ we search for a submatrix with the
maximum determinant module:

$\begin{split}{\\{A((h_{1}^{2}(k),h_{2}^{2}(k)),h_{3}^{1}(j),\ldots,}\\\
{h_{d}^{1}(j))\\}}_{j=1,k=1}^{j=r,k=r}\end{split}$ (5)

$r$ combinations ${\\{(h_{1}^{2}(k),h_{2}^{2}(k))\\}}_{k=1}^{k=r}$ of the
first and the second hyperparameters are fixed.

  6. 6.

The $d-1$ TT core evaluation:

     * •

We fix $r$ combinations
${\\{(h_{1}^{d-2}(k),h_{2}^{d-2}(k),\ldots,h_{d-2}^{d-2}(k))\\}}_{k=1}^{k=r}$
of the $d-2$ TT core as well as $r$ combinations
${\\{h_{d}^{1}(j)\\}}_{j=1}^{j=r}$ of the third step. We, then, estimate the
accuracy of the $nr^{2}$ elements with all $n$ values of the
${\\{h_{d-1}(i_{d})\\}}_{i_{d}=1}^{i_{d}=n}$:

$\begin{split}{\\{A((h_{1}^{d-2}(k),\ldots,h_{d-2}^{d-2}(k)),}\\\
{h_{d-1}(i_{d-1}),h_{d}^{1}(j))\\}}_{k=1,i_{d-1}=1,j=1}^{k=r,i_{d-1}=n,j=r}\end{split}$
(6)

     * •

Again, in this matrix of size $nr\times r$ we search for a submatrix with the
maximum determinant module:

$\begin{split}{\\{A((h_{1}^{d-1}(k),h_{2}^{d-1}(k),\ldots,}\\\
{h_{d-1}^{d-1}(k)),h_{d}^{1}(j))\\}}_{k=1,j=1}^{k=r,j=r}\end{split}$ (7)

$r$ combinations of
${\\{(h_{1}^{d-1}(k),h_{2}^{d-1}(k),\ldots,h_{d-1}^{d-1}(k)\\}}_{k=1}^{k=r}$
hyperparameters are fixed.

The end of one “sweep right” is reached.

  7. 7.

Similar to step 3, we have $r$ combinations of hyperparameters, but they are
not random anymore. We next perform a similar procedure in the reverse
direction (from the last hyperparameter to the first). The process is called
the “sweep left”.

One first changes the index order:

$\quad{\\{(h_{1}^{d-1}(k),h_{2}^{d-1}(k),\ldots,h_{d-1}^{d-1}(k)\\}}_{k=1}^{k=r}\Longrightarrow\mathrm{relabel}$
${\\{(h_{d-1}^{d-1}(k),h_{d-2}^{d-1}(k),\ldots,h_{2}^{d-1}(k)\\}}_{j=1}^{j=r}$
(8)

And then, continues from the fourth step of the TTO algorithm.

  8. 8.

A combination of the “sweep right” and the “sweep left” is a full sweep. We do
$n_{\mathrm{swp}}$ full sweeps in this algorithm.

  9. 9.

During all the iterations, we record it if we estimate a new maximum score.

###  .3 Benchmarking HPO Methods

[width=1]Fig2-Benchmark.png

Figure 2: Tensor Train (TT) and Grid Search (GS): Expected Runtime in maximum
objective function evaluations vs. growth of problem dimension $d$.

In order to ascertain the solution quality in our proposed method for
hyperparameter optimization, we tested over three black-box objective
functions. These functions included the Schwefel, Fletcher-Powell, and Vincent
functions from the optproblems Python library OptProblems . We ran 100
randomly initialized trails and recorded average fitness and maximum number of
function evaluations in response to the change in the problem size $d$ for
each objective function. We compared grid search (GS) and tensor train (TT) -
both tabular methods for hyperparameter optimization. For tensor train and
grid search, we partitioned the hyperparameter ranges with 4 discrete points
per hyperparameter. For tensor train we set the rank parameter $r=2$.

Schwefel |  |
---|---|---
HPO Method | Average Fitness | $d$ | ER
TT | -541.76 | 3 | 32
GS | -541.76 | 3 | 64
TT | -1083.53 | 6 | 80
GS | -1083.53 | 6 | 4092
TT | -1805.89 | 10 | 144
GS | -1805.89 | 10 | 10000
Fletcher-Powell |  |
HPO Method | Average Fitness | $d$ | ER
TT | 5136.64 | 3 | 32
GS | 4113.78 | 3 | 64
TT | 23954.5 | 6 | 80
GS | 14295.2 | 6 | 4092
TT | 78101.4 | 10 | 144
GS | 36890.11 | 10 | 10000
Vincent |  |
HPO Method | Average Fitness | $d$ | ER
TT | -0.232 | 3 | 32
GS | -0.243 | 3 | 64
TT | -0.242 | 6 | 80
GS | -0.243 | 6 | 4092
TT | -0.241 | 10 | 144
GS | -0.243 | 10 | 10000
Table 1: Table of results comparing HPO methods for Schwefel, Fletcher-Powell,
and Vincent objective functions. Average fitness values and Expected Runtimes
(ER) in maximum function evaluations were calculated over 100 runs for varying
sizes of problem dimension $d$ (lower is better). Methods obtaining the best
average fitness are highlighted in bold, with ties broken by lower ER.

[width=1]Fig3-HQNN.pdf

Figure 3: Classical (a) and Hybrid (b) quantum neural network architectures.

###  .4 Car Classification with Hybrid Quantum Neural Networks

Computer vision and classification systems are ubiquitous within the mobility
and automotive industries. In this article, we investigate the car
classification problem using the Car data set Cars  provided by Stanford CS
Department. Examples of cars in the data set are shown in Fig. 3. The Stanford
Cars data set contains 16,185 images of 196 classes of cars. The data is split
into 8,144 training images and 8,041 testing images. The classes are typically
at the combination of Make, Model, Year, e.g., Volkswagen Golf Hatchback 1991
or Volkswagen Beetle Hatchback 2012. Since the images in this data set have
different sizes, we resized all images to 400 by 400 pixels. In addition, we
apply random rotations by maximum 15°, random horizontal flips, and
normalization to the training data. For testing data, only normalization has
been applied.

We use transfer learning to solve the car classification problem. Transfer
learning is a powerful method for training neural networks in which experience
in solving one problem helps in solving another problem Transfer_Learning . In
our case, the ResNet (Residual Neural Network) ResNet  is pretrained on the
ImageNet data set ImageNet , and is used as a base model. One can fix the
weights of the base model, but if the base model is not flexible enough, one
can “unfreeze” certain layers and make it trainable. Training deep networks is
challenging due to the vanishing gradient problem, but ResNet solves this
problem with so-called residual blocks: inputs are passed to the next layer in
the residual block. In this way, deeper layers can see information about the
input data. ResNet has established itself as a robust network architecture for
solving image classification problems. We dowloaded ResNet34 via PyTorch
PyTorch , where the number after the model name, 34, indicates the number of
layers in the network.

As shown in the Fig. 3(a), in the classical network after ResNet34 we add
three fully-connected layers. Each output neuron corresponds to a particular
class of the classification problem, e.g., Volkswagen Golf Hatchback 1991 or
Volkswagen Beetle Hatchback 2012. The output neuron with the largest value
determines the output class. Since the output from the ResNet34 is composed of
512 features, the first fully-connected layer consists of 512 input neurons
and a bias neuron and $n$ output features. The second fully-connected layer
connects $n$ input neurons and a bias neuron with $nq$ output features. The
value of $n$ and $q$ can vary, thus changing the number of weights in the
classical network. Since the network classifies $k$ classes in the general
case, the third fully-connected layer takes $nq$ neurons and a bias neuron as
input and feeds $k$ neurons as output.

In the hybrid analog as shown in Fig. 3(b) we replace the second fully-
connected layer with a quantum one. It is worth noting that the number of
qubits used for the efficient operation of the model is initially unknown. In
the quantum layer, the Hadamard transform is applied to each qubit, then the
input data is encoded into the angles of rotation along the $y$-axis. The
variational layer consists of the application of the CNOT gate and rotation
along $x$, $y$, $z$-axes. The number of variational layers can vary.
Accordingly, the number of weights in the hybrid network can also change. The
measurement is made in the $X$-basis. For each qubit, the local expectation
value of the $X$ operator is measured. This produces a classical output
vector, suitable for additional post-processing. Since the optimal number of
variational layers ($q$, depth of quantum circuit) and the optimal number of
qubits $n$ are not known in advance, we choose these values as
hyperparameters.

We use the cross-entropy as a loss function

$l=-\sum_{c=1}^{k}{y_{c}\log p_{c}}$ (9)

where $p_{c}$ is the prediction probability, $y_{c}$ is 0 or 1, determining
respectively if the image belongs to the prediction class, and $k$ is the
number of classes. We run our model for 10 epochs and apply weight decay and
gradient clipping to prevent interference from large gradient or weight
values. We use the Adam optimizer Adam ; kingma2014adam  and reduce the
learning rate after several epochs. There is no one-size-fits-all rule of how
to choose a learning rate. Moreover, in most cases, dynamic control of the
learning rate of a neural network can significantly improve the efficiency of
the backpropagation algorithm. For these reasons, we choose the initial
learning rate, the period of learning rate decay, and the multiplicative
factor of the learning rate decay as hyperparameters. In total, together with
number of variational layers and number of qubits, we optimize five
hyperparameters presented in Table 2 to improve the accuracy of solving the
problem of car classification.

