# KANQAS: Kolmogorov-Arnold Network for Quantum Architecture Search

Akash Kundu1,,a, Aritra Sarkar2,b, Abhishek Sadhu3,c 1 QTF Centre of
Excellence, Department of Physics, University of Helsinki, Finland  1
Institute of Theoretical and Applied Informatics, Polish Academy of Sciences,
Gliwice, Poland  1 Joint Doctoral School, Silesian University of Technology,
Gliwice, Poland  2 QuTech, Delft University of Technology, Delft, The
Netherlands  3 Raman Research Institute, Bengaluru, India
3 Centre for Quantum Science and Technology (CQST), International Institute of
Information Technology, Hyderabad, Telangana, India  aCorresponding Author:
<EMAIL_ADDRESS>b<EMAIL_ADDRESS>c<EMAIL_ADDRESS>
###### Abstract

Quantum architecture search (QAS) is a promising direction for optimization
and automated design of quantum circuits towards quantum advantage. Recent
techniques in QAS focus on machine learning-based approaches from
reinforcement learning, like deep Q-network. While multi-layer perceptron-
based deep Q-networks have been applied for QAS, their interpretability
remains challenging due to the high number of parameters. In this work, we
evaluate the practicality of Kolmogorov-Arnold Networks (KANs) in QAS
problems, analyzing their efficiency in the task of quantum state preparation
and quantum chemistry. In quantum state preparation, our results show that in
a noiseless scenario, the probability of success and the number of optimal
quantum circuit configurations to generate the multi-qubit maximally entangled
states are $2\times$ to $5\times$ higher than Multi-Layer perceptions (MLPs).
Moreover, in noisy scenarios, KAN can achieve a better fidelity in
approximating maximally entangled state than MLPs, where the performance of
the MLP significantly depends on the choice of activation function. In
tackling quantum chemistry problems, we enhance the recently proposed QAS
algorithm by integrating Curriculum Reinforcement Learning (CRL) with a KAN
structure instead of the traditional MLP. This modification allows us to
design a parameterized quantum circuit that contains fewer 2-qubit gates and
has a shallower depth, thereby improving the efficiency of finding the ground
state of a chemical Hamiltonian. Further investigation reveals that KAN
requires a significantly smaller number of learnable parameters compared to
MLPs; however, the average time of executing each episode for KAN is higher.

##  I Introduction

Recent research has advanced quantum computing concepts and software
development, yet significant challenges remain before real-world applications
are feasible. Automating quantum algorithm design via Machine Learning (ML)
and optimization algorithms offers promising solutions to enhance quantum
hardware and programming capabilities for complex problems. Strongly inspired
by Neural Architecture Search (NAS) [1], Quantum Architecture Search (QAS) [2,
3] is introduced.

QAS encompasses techniques to automate the optimization of quantum circuits
and it typically consists of two parts. In the first part, a template of the
circuits is built where the circuit can have parameterized quantum gates,
e.g., rotation angles. Then, these parameters are determined via the
variational principle using a classical optimizer in a feedback loop.
Algorithms constructed via this technique are called Variational Quantum
Algorithms (VQA) [4, 5]. The parameterized circuit design in VQAs is critical,
as it directly influences the expressivity and efficiency of the quantum
solution. Recently, QAS has been utilized to automate