Hyperparameter |  Label |  Range |  Hybrid HPO values |  Classical HPO values
---|---|---|---|---
number of qubits, number of neurons |  $n$ |  $4-16$ |  $13$ |  $5$
depth of quantum circuit |  $q$ |  $1-5$ |  $4$ |  $\times$
number of neurons |  $nq$ |  $4-80$ |  $\times$ |  $80$
initial learning rate |  $\alpha_{0}$ |  $1-10\times 10^{-4}$ |  $5\times 10^{-4}$ |  $5\times 10^{-4}$
step of learning rate |  $\alpha_{\delta}$ |  $1-8$ |  $8$ |  $5$
multiplicative factor of learning rate decay |  $\alpha_{r}$ |  $0.1-0.2$ |  $0.1$ |  $0.2$
Table 2: The table shows which hyperparameters are being optimized, their
labels, limits of change, and the best values found during HPO.

[width=1]Fig4-Results.pdf

Figure 4: (a) Dependence of accuracy on the number of iterations HPO. TTO for
the hybrid model found a set of hyperparameters that gives an accuracy of
0.852 after 6 iterations, 0.977 after 18 iterations, for the classical model
found 0.977 after 6 iterations. Grid search for the hybrid model found a set
of hyperparameters that gives an accuracy of 0.989 after 75 iterations, for
the classical model found 0.920 after 75 iterations. (b) Dependence of
accuracy on the number of epochs with the found optimal set of
hyperparameters.

###  .5 Simulation Results

We next perform a simulation of the hybrid quantum residual neural network
described in the previous section. The simulation is compared to its classical
analog, the residual neural network, in a test car classification task.
Because of the limited number of qubits available and computational time
constraints, we used a classification between two classes, Volkswagen Golf
Hatchback 1991 and Volkswagen Beetle Hatchback 2012, to compare the classical
and hybrid networks fairly. In total, we used 88 testing images and 89
training images. Both the hybrid quantum HQNN model, and the classical NN
model, were used together with the GS and TTO methods for hyperparameter
optimization. All machine learning simulations were carried out in the QMware
cloud, on which the classical part was implemented with the PyTorch framework,
and the quantum part was implemented with the $<$basiq$>$ SDK QMWare . The
results of the simulations are shown in Fig. 4.

Fig. 4(a) shows the dependence of accuracy on the number of HPO iterations on
the test data, where one iteration of HPO is one run of the model. Green color
shows the dependence of accuracy on the number of iterations for the HQNN,
blue color shows for the classical NN. As one can see from Fig. 4(a), TTO
works more efficiently than GS and in fewer iterations finds hyperparameters
that give an accuracy above 0.9. HQNN with TTO (marked with green crosses)
finds a set of hyperparameters that yields 97.7% accuracy over 18 iterations.
As for the GS (marked solid green line), it took 44 iterations to pass the
threshold of 98% accuracy.

TTO finds in 6 iterations a set of hyperparameters for the classical NN, which
gives an accuracy of 97.7%, which is the same as the accuracy given by the set
of hyperparameters for the HQNN that found in 18 iterations. As for the GS, it
is clear that the optimization for the HQNN works more efficiently than for
the classical one. And the optimization of the HQNN requires fewer iterations
to achieve higher accuracy compared to the optimization of the classical NN. A
possible reason is that a quantum layer with a relatively large number of
qubits and a greater depth works better than its classical counterpart.

[width=0.6]Fig5-Result-Cars.pdf

Figure 5: Examples of test car images that were correctly classified by the
hybrid quantum residual neural network.

The best values found during HPO are displayed in Table 2. The quantum circuit
corresponding to the optimal set of hyperparameters has 52 variational
parameters, leading to a total of 6749 weights in the HQNN. In the classical
NN there are 9730 weights. Therefore, there are significantly fewer weights in
a HQNN compared to a classical NN. Nevertheless, as can be seen from the Fig.
4(b), the HQNN, with the hyperparameters found using the GS, reaches the
highest overall accuracy (98.9%). The Fig. 5 shows examples of car images that
were classified correctly by the HQNN model. The HQNN with an optimized set of
hyperparameters achieved an accuracy of 0.989.

## DISCUSSION

We introduced two new ML developments to image recognition. First, we
presented a quantum-inspired method of tensor train decomposition for choosing
ML model hyperparameters. This decomposition enabled us to optimize
hyperparameters similar to other tabular search methods, e.g., grid search,
but required only $\mathcal{O}(dnr^{2})$ hyperparameter choices instead of
$\mathcal{O}(n^{d})$ in the grid search method. We verified this method over
various black box functions and found that the tensor train method achieved
comparable results in average fitness, with a reduced expected run time for
most of the test functions compared to grid search. This indicates that this
method may be useful for high dimensional hyperparameter searches for
expensive black-box functions. Future work could investigate using this method
in combination with local search heuristic, where the tensor train optimizer
performs a sweep over a larger search space within a budget, and seeds another
optimization routine for a local search around this region. This method could
also be applied to the $B/n$ problem for successive halving algorithm by
decomposing the search space to find the optimal ratio of budget $B$ over
configurations $n$. Future work could investigate these applications in more
detail.

Second, we presented a hybrid quantum neural network model for supervised
learning. The hybrid model consisted of the combination of ResNet34 and a
quantum circuit part, whose size and depth became the hyperparameters. The
size and flexibility of the hybrid ML model allowed us to apply it to car
image classification. The hybrid ML model with GS showed an accuracy of 0.989
after 75 iterations in our binary classification tests with images of
Volkswagen Golf Hatchback 1991 and Volkswagen Beetle Hatchback 2012. This
accuracy was better than of a comparable classical ML model with GS showed an
accuracy of 0.920 after 75 iterations. In the same test, the hybrid ML model
with TTO showed an accuracy of 0.977 after 18 iterations, whereas the
comparable classical ML model with TTO, which showed the same accuracy of
0.977 after 6 iterations. Our developments provide new ways of using quantum
and quantum-inspired methods in practical industry problems. In future
research, exploring the sample complexity of the hybrid quantum model is of
importance, in addition to generalization bounds of the quantum models similar
to research in Ref. caro2021generalization . Future work could also entail
investigating state-of-the-art improvements in hyperparameter optimization for
classical and quantum-hybrid neural networks and other machine learning models
by leveraging quantum-inspired or quantum-enhanced methods.

## References

  * (1) Mark W Johnson, Mohammad HS Amin, Suzanne Gildert, Trevor Lanting, Firas Hamze, Neil Dickson, Richard Harris, Andrew J Berkley, Jan Johansson, Paul Bunyk, et al.  Quantum annealing with manufactured spins.  Nature, 473(7346):194–198, 2011.
  * (2) Florian Neukart, Gabriele Compostella, Christian Seidel, David Von Dollen, Sheir Yarkoni, and Bob Parney.  Traffic flow optimization using a quantum annealer.  Frontiers in Information and Communication Technologies, 4:29, 2017\.
  * (3) Arpit Mehta, Murad Muradi, and Selam Woldetsadick.  Quantum annealing based optimization of robotic movement in manufacturing.  In International Workshop on Quantum Technology and Optimization Problems, pages 136–144. Springer, 2019.
  * (4) Masayuki Ohzeki, Akira Miki, Masamichi J Miyama, and Masayoshi Terabe.  Control of automated guided vehicles without collision by quantum annealer and digital devices.  Frontiers in Computer Science, 1:9, 2019.
  * (5) Sheir Yarkoni, Alex Alekseyenko, Michael Streif, David Von Dollen, Florian Neukart, and Thomas Bäck.  Multi-car paint shop optimization with quantum annealing.  In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 35–41. IEEE, 2021.
  * (6) Michael Streif, Florian Neukart, and Martin Leib.  Solving quantum chemistry problems with a d-wave quantum annealer.  In International Workshop on Quantum Technology and Optimization Problems, pages 111–122. Springer, 2019.
  * (7) Rongxin Xia, Teng Bian, and Sabre Kais.  Electronic structure calculations and the ising hamiltonian.  The Journal of Physical Chemistry B, 122(13):3384–3395, 2017.
  * (8) Tobias Stollenwerk, Bryan O’Gorman, Davide Venturelli, Salvatore Mandra, Olga Rodionova, Hokkwan Ng, Banavar Sridhar, Eleanor Gilbert Rieffel, and Rupak Biswas.  Quantum annealing applied to de-conflicting optimal trajectories for air traffic management.  IEEE Transactions on Intelligent Transportation Systems, 21(1):285–297, 2019.
  * (9) Sebastian Feld, Christoph Roch, Thomas Gabor, Christian Seidel, Florian Neukart, Isabella Galter, Wolfgang Mauerer, and Claudia Linnhoff-Popien.  A hybrid solution method for the capacitated vehicle routing problem using a quantum annealer.  Frontiers in Information and Communication Technologies, 6:13, 2019\.
  * (10) Erica Grant, Travis S Humble, and Benjamin Stump.  Benchmarking quantum annealing controls with portfolio optimization.  Physical Review Applied, 15(1):014012, 2021.
  * (11) Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al.  Quantum supremacy using a programmable superconducting processor.  Nature, 574(7779):505–510, 2019.
  * (12) Michael Streif, Sheir Yarkoni, Andrea Skolik, Florian Neukart, and Martin Leib.  Beating classical heuristics for the binary paint shop problem with the quantum approximate optimization algorithm.  Physical Review A, 104(1):012403, 2021.
  * (13) Michael Streif and Martin Leib.  Training the quantum approximate optimization algorithm without access to a quantum processing unit.  Quantum Science and Technology, 5(3):034008, 2020.
  * (14) David Amaro, Matthias Rosenkranz, Nathan Fitzpatrick, Koji Hirano, and Mattia Fiorentini.  A case study of variational quantum algorithms for a job shop scheduling problem.  arXiv preprint arXiv:2109.03745, 2021.
  * (15) Constantin Dalyac, Loïc Henriet, Emmanuel Jeandel, Wolfgang Lechner, Simon Perdrix, Marc Porcheron, and Margarita Veshchezerova.  Qualifying quantum approaches for hard industrial optimization problems. a case study in the field of smart-charging of electric vehicles.  EPJ Quantum Technology, 8(1):12, 2021.
  * (16) Andre Luckow, Johannes Klepsch, and Josef Pichlmeier.  Quantum computing: Towards industry reference problems.  arXiv preprint arXiv:2103.07433, 2021.
  * (17) Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, David A Buell, et al.  Hartree-fock on a superconducting qubit quantum computer.  Science, 369(6507):1084–1089, 2020.
  * (18) Fionn D Malone, Robert M Parrish, Alicia R Welden, Thomas Fox, Matthias Degroote, Elica Kyoseva, Nikolaj Moll, Raffaele Santagati, and Michael Streif.  Towards the simulation of large scale protein-ligand interactions on nisq-era quantum computers.  arXiv preprint arXiv:2110.01589, 2021.
  * (19) Manuel S Rudolph, Ntwali Bashige Toussaint, Amara Katabarwa, Sonika Johri, Borja Peropadre, and Alejandro Perdomo-Ortiz.  Generation of high-resolution handwritten digits with an ion-trap quantum computer.  arXiv preprint arXiv:2012.03924, 2020.
  * (20) Andrea Skolik, Jarrod R McClean, Masoud Mohseni, Patrick van der Smagt, and Martin Leib.  Layerwise learning for quantum neural networks.  Quantum Machine Intelligence, 3(1):1–11, 2021.
  * (21) Andrea Skolik, Sofiene Jerbi, and Vedran Dunjko.  Quantum agents in the gym: a variational quantum algorithm for deep q-learning.  arXiv preprint arXiv:2103.15084, 2021.
  * (22) Evan Peters, Joao Caldeira, Alan Ho, Stefan Leichenauer, Masoud Mohseni, Hartmut Neven, Panagiotis Spentzouris, Doug Strain, and Gabriel N Perdue.  Machine learning of high dimensional data on a noisy quantum processor.  arXiv preprint arXiv:2101.09581, 2021.
  * (23) Javier Alcazar, Vicente Leyton-Ortega, and Alejandro Perdomo-Ortiz.  Classical versus quantum models in machine learning: insights from a finance application.  Machine Learning: Science and Technology, 1(3):035003, 2020.
  * (24) Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al.  Variational quantum algorithms.  Nature Reviews Physics, pages 1–20, 2021.
  * (25) Shi-Xin Zhang, Zhou-Quan Wan, Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao.  Variational quantum-neural hybrid eigensolver.  arXiv preprint arXiv:2106.05105, 2021.
  * (26) Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, and Nathan Killoran.  Transfer learning in hybrid classical-quantum neural networks.  Quantum, 4:340, 2020.
  * (27) Chen Zhao and Xiao-Shan Gao.  Qdnn: Dnn with quantum neural network layers.  arXiv preprint arXiv:1912.12660, 2019.
  * (28) Tong Dou, Kaiwei Wang, Zhenwei Zhou, Shilu Yan, and Wei Cui.  An unsupervised feature learning for quantum-classical convolutional network with applications to fault detection.  In 2021 40th Chinese Control Conference (CCC), pages 6351–6355. IEEE, 2021.
  * (29) Alessandro Sebastianelli, Daniela Alessandra Zaidenberg, Dario Spiller, Bertrand Le Saux, and Silvia Liberata Ullo.  On circuit-based hybrid quantum neural networks for remote sensing imagery classification.  IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 15:565–580, 2021.
  * (30) Sayantan Pramanik, M Girish Chandra, CV Sridhar, Aniket Kulkarni, Prabin Sahoo, Vishwa Chethan DV, Hrishikesh Sharma, Ashutosh Paliwal, Vidyut Navelkar, Sudhakara Poojary, et al.  A quantum-classical hybrid method for image classification and segmentation.  arXiv preprint arXiv:2109.14431, 2021.
  * (31) Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei.  3d object representations for fine-grained categorization.  In 4th International IEEE Workshop on 3D Representation and Recognition (3dRR-13), Sydney, Australia, 2013.
  * (32) Vojtěch Havlíček, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta.  Supervised learning with quantum-enhanced feature spaces.  Nature, 567(7747):209–212, 2019.
  * (33) Maria Schuld and Nathan Killoran.  Quantum machine learning in feature hilbert spaces.  Physical Review Letters, 122(4):040504, 2019.
  * (34) Maria Schuld, Alex Bocharov, Krysta M Svore, and Nathan Wiebe.  Circuit-centric quantum classifiers.  Physical Review A, 101(3):032308, 2020.
  * (35) Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd.  Quantum support vector machine for big data classification.  Physical Review Letters, 113(13):130503, 2014.
  * (36) Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme.  A rigorous and robust quantum speed-up in supervised machine learning.  Nature Physics, 17(9):1013–1017, 2021.
  * (37) Ron Kohavi and George H. John.  Automatic parameter selection by minimizing estimated error.  In Armand Prieditis and Stuart Russell, editors, Machine Learning Proceedings 1995, pages 304–312. Morgan Kaufmann, San Francisco (CA), 1995.
  * (38) Lisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar.  Hyperband: A novel bandit-based approach to hyperparameter optimization.  The Journal of Machine Learning Research, 18(1):6765–6816, 2017\.
  * (39) Zohar Karnin, Tomer Koren, and Oren Somekh.  Almost optimal exploration in multi-armed bandits.  In Sanjoy Dasgupta and David McAllester, editors, Proceedings of the 30th International Conference on Machine Learning, volume 28 of Proceedings of Machine Learning Research, pages 1238–1246, Atlanta, Georgia, USA, 2013. PMLR.
  * (40) Liam Li, Kevin G. Jamieson, Afshin Rostamizadeh, Ekaterina Gonina, Moritz Hardt, Benjamin Recht, and Ameet Talwalkar.  Massively parallel hyperparameter tuning.  arXiv preprint arXiv:1810.05934, 2018.
  * (41) Frank Hutter, Holger H. Hoos, and Kevin Leyton-Brown.  Sequential model-based optimization for general algorithm configuration.  In Carlos A. Coello Coello, editor, Learning and Intelligent Optimization, pages 507–523. Springer Berlin Heidelberg, 2011.
  * (42) Marius Lindauer and Frank Hutter.  Warmstarting of model-based algorithm configuration.  Proceedings of the AAAI Conference on Artificial Intelligence, 32(1), 2018.
  * (43) Diederick Vermetten, Hao Wang, Carola Doerr, and Thomas Bäck.  Sequential vs. integrated algorithm selection and configuration: A case study for the modular cma-es.  arXiv preprint arXiv:1912.05899, 2020.
  * (44) Thomas Bäck.  Evolutionary Algorithms in Theory and Practice: Evolution Strategies, Evolutionary Programming, Genetic Algorithms.  Oxford University Press, Inc., USA, 1996.
  * (45) Noor Awad, Gresa Shala, Difan Deng, Neeratyoy Mallik, Matthias Feurer, Katharina Eggensperger, Andre’ Biedenkapp, Diederick Vermetten, Hao Wang, Carola Doerr, Marius Lindauer, and Frank Hutter.  Squirrel: A switching hyperparameter optimizer.  arXiv preprint arXiv:2012.08180, 2020.
  * (46) Thomas Elsken, Jan Hendrik Metzen, and Frank Hutter.  Neural architecture search: A survey.  The Journal of Machine Learning Research, 20(1):1997–2017, 2019\.
  * (47) Frank Hutter, Lars Kotthoff, and Joaquin Vanschoren.  Automated Machine Learning: Methods, Systems, Challenges.  Springer Nature, 2019.
  * (48) Raúl Berganza Gómez, Corey O’Meara, Giorgio Cortiana, Christian B. Mendl, and Juan Bernabé-Moreno.  Towards autoqml: A cloud-based automated circuit architecture search framework.  arXiv preprint arXiv:2202.08024, 2022.
  * (49) Dmitry Zheltkov and Alexander Osinsky.  Global optimization algorithms using tensor trains.  Lecture Notes in Computer Science, 11958:197–202, 2020.
  * (50) Sergey Kabanikhin, Olga Krivorotko, Shuhua Zhang, Victoriya Kashtanova, and Yufang Wang.  Tensor train optimization for mathematical model of social networks.  arXiv preprint arXiv:1906.05246, 2019.
  * (51) Dingheng Wang, Guangshe Zhao, Hengnu Chen, Zhexian Liu, Lei Deng, and Guoqi Li.  Nonlinear tensor train format for deep neural network compression.  Neural Networks, 144:320–333, 2021.
  * (52) Steven R. White.  Density matrix formulation for quantum renormalization groups.  Physical Review Letters, 69:2863–2866, 1992.
  * (53) J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete.  Matrix product states and projected entangled pair states: Concepts, symmetries, theorems.  Reviews of Modern Physics, 93(4), 2021.
  * (54) Ivan Oseledets and Eugene Tyrtyshnikov.  Tt-cross approximation for multidimensional arrays.  Linear Algebra and its Applications, 432(1):70–88, 2010.
  * (55) Dmitry Zheltkov and Eugene Tyrtyshnikov.  Global optimization based on tt-decomposition.  Russian Journal of Numerical Analysis and Mathematical Modelling, 35(4):247–261, 2020.
  * (56) Sergei Goreinov, Ivan Oseledets, D. Savostyanov, E. Tyrtyshnikov, and Nickolai Zamarashkin.  How to find a good submatrix.  Matrix Methods: Theory, Algorithms and Applications, 2010.
  * (57) optproblems.  https://pypi.org/project/optproblems/, 2022.
  * (58) Jonathan Krause, Michael Stark, Jia Deng, and Li Fei-Fei.  3d object representations for fine-grained categorization.  In 4th International IEEE Workshop on 3D Representation and Recognition (3dRR-13), Sydney, Australia, 2013.
  * (59) Behnam Neyshabur, Hanie Sedghi, and Chiyuan Zhang.  What is being transferred in transfer learning?  arXiv preprint arXiv:2008.11687, 2020.
  * (60) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  arXiv preprint arXiv:1512.03385, 2015.
  * (61) Imagenet dataset.  https://image-net.org/, 2022.
  * (62) PyTorch.  https://pytorch.org/, 2022.
  * (63) Adam optimizer.  https://pytorch.org/docs/stable/generated/torch.optim.Adam.html, 2022.
  * (64) Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * (65) QMware, The first global quantum cloud.  https://qm-ware.com, 2022.
  * (66) Matthias C Caro, Hsin-Yuan Huang, M Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J Coles.  Generalization in quantum machine learning from few training data.  arXiv preprint arXiv:2111.05292, 2021.


# Entanglement Entropy in Horndeski Gravity

F. F. Santos1<EMAIL_ADDRESS>1Instituto de Física, Universidade
Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, Brazil.

(August 27, 2024)

###### Abstract

In this work, we explore the holographic entanglement entropy with an infinite
strip region of the boundary in Horndeski gravity. In our prescription we
consider the spherically and planar topologies black holes in the AdS4/CFT3
scenario. In such framework, we show the behavior of the entanglement entropy
in function of the Horndeski parameters. Such parameters modify the
information store of subsystem A, especially when the parameter $\gamma$
increases the information about the subsystem will also increase or decrease
when it decreases. Thus, with this scheme we compute the “first law of
entanglement thermodynamics” in Horndeski gravity and we show that a very
small subsystem obeys the analogous property of the first law of
thermodynamics if we excite the system.

##  I Introduction

In recent years, the scenario of entanglement entropy have call attention in
special the works of Takayanagi:2012kg ; Bhattacharya:2019zkb ;
Bhattacharya:2017gzt ; Bhattacharya:2012mi ; Calabrese:2004eu ; Holzhey:1994we
; Ryu:2006ef ; Ryu:2006bv ; Susskind:1994sm ; Chaturvedi:2016kbk ;
Tonni:2010pv ; Mansoori:2015sit ; Caputa:2013lfa ; Blanco:2013joa ;
Susskind:2021esx ; Kay:2016kbi ; Park:2015afa ; He:2014lfa  where the $S_{A}$
as the entropy for an observer accessible only to the $A$ subsystem and it
cannot receive any $B$ signal. Such subsystem $B$ is analogous to the interior
of the black hole horizon. For an observer who is sitting on $A$, that is,
outside the horizon. On the other hand, because this analogy is not correct,
the quantum correction involving a loop for the entropy BH in the presence of
fields of the matter is known to be equal to the entanglement entropy
Susskind:1994sm . Thus, we have that this interesting relation provides us
with an important tip to find the holographic dual of the entanglement
entropy. In the context of the AdS3/CFT2 correspondence presented by
Takayanagi:2012kg ; Calabrese:2004eu ; Ryu:2006ef  is possible to calculate
the entropy SA in a CFT2, holographically — See Fig. 1. This entropy is
calculated as follows:

$\displaystyle
S_{A}=Min_{\Sigma_{A}}\left[\frac{Area(\Sigma_{A})}{4G_{N}}\right]$ (1)

Note that $\Sigma_{A}$ represents a two-dimensional surface, that is, four-
dimensional in AdS3 satisfying $\partial\Sigma_{A}=\partial A$ since
$\Sigma_{A}$ is a counterpart to $A$. In addition to these considerations, we
have that (1) is obtained for all surfaces, $\Sigma_{A}$, which is a minimal
surface. The equation (1) is applied to any static configuration. Besides, as
we know, the minimum surface surrounding the area is well defined in the
static case, thus making it possible to consider Euclidean AdS space
equivalently.

Figure 1: The calculation of holographic entanglement entropy.

It has an interesting feature that is a striking similarity to the Bekenstein-
Hawking (BH) entropy of black holes Takayanagi:2012kg ; Calabrese:2004eu ;
Ryu:2006ef ; Ryu:2006bv ; Susskind:1994sm . Besides, as we know, the minimum
surface surrounding the area is well defined in the static case, thus making
it possible to consider Euclidean AdS space equivalently. Motivated by the
applications of Chaturvedi:2016kbk , we propose a scenario of entanglement
entropy in Horndeski gravity where is considered an analytically the
entanglement entropy of the subsystem $A$ in the ($2+1$)-dimensional boundary
field theory. In our case we consider the duality AdS4/CFT3, where on the
gravity side, we have a planar black hole solutions in Horndeski gravity and
with this solution we extract the length and the area integral to the
subsystem $A$.

The main motivation to investigate the play role of Horndeski gravity in the
entanglement entropy scenario is due the recent investigations of the AdS/CFT
correspondence in this gravity Santos:2020egn ; Brito:2019ose ; Santos:2021orr
; Jiang:2017imk ; Baggioli:2017ojd ; Liu:2018hzo ; Li:2018kqp ; Li:2018rgn ;
Feng:2015oea ; Caceres:2017lbr ; Hajian:2020dcq ; MohammadiMozaffar:2016vpf .
Besides, beyond to the classes of boundary field theories that were studied by
Ryu and Takayanagi Ryu:2006ef ; Ryu:2006bv  other classes were proposed using
this conjecture, such as AdS black holes with dual charge in the bulk
Tonni:2010pv ; Mansoori:2015sit ; Caputa:2013lfa . But in our case, we compute
the entanglement entropy in AdS4/CFT3 within Horndeski gravity for spherically
and planar black holes. For this boundary field theory at finite temperature,
we address the issues to related to entanglement thermodynamics within
Horndeski gravity in our setup, for more discussion about this see
Blanco:2013joa ; Park:2015afa ; He:2014lfa ; Chaturvedi:2016kbk . For the
avaible the excited states we compute the stress-energy tensor of boundary
field theory in Horndeski gravity following the prescription of Santos:2021orr
; Balasubramanian:1999re .

This work is summarized as follows. In Sec. II we address the issue of finding
black hole solutions in Horndeski gravity. In Sec. III, we present the
entanglement entropy. In Sec. IV we compute the entanglement thermodynamics in
Horndeski gravity. In section Sec. IV, we present our conclusions.

##  II Black hole solutions in Horndeski gravity

In this section we address the issue of finding black hole solutions in
Horndeski gravity Santos:2020egn ; Brito:2019ose ; Santos:2021orr ;
Horndeski:1974wa ; Santos:2020xox ; Cisterna:2014nua ; Bravo-Gaete:2014haa ;
Anabalon:2013oea . Black holes in Horndeski’s theory have been previously
studied in Santos:2020egn ; Brito:2019ose ; Santos:2021orr ; Santos:2020xox ;
Cisterna:2014nua ; Anabalon:2013oea . The Horndeski Lagrangian is given by

$\displaystyle\mathcal{L}_{H}=\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}+\mathcal{L}_{5},$
(2) $\displaystyle\mathcal{L}_{2}=G_{2}(X,\phi),$ (3)
$\displaystyle\mathcal{L}_{3}=-G_{3}(X,\phi)\Box\phi,$ (4)
$\displaystyle\mathcal{L}_{4}=G_{4}(X,\phi)R+\partial_{X}G_{4}(X,\phi)\delta^{\mu\nu}_{\alpha\beta}\nabla^{\alpha}_{\mu}\phi\nabla^{\beta}_{\nu}\phi,$
(5)
$\displaystyle\mathcal{L}_{5}=G_{5}(X,\phi)G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi$
$\displaystyle-\frac{1}{6}\partial_{X}G_{5}(X,\phi)\delta^{\mu\nu\rho}_{\alpha\beta\gamma}\nabla^{\alpha}_{\mu}\phi\nabla^{\beta}_{\nu}\phi\nabla^{\gamma}_{\rho}\phi,$
(6)

where $X\equiv-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\nu}\phi$. Furthermore, an
interesting special truncation of this theory was presented by
Charmousis:2011bf ; Charmousis:2011ea ; Starobinsky:2016kua ; Bruneton:2012zk
,where the idea is realize a constrain the coefficients $G_{k}(X,\phi)$.
Through this truncation and considering the non-minimal kinetic coupling, we
have

$\displaystyle I[g_{\mu\nu},\phi]=\int{\sqrt{-g}d^{4}x\mathcal{L}}.$ (7)
$\displaystyle\mathcal{L}=\kappa(R-2\Lambda)-\frac{1}{2}(\alpha
g_{\mu\nu}-\gamma G_{\mu\nu})\nabla^{\mu}\phi\nabla^{\nu}\phi$

Here the action (7) $\kappa=(16\pi G)^{-1}$. Such action have a non-minimal
scalar-tensor coupling and we can define a new field
$\phi^{{}^{\prime}}\equiv\psi$. The field has a dimension of $(mass)^{2}$
controlled by the parameters $\alpha$ and $\gamma$ where $\alpha$ is
dimensionless and $\gamma$ has a dimension of $(mass)^{-2}$. The equations of
motion are:

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{1}{2\kappa}T_{\mu\nu},$ (8)

where $T_{\mu\nu}=\alpha T^{(1)}_{\mu\nu}+\gamma T^{(2)}_{\mu\nu}$. The
energy-momentum tensors $T^{(1)}_{\mu\nu}$ and $T^{(2)}_{\mu\nu}$ take the
following form

$\displaystyle
T^{(1)}_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi$
$\displaystyle T^{(2)}_{\mu\nu}=\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi
R-2\nabla_{\lambda}\phi\nabla_{(\mu}\phi
R^{\lambda}_{\nu)}-\nabla^{\lambda}\phi\nabla^{\rho}\phi
R_{\mu\lambda\nu\rho}$ $\displaystyle-
g_{\mu\nu}\left[-\frac{1}{2}(\nabla^{\lambda}\nabla^{\rho}\phi)(\nabla_{\lambda}\nabla_{\rho}\phi)+\frac{1}{2}(\Box\phi)^{2}-(\nabla_{\lambda}\phi\nabla_{\rho}\phi)R^{\lambda\rho}\right]$
$\displaystyle-(\nabla_{\mu}\nabla^{\lambda}\phi)(\nabla_{\nu}\nabla_{\lambda}\phi)+(\nabla_{\mu}\nabla_{\nu}\phi)\Box\phi+\frac{1}{2}G_{\mu\nu}(\nabla\phi)^{2}.$
(9)

And the scalar field equation is given by

$\nabla_{\mu}[(\alpha g^{\mu\nu}-\gamma G^{\mu\nu})\nabla_{\nu}\phi]=0.$ (10)

In our case for Einstein-Horndeski gravity, we consider the following Ansatz
for a four-dimensional black hole of the form

$ds^{2}=\mathcal{R}^{2}\left(-r^{2}f(r)dt^{2}+r^{2}(dx^{2}+dy^{2})+\frac{dr^{2}}{r^{2}f(r)}\right).$
(11)

Now, following the results of Santos:2020egn ; Brito:2019ose ; Santos:2021orr
; Santos:2020xox ; Cisterna:2014nua ; Anabalon:2013oea , we can find the black
hole solution through the imposing of the radial component with the conserved
current that vanishes identically without restricting the radial dependence of
the scalar field Hui:2012qt ; Bravo-Gaete:2013dca ; Babichev:2013cya :

$\alpha g_{rr}-\gamma G_{rr}=0.$ (12)

Taking $\phi^{\prime}(r)\equiv\psi(r)$ and we can easily note that this
condition annihilates $\psi^{2}(r)$ regardless of its behavior at the horizon.
Now, using the equation (12) the metric function $f(r)$ can be found as
following:

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha\mathcal{R}^{2}}{3\gamma}-\left(\frac{r_{h}}{r}\right)^{3},$
(13) $\displaystyle\psi^{2}(r)$ $\displaystyle=$
$\displaystyle-\frac{2\mathcal{R}^{2}\kappa(\alpha+\gamma\Lambda)}{\alpha\gamma
r^{2}f(r)},$ (14)

The equation of motion (8) are satisfied by these equations. The solution (13)
corresponds to black hole solution for asymptotically AdS4 spacetime
Anabalon:2013oea . Beyond, through $\Lambda=-3/\mathcal{R}^{2}$ and
considering the analyses of Santos:2020egn ; Brito:2019ose ; Santos:2021orr
in our prescription, we can write the black hole solution

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{\gamma\Lambda}-\left(\frac{r_{h}}{r}\right)^{3},$
(15) $\displaystyle\psi^{2}(r)$ $\displaystyle=$
$\displaystyle\frac{6\kappa(\alpha+\gamma\Lambda)}{\alpha\gamma\Lambda
r^{2}f(r)},$ (16)

we can note that the parameters are defined in the range
$-\infty<\alpha/\gamma\Lambda\leq-1$, with $\alpha,\gamma<0$, or
$-1\leq\alpha/\gamma\Lambda<0$, with $\alpha,\gamma>0$. Furthermore, we have
that the surface located at $r=r_{h}$ is infinitely shifted to red in relation
to an asymptotic observer. On the other hand, we can see by looking at the
equation (16), which $(\alpha+\gamma\Lambda)>0$, indicates ghost freedom.
Thus, stability requires that $(\alpha+\gamma\Lambda)$ is not negative, this
leads to an interval of the form $-\infty<\gamma\leq\alpha/(-\Lambda)$. Thus,
following these transformation in the metric (11) with solution (15), we can
write:

$\displaystyle
f(r)\to-\frac{\alpha}{\gamma\Lambda}{f}(r),r_{h}^{3}\to-\frac{\alpha}{\gamma\Lambda}\;r_{h}^{3},$
$\displaystyle\mathcal{R}\to\left(-\frac{{\alpha}}{\gamma\Lambda}\right)^{1/2}\mathcal{R},t\to-\frac{\gamma\Lambda}{\alpha}t,$
$\displaystyle
x\to\left(-\frac{{\gamma\Lambda}}{\alpha}\right)^{1/2}x,y\to\left(-\frac{{\gamma\Lambda}}{\alpha}\right)^{1/2}y,$
(17)

in order to put the black hole solution in the standard form

$\displaystyle{f}(r)$ $\displaystyle=$ $\displaystyle
1-\left(\frac{r_{h}}{r}\right)^{3},$ (18) $\displaystyle\psi^{2}(r)$
$\displaystyle=$
$\displaystyle-\frac{6\kappa(\alpha+\gamma\Lambda)}{\alpha^{2}r^{2}{f}(r)}.$
(19)

In the limit $\psi^{2}(r\to\infty)=0$ into the action (7), provide that this
is a genuine vacuum solution. The equations (18)-(19), provide that the black
hole geometry is regular everywhere (except at the central singularity), the
scalar field derivative $\psi(r)$ diverges at horizon Anabalon:2013oea ;
Feng:2015oea ; Babichev:2013cya , but the scalar field does not explode at
horizon since it approaches to a constant near the horizon as:

$\displaystyle\phi^{2}(r)\sim\left((2\Lambda\mathcal{R}^{2}(\alpha+\gamma\Lambda)/\alpha^{2}r^{2}_{h}f^{{}^{\prime}}(r_{h}))(r-r_{h})\right)+const.$
(20)

Thus, we agree with the no-hair theorem, such discussions were presented by
Babichev:2013cya . An interesting characterize is that the scalar field
equation (19) is a real function outside the horizon for $r>r_{h}$, namely,
$f(r>r_{h})>0$, and the scalar field is real in the interval
$-1<\alpha/\gamma\Lambda<0$, with $\alpha,\gamma>0$. Besides of this analyzes,
we have that in the infinity the scalar field itself diverges as
$\phi(r)\sim\ln{r}$, but not its derivatives $\psi$ that are the ones present
in the action (7), which are finite at asymptotic infinity Babichev:2013cya .

In fact, we have that the appearance of a black hole, which has a flat horizon
$\Re^{2}$, leads us to a physics in IR that corresponds to placing the
invariant theory of scale at a finite temperature. So, we have that the
temperature of the black hole is given by

$\displaystyle T(r_{h})=\frac{f^{{}^{\prime}}(r=r_{h})}{4\pi}=\frac{3}{4\pi
r_{h}}$ (21)

##  III Holographic Entanglement Entropy in Horndeski Gravity

In this section, we present the computations of the entanglement entropy to a
subsystem $A$ in Horndeski gravity following the procedures of
Takayanagi:2012kg ; Calabrese:2004eu ; Ryu:2006ef ; Ryu:2006bv ;
Susskind:1994sm . For this, considering the metric (11) the three-dimensional
CFT lives in the space measured by $t$ and $x$. Thus, we can choose the
subsystem $A$ (which are the two charges) to be the length $l$ interval
$x\epsilon[-l/2,l/2]$, $y\epsilon[L/2,L/2]$ in the infinitely long total space
$-\infty<x<\infty$, see Fig.$\sim$2.

Figure 2: The figure show the schematic of a extremal surface where $l$ is the
length of the subsystem $A$, which is anchored on the subsystem living on the
boundary.

Now, we will provide the entanglement entropy for the Horndeski gravity
following the steps by steps of Caceres:2017lbr ; Hajian:2020dcq  and
motivated by recent studies of Santos:2021orr , we have to the action (7) that
the action with boundary counterterms is given by:

$\displaystyle
I_{E}=I_{bulk}=-\int{\sqrt{g}d^{4}x\mathcal{L}}-2\kappa\int{d^{3}x\sqrt{\bar{\gamma}}\mathcal{L}_{b}}+2\kappa\int{d^{3}x\sqrt{\bar{\gamma}}\mathcal{L}_{ct}},$
(22)
$\displaystyle\mathcal{L}=\kappa(R-2\Lambda)+\frac{\gamma}{2}G_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi$
(23)
$\displaystyle\mathcal{L}_{b}=K^{({\bar{\gamma}})}-\Sigma^{(\bar{\gamma})}+\frac{\gamma}{4}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi\,n^{\mu}n^{\nu}-(\nabla\phi)^{2}\right)K^{(\bar{\gamma})}+\frac{\gamma}{4}\nabla^{\mu}\phi\nabla^{\nu}\phi
K^{(\bar{\gamma})}_{\mu\nu}$ (24) $\displaystyle{\cal
L}_{ct}=c_{0}+c_{1}R+c_{2}R^{ij}R_{ij}+c_{3}R^{2}+b_{1}(\partial_{i}\phi\partial^{i}\phi)^{2}$
(25)

$\mathcal{L}_{b}$ corresponds to the Gibbons-Hawking $\gamma$-dependent terms
associated with the Horndeski gravity, where $n^{\mu}$ is an outward pointing
unit normal vector to the boundary,
$K^{(\bar{\gamma})}=\bar{\gamma}^{\mu\nu}K^{({\bar{\gamma}})}_{\mu\nu}$ is the
trace of the extrinsic curvature and $\bar{\gamma}_{\mu\nu}$ is the induced
metric on the boundary $r\to\infty$. The Lagragian ${\cal L}_{ct}$ is related
to the boundary counterterms, they do not affect the bulk dynamics and will be
neglect Balasubramanian:1999re . Thus, the induced metric to (11) is written
as

$\displaystyle
ds^{2}_{ind}=\bar{\gamma}_{\mu\nu}dx^{\mu}dx^{\nu}=\mathcal{R}^{2}\left(r^{2}f(r)d\tau^{2}+r^{2}(dx^{2}+dy^{2})+\frac{dr^{2}}{r^{2}f(r)}\right),$
(26)

With the above, we can provide that the Ryu-Takayanagi formula Ryu:2006ef ;
Ryu:2006bv  is given by:

$\displaystyle S_{A}=\frac{\mathcal{A}}{4G_{N}},$ (27)
$\displaystyle\mathcal{A}=\int{ds_{ind}\chi},$ (28)
$\displaystyle\chi=1-2\gamma(\bar{\gamma}^{\lambda\sigma}\nabla_{\lambda}\phi\nabla_{\sigma}\phi).$
(29)

Here this entropy is equivalently to the cases of Feng:2015oea ;
Caceres:2017lbr . For AdS4/CFT3, we have that the minimum surface is given by
the geodesic line in AdS4. That is, we have following the prescription of
Chaturvedi:2016kbk , we have that the area of the surface anchored on the
boundary to the subsystem (A), can be expressed as:

$\displaystyle\mathcal{A}=2\chi\mathcal{R}Lr_{c}\int^{1}_{0}{\frac{du}{u^{2}\sqrt{(1-u^{4})f(u)}}};\quad
f(u)=1-\left(\frac{r_{h}u}{r_{c}}\right)^{3},$ (30)

where $u=r_{c}/r$. In our setup to holographic entanglement entropy in
Horndeski gravity $r_{c}$ is a constant of integration, which represents the
turning point of the extremal surface in the higher dimensional AdS4 bulk
spacetime, see Fig.$\sim$2. We can see that through the equation (30) the area
integral is divergent at the point $u=1$ and must be regularized introducing
an infrared cutoff ($r_{b}$). From the point of view of the holographic
dictionary, we have a relation between the UV cutoff of the boundary field
theory ($\epsilon$) to bulk IR cutoff. Such relation is inversely related
through the AdS length scale $\mathcal{R}$ and can be establish as,
$r_{b}=\mathcal{R}/\epsilon$. But note that the finite part of entanglement
entropy can then be used to study the high and low temperature behavior of the
boundary field theory which is dual to black hole as covered in

$\displaystyle
S^{finite}_{A}=S_{A}-S^{divergent}_{A}=\frac{\mathcal{A}^{finite}}{4G_{N}}\chi.$
(31)
$\displaystyle\chi=1+\frac{12\kappa\gamma(\alpha+\gamma\Lambda)}{\alpha^{2}\mathcal{R}^{2}}.$
(32)

We can to obtain the quantity $r_{c}$ inverting the equation of motion

$\displaystyle\frac{l}{2}=\frac{1}{r_{c}}\int^{1}_{0}{\frac{u^{2}du}{\sqrt{(1-u^{4})f(u)}}}$
(33)

As we know, the horizon radius ($r_{h}$) is very small, we have that the black
hole remains deep inside the bulk. Thus, far away from the extremal surface,
namely, $r_{h}<<r_{c}$.For this limit, we can performing Taylor expand. In
this case the quantity $1/\sqrt{f(u)}$ will be expanded around $r_{h}/r_{c}=0$
as:

$\displaystyle\frac{1}{\sqrt{f(u)}}=1+\frac{1}{2}\left(\frac{r_{h}u}{r_{c}}\right)^{3}+\mathcal{O}\left[\left(\frac{r_{h}u}{r_{c}}\right)^{3}\right]^{4}$
(34)

Now, replacing this expansion in the equation (33), we can wrote

$\displaystyle\frac{lr_{c}}{2}=\int^{1}_{0}{\frac{u^{2}du}{\sqrt{(1-u^{4})}}}+\frac{1}{2}\left(\frac{r_{h}}{r_{c}}\right)^{3}\int^{1}_{0}{\frac{u^{5}du}{\sqrt{(1-u^{4})}}}$
(35)
$\displaystyle\frac{lr_{c}}{2}=\pi\left(\frac{r_{h}}{r_{c}}\right)^{3}+\frac{2\sqrt{\pi}\Gamma(3/4)}{\Gamma(1/4)}$
(36)

However, solving the equation (36) in terms of $r_{h}l$, we have

$\displaystyle
r_{c}=\frac{2\pi\Gamma(3/4)}{l\Gamma(1/4)}+\frac{l^{2}\Gamma^{3}(3/4)}{16\sqrt{\pi}\Gamma^{3}(1/4)}$
(37)

For the extremal area, we can write following the same steps

$\displaystyle\mathcal{A}=2\chi\mathcal{R}Lr_{c}\left(\int^{1}_{0}{\frac{u^{2}du}{\sqrt{(1-u^{4})}}}+\frac{1}{2}\left(\frac{r_{h}}{r_{c}}\right)^{3}\int^{1}_{0}{\frac{udu}{\sqrt{(1-u^{4})}}}\right)$
(38)

Through the equation (38), we have that first integral is same of a pure AdS,
which is divergent. For this reason, we need regularize her, introducing an UV
cutoff $1/r_{b}$ and add a counter term ($-2\mathcal{R}Lr_{b}$), after this
considerations we provide the finite part of the extremal area as

$\displaystyle\mathcal{A}^{finite}=2\chi\mathcal{R}Lr_{c}\left(\int^{1}_{0}{\frac{u^{2}du}{\sqrt{(1-u^{4})}}}-2\mathcal{R}Lr_{b}+\frac{1}{2}\left(\frac{r_{h}}{r_{c}}\right)^{3}\int^{1}_{0}{\frac{udu}{\sqrt{(1-u^{4})}}}\right)$
(39)
$\displaystyle\mathcal{A}^{finite}=\chi\mathcal{R}Lr_{c}\left(\sqrt{\pi}\frac{\Gamma(-1/4)}{\Gamma(1/4)}+\pi\left(\frac{r_{h}}{r_{c}}\right)^{3}\right)$
(40)

Combining the equation (40) with (37) the entanglement entropy is given by:

$\displaystyle
S^{finite}_{A}=\frac{\chi\mathcal{R}L}{4lG_{N}}\left(-\frac{4\pi\Gamma^{2}(3/4)}{\Gamma^{2}(1/4)}+\frac{l^{3}r^{3}_{h}\Gamma^{2}(1/4)}{8\Gamma^{2}(3/4)}\right)$
(41)

For case of extremal black holes we consider $r^{3}_{h}=\mathcal{M}_{ext}/4$
where $\mathcal{M}_{ext}$ is the mass of the extremal black hole, we have

$\displaystyle S^{finite}_{A}=[S^{AdS}_{A}+k\mathcal{M}_{ext}l^{2}L]\chi,\quad
k=\frac{\mathcal{R}L}{32G_{N}}\frac{\Gamma^{2}(1/4)}{\Gamma^{2}(3/4)}$ (42)
$\displaystyle
S^{AdS}_{A}=-\frac{4\pi\mathcal{R}L}{4lG_{N}}\frac{\Gamma^{2}(3/4)}{\Gamma^{2}(1/4)}$
(43)

The results show in the equations (42) and (43) are similar to the case of
Chaturvedi:2016kbk , but in our case, we have the presence of Horndeski
parameters, where $S^{AdS}_{A}$ is the entanglement entropy of the subsystem
(A), when the bulk theory is pure AdS as described by Fischler:2012ca . On the
other hand, if $\gamma$ is large the Horndeski contribution through the
$\chi$-term in the equation (42), this imply that $S^{finite}_{A}$ becomes
large and the storage of information show that, we have more information about
the subsystem (A). Furthermore, at the critical point
$\alpha=3\gamma/\mathcal{R}^{2}$, for more discussion see Li:2018kqp , the
entropy in equation (42) reduces to the usual case of Chaturvedi:2016kbk
where sub-leading correction term in the equation (42), becomes important in
defining the first law like relation.

###  III.1 Planar black hole

We address the issue of finding planar black hole solutions in Horndeski
gravity. For this, we consider for Einstein-Horndeski gravity the following
Ansatz:

$ds^{2}=-f(r)dt^{2}+r^{2}(dx^{2}+dy^{2})+\frac{dr^{2}}{f(r)}.$ (44)

One can show that the equations (8) and (10) are satisfied by the following
solution

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle\frac{\alpha
r^{2}}{3\gamma}-\frac{r_{h}}{r},$ (45) $\displaystyle\psi^{2}(r)$
$\displaystyle=$
$\displaystyle-\frac{2\kappa(\alpha+\gamma\Lambda)}{\alpha\gamma
r^{2}}\frac{1}{f(r)}.$ (46)

Following the previous procedures, we have

$\displaystyle\frac{l}{2}=\int^{1}_{0}{\frac{du}{u\sqrt{(1-u^{4})f(u)}}};\quad
f(u)=\frac{\alpha r^{2}_{c}}{3\gamma u^{2}}-\frac{r_{h}u}{r_{c}},$ (47)
$\displaystyle\mathcal{A}=2\chi
Lr_{c}\int^{1}_{0}{\frac{du}{u^{2}\sqrt{(1-u^{4})f(u)}}}$ (48)

Again through the expansion of $1/\sqrt{f(u)}$, we have

$\displaystyle\frac{1}{\sqrt{f(u)}}=\sqrt{\frac{3\gamma}{\alpha}}\frac{u}{r_{c}}+\sqrt{\left(\frac{3\gamma}{\alpha}\right)^{3}}\frac{u^{4}r_{h}}{r^{4}_{c}}$
(49)

Now, we can write for the equations (47) and (48) as:

$\displaystyle\frac{l}{2}=\frac{1}{r_{c}}\sqrt{\frac{3\gamma}{\alpha}}\frac{\sqrt{\pi}\Gamma(5/4)}{\Gamma(3/4)}+\frac{r_{h}}{4r^{4}_{c}}\sqrt{\left(\frac{3\gamma}{\alpha}\right)^{3}},$
(50) $\displaystyle\mathcal{A}^{finite}\approx\chi
L\sqrt{\frac{3\gamma}{\alpha}}\ln\left(\frac{l}{r_{b}}\right)-\frac{3\sqrt{3\pi}l\chi
L}{4}\sqrt{\frac{\gamma}{\alpha}}\frac{\Gamma(5/4)}{\Gamma(1/4)}$ (51)

Thus, the entanglement entropy to the planar black is

$\displaystyle S^{finite}_{A}=\frac{\mathcal{A}^{finite}}{4G_{N}}=\frac{\chi
L}{4G_{N}}\sqrt{\frac{3\gamma}{\alpha}}\ln\left(\frac{l}{r_{b}}\right)-\frac{3\sqrt{\pi}l\chi
L}{16G_{N}}\sqrt{\frac{3\gamma}{\alpha}}\frac{\Gamma(5/4)}{\Gamma(1/4)}$ (52)

When we compare the equation (52) with the equation (42), we have that
entanglement entropy to the subsystem (A) have a logarithmic term
Calabrese:2004eu ; Holzhey:1994we  and a sub-leading correction with Horndeski
parameters. Besides, if $\gamma$ is small, the $S^{finite}_{A}\to 0$ for the
planar black hole and all information about the subsystem (A) is destroyed,
but not at the critical point $\alpha=3\gamma/\mathcal{R}^{2}$ Li:2018kqp
where we have a behavior like area law, which is corrected by a logarithmic
factor. Such behavior is find fermionic systems with the presence of a finite
Fermi surface Gioev:2006zz ; Wolf:2006zzb .

##  IV Entanglement thermodynamics in Horndeski gravity

In this section, we present the "first law of entanglement thermodynamics".
For this, we need of the stress-energy tensor of boundary field theory in
Horndeski gravity. Through the renormalization procedure
Balasubramanian:1999re  the form of stress-energy tensor $T_{\alpha\beta}$ can
be write as:

$\displaystyle
T_{\alpha\beta}=-\frac{r^{3}}{16\pi\mathcal{R}^{3}G_{N}}\left[K^{({\bar{\gamma}})}_{\alpha\beta}-\bar{\gamma}_{\alpha\beta}(K^{({\bar{\gamma}})}-\Sigma^{(\bar{\gamma})})+\frac{\gamma}{4}H_{\alpha\beta}-\kappa
T^{R}_{\alpha\beta}-\kappa T^{ct}_{\alpha\beta}\right],$ (53) $\displaystyle
H_{\alpha\beta}=(\nabla_{\alpha}\phi\nabla_{\beta}\phi
n^{\alpha}n^{\beta}-(\nabla\phi)^{2})(K^{({\bar{\gamma}})}_{\alpha\beta}-\bar{\gamma}_{\alpha\beta}K^{({\bar{\gamma}})})-(\nabla_{\alpha}\phi\nabla_{\beta}\phi)\bar{\gamma}_{\alpha\beta}K^{({\bar{\gamma}})}.$
(54)

Here $T^{R}_{\alpha\beta}$ and $T^{ct}_{\alpha\beta}$ are possible
contribution of extrinsic curvature and counter term, respectively. However,
fixing the energy-momentum tensor on the boundary with
$T^{R}_{\alpha\beta}=T^{ct}_{\alpha\beta}=0$, we have

$\displaystyle
T_{\alpha\beta}=-\frac{r^{3}}{16\pi\mathcal{R}^{3}G_{N}}\left[K^{({\bar{\gamma}})}_{\alpha\beta}-\bar{\gamma}_{\alpha\beta}(K^{({\bar{\gamma}})}-\Sigma^{(\bar{\gamma})})+\frac{\gamma}{4}H_{\alpha\beta}\right].$
(55)

However, in order to obtain the “first law of entanglement thermodynamics” in
Horndeski gravity we need compute the following quantity:

$\displaystyle\Delta S_{A}=\frac{\Delta E_{A}}{T_{en}},$ (56)
$\displaystyle\Delta E_{A}=\int_{A}{dxdyT^{Temp\neq
0}_{tt}}-\int_{A}{dxdyT^{Temp=0}_{tt}}.$ (57)

But one interesting physical observables is the mass written as:

$\displaystyle\mathcal{M}=\int_{A}{dxdyT^{Temp\neq 0}_{tt}}.$ (58)

With

$\displaystyle T_{tt}=-\chi\frac{r^{3}}{16\mathcal{R}^{3}\pi
G_{N}}\left[K^{({\bar{\gamma}})}_{tt}-\bar{\gamma}_{tt}(K^{({\bar{\gamma}})}-\Sigma^{(\bar{\gamma})})\right],$
(59) $\displaystyle T_{tt}=\chi\frac{r^{3}}{8\pi\mathcal{R}^{4}G_{N}}.$ (60)

The result presented by the equation (60) is the same of
Balasubramanian:1999re  where the equation (60) corresponds to a excited state
in the CFT3 Bhattacharya:2012mi . In this way, we can express the first law of
entanglement thermodynamics as:

$\displaystyle\Delta E_{A}=\frac{l\chi L}{8\pi
G_{N}}(\mathcal{M}-\mathcal{M}_{ext}),$ (61) $\displaystyle
T_{en}=\frac{\pi}{16l}\frac{\Gamma^{2}(3/4)}{\Gamma^{2}(1/4)}.$ (62)

Where $T_{en}$ is the entanglement temperature Bhattacharya:2012mi ;
Chaturvedi:2016kbk . Such result are in perfect agreement with Blanco:2013joa
. But here this result and also as shown in Bhattacharya:2012mi , it differs
from our results, because we consider a canonical ensemble where the ground
state of the boundary field theory is dual to the extremal AdS4 black hole in
the bulk.

###  IV.1 Planar black hole

Now, for the planar black hole, following the step by step as done to the
spherically case we compute the increased amount of energy for such black
hole. Thus, through the equation (55), we can show that:

$\displaystyle
T_{tt}=\frac{\chi}{16\pi\mathcal{R}G_{N}}\sqrt{\frac{3\gamma}{\alpha}}\left(\frac{4}{r}-\frac{1}{R}\right).$
(63)

The equation (63), provide that we can must be not need to make any
assumptions in the infrared region $r\to\infty$ for this excited state in the
CFT3. Thus, objects such as black branes or stars in the infrared region can
be find. The first has a horizon and is a thermal state, but the second does
not have any horizon, however, is dual to a zero temperature state
Bhattacharya:2012mi . Furthermore, the increased amount of energy in the
subsystem (A) can be written as

$\displaystyle\Delta
E_{A}=\frac{\chi}{16\pi\mathcal{R}G_{N}}\sqrt{\frac{\alpha}{3\gamma}}\int_{A}{dydr\frac{4}{r}}-\frac{\chi}{16\pi\mathcal{R}^{2}G_{N}}\sqrt{\frac{\alpha}{3\gamma}}\int_{A}{dxdy},$
(64) $\displaystyle\Delta E_{A}=\frac{\chi
L}{4\pi\mathcal{R}G_{N}}\sqrt{\frac{3\gamma}{\alpha}}\ln\left(\frac{l}{r_{b}}\right)-\frac{l\chi
L}{16\pi\mathcal{R}^{2}G_{N}}\sqrt{\frac{3\gamma}{\alpha}},$ (65)
$\displaystyle T_{en}=\frac{1}{3\sqrt{\pi}l}\frac{\Gamma(1/4)}{\Gamma(5/4)}.$
(66)

where the entropy in equation (65) to the planar black hole is the area of the
boundary in the region $r\to\infty$. In summary, this result is very similar
to the equation (61). On the other hand, when we compare the equation (65)
with the equation (52), we see that are very similar. This convergence of
results can be done through the renormalization procedure, which removes the
logarithmic correction.

##  V Conclusion

We show in four-dimensions that the study of the entanglement entropy in
Horndeski gravity for planar and spherically topologies for a strip like
region denoted by the subsystem (A), which is Boundary Conformal Field Theory
dual to bulk black holes in an AdS4/CFT3 scenario, provided interesting
results, respectively. For this two topologies, we find two interesting
aspects of the holographic entanglement entropy. The first behavior of the
entanglement entropy with the Horndeski parameters of the AdS black hole can
be becomes pure AdS, if kinematic coupling $\gamma$ is very small, that is,
$\gamma\to 0$. Thus, such result show that there is similar behavior of study
in the field of AdS/CFT correspondence within Horndeski gravity, when if
explore the black hole thermodynamics of the black holes, as for example in
the studies of Santos:2021orr ; Feng:2015oea . For $\gamma=0$ we recover the
usual entropy of the AdS space.

The second behavior for the entanglement entropy is very interesting, because
the values of parameters of Horndeski gravity impose limitations in storage of
information to the planar black holes, if the entanglement entropy of him
becomes null, that is, $S^{finite}_{A}\to 0$ when $\gamma\to 0$ the storage
information are completely destroyed, but in the critical points Li:2018kqp
where $\alpha=3\gamma/\mathcal{R}^{2}$, we have that the entanglement
entropy’s described by equation (52) preserve the storage information of the
subsystem (A), because are not constraint by the parameters at this point.

Finally, for the “first law of entanglement thermodynamics” in Horndeski
gravity, we show that for spherically case, it has extremal black holes
solution. The first topology of extremal black hole the large implies a large
horizon radius. For the second topology, we have that the increasing of energy
$\Delta E_{A}$ is constraint by the Horndeski parameters. This fact, agree
with Ryu-Takayanagi formula as shown in (52). This first law of entanglement
thermodynamics in Horndeski gravity in fact agree with the result of
Santos:2021orr ; Feng:2015oea ; Caceres:2017lbr ; Hajian:2020dcq .

## Acknowledgment

Fabiano F. Santos would like to thank CNPq and CAPES for partial financial
support. I would like to thank Mohammad Hassan Vahidinia for valuable comments
and discussions.

## References

  * (1) T. Takayanagi, Entanglement Entropy from a Holographic Viewpoint, Class. Quant. Grav. 29 (2012), 153001, [arXiv:1204.2450 [gr-qc]].
  * (2) A. Bhattacharya, K. T. Grosvenor and S. Roy, Entanglement Entropy and Subregion Complexity in Thermal Perturbations around Pure-AdS Spacetime, Phys. Rev. D 100, no.12, 126004 (2019), arXiv:1905.02220 [hep-th]].
  * (3) A. Bhattacharya and S. Roy, Holographic entanglement entropy and entanglement thermodynamics of ‘black’ non-susy D3 brane, Phys. Lett. B 781, 232-237 (2018), [arXiv:1712.03740 [hep-th]].
  * (4) J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical Property of Entanglement Entropy for Excited States, Phys. Rev. Lett. 110, no.9, 091602 (2013), [arXiv:1212.1164 [hep-th]].
  * (5) P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004), P06002, [arXiv:hep-th/0405152 [hep-th]].
  * (6) C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424, 443-467 (1994), [arXiv:hep-th/9403108 [hep-th]].
  * (7) S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006), 045, [arXiv:hep-th/0605073 [hep-th]].
  * (8) S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006), 181602, [arXiv:hep-th/0603001 [hep-th]].
  * (9) L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994), 2700-2711, [arXiv:hep-th/9401070 [hep-th]].
  * (10) P. Chaturvedi, V. Malvimat and G. Sengupta, Entanglement thermodynamics for charged black holes, Phys. Rev. D 94, no.6, 066004 (2016), [arXiv:1601.00303 [hep-th]].
  * (11) E. Tonni, Holographic entanglement entropy: near horizon geometry and disconnected regions, JHEP 05 (2011), 004, [arXiv:1011.0166 [hep-th]].
  * (12) S. A. H. Mansoori, B. Mirza, M. D. Darareh and S. Janbaz, Entanglement Thermodynamics of the Generalized Charged BTZ Black Hole, Int. J. Mod. Phys. A 31 (2016) no.12, 1650067, [arXiv:1512.00096 [gr-qc]].
  * (13) P. Caputa, V. Jejjala and H. Soltanpanahi, Entanglement entropy of extremal BTZ black holes, Phys. Rev. D 89 (2014) no.4, 046006, [arXiv:1309.7852 [hep-th]].
  * (14) D. D. Blanco, H. Casini, L. Y. Hung and R. C. Myers, Relative Entropy and Holography, JHEP 08 (2013), 060, [arXiv:1305.3182 [hep-th]].
  * (15) L. Susskind, Entanglement and Chaos in De Sitter Holography: An SYK Example, [arXiv:2109.14104 [hep-th]].
  * (16) B. S. Kay, Entanglement entropy and algebraic holography, [arXiv:1605.07872 [hep-th]].
  * (17) C. Park, Holographic entanglement entropy in the nonconformal medium, Phys. Rev. D 91 (2015) no.12, 126003, [arXiv:1501.02908 [hep-th]].
  * (18) S. He, J. R. Sun and H. Q. Zhang, On Holographic Entanglement Entropy with Second Order Excitations, Nucl. Phys. B 928 (2018), 160-181, [arXiv:1411.6213 [hep-th]].
  * (19) F. F. Santos, Aplicações do Setor John da Gravidade de Horndeski nos Cenários de Brana Negra e Relação de viscosidade/entropia, Mundo Brana e Cosmologia (In Portuguese), [arXiv:2006.06550 [hep-th]].
  * (20) F. A. Brito and F. F. Santos, Black brane in asymptotically Lifshitz spacetime and viscosity/entropy ratios in Horndeski gravity, EPL 129, no.5, 50003 (2020), [arXiv:1901.06770 [hep-th]].
  * (21) F. F. Santos, E. F. Capossoli and H. Boschi-Filho, AdS/BCFT correspondence and BTZ black hole thermodynamics within Horndeski gravity, Phys. Rev. D 104, no.6, 066014 (2021), [arXiv:2105.03802 [hep-th]].
  * (22) W. J. Jiang, H. S. Liu, H. Lu and C. N. Pope, DC Conductivities with Momentum Dissipation in Horndeski Theories, JHEP 1707, 084 (2017), [arXiv:1703.00922 [hep-th]].
  * (23) M. Baggioli and W. J. Li, Diffusivities bounds and chaos in holographic Horndeski theories, JHEP 1707, 055 (2017), [arXiv:1705.01766 [hep-th]].
  * (24) H. S. Liu, Violation of Thermal Conductivity Bound in Horndeski Theory, Phys. Rev. D 98, no. 6, 061902 (2018), [arXiv:1804.06502 [hep-th]].
  * (25) Y. Z. Li and H. Lu, $a$-theorem for Horndeski gravity at the critical point, Phys. Rev. D 97, no. 12, 126008 (2018), [arXiv:1803.08088 [hep-th]].
  * (26) Y. Z. Li, H. Lu and H. Y. Zhang, Scale Invariance vs. Conformal Invariance: Holographic Two-Point Functions in Horndeski Gravity, arXiv:1812.05123 [hep-th].
  * (27) X. H. Feng, H. S. Liu, H. Lü and C. N. Pope, Black Hole Entropy and Viscosity Bound in Horndeski Gravity, JHEP 1511, 176 (2015), [arXiv:1509.07142 [hep-th]].
  * (28) E. Caceres, R. Mohan and P. H. Nguyen, On holographic entanglement entropy of Horndeski black holes, JHEP 10, 145 (2017) doi:10.1007/JHEP10(2017)145 [arXiv:1707.06322 [hep-th]].
  * (29) K. Hajian, S. Liberati, M. M. Sheikh-Jabbari and M. H. Vahidinia, On Black Hole Temperature in Horndeski Gravity, Phys. Lett. B 812, 136002 (2021) doi:10.1016/j.physletb.2020.136002 [arXiv:2005.12985 [gr-qc]].
  * (30) M. R. Mohammadi Mozaffar, A. Mollabashi, M. M. Sheikh-Jabbari and M. H. Vahidinia, Holographic Entanglement Entropy, Field Redefinition Invariance and Higher Derivative Gravity Theories, Phys. Rev. D 94, no.4, 046002 (2016), [arXiv:1603.05713 [hep-th]].
  * (31) V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208, 413 (1999), [hep-th/9902121].
  * (32) G. W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974).
  * (33) F. F. Santos, Rotating black hole with a probe string in Horndeski Gravity, Eur. Phys. J. Plus 135, no.10, 810 (2020), [arXiv:2005.10983 [hep-th]].
  * (34) A. Cisterna and C. Erices, Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario, Phys. Rev. D 89, 084038 (2014), [arXiv:gr-qc/1401.4479].
  * (35) M. Bravo-Gaete and M. Hassaine, Thermodynamics of a BTZ black hole solution with an Horndeski source, Phys. Rev. D 90, no.2, 024008 (2014), [arXiv:1405.4935 [hep-th]].
  * (36) A. Anabalon, A. Cisterna and J. Oliva, Asymptotically locally AdS and flat black holes in Horndeski theory, Phys. Rev. D 89, 084050 (2014), [arXiv:gr-qc/1312.3597 [gr-qc]].
  * (37) C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, General second order scalar-tensor theory, self tuning, and the Fab Four, Phys. Rev. Lett. 108, 051101 (2012), [arXiv:1106.2000 [hep-th]].
  * (38) C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, Self-tuning and the derivation of a class of scalar-tensor theories, Phys. Rev. D 85, 104040 (2012), [arXiv:1112.4866 [hep-th]].
  * (39) A. A. Starobinsky, S. V. Sushkov and M. S. Volkov, The screening Horndeski cosmologies, JCAP 1606, 007 (2016), [arXiv:1604.06085 [hep-th]].
  * (40) J. P. Bruneton, M. Rinaldi, A. Kanfon, A. Hees, S. Schlogel and A. Fuzfa, Fab Four: When John and George play gravitation and cosmology, Adv. Astron. 2012, 430694 (2012), [arXiv:1203.4446 [gr-qc]].
  * (41) L. Hui and A. Nicolis, No-Hair Theorem for the Galileon, Phys. Rev. Lett. 110, 241104 (2013), [arXiv:1202.1296 [hep-th]].
  * (42) M. Bravo-Gaete and M. Hassaine, Lifshitz black holes with a time-dependent scalar field in a Horndeski theory, Phys. Rev. D 89, 104028 (2014), [arXiv:1312.7736 [hep-th]].
  * (43) E. Babichev and C. Charmousis, Dressing a black hole with a time-dependent Galileon, JHEP 1408, 106 (2014), [arXiv:1312.3204 [gr-qc]].
  * (44) W. Fischler and S. Kundu, Strongly Coupled Gauge Theories: High and Low Temperature Behavior of Non-local Observables, JHEP 05, 098 (2013), [arXiv:1212.2643 [hep-th]].
  * (45) D. Gioev and I. Klich, Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture, Phys. Rev. Lett. 96, 100503 (2006), [arXiv:quant-ph/0504151 [quant-ph]].
  * (46) M. M. Wolf, Violation of the entropic area law for Fermions, Phys. Rev. Lett. 96, 010404 (2006), [arXiv:quant-ph/0503219 [quant-ph]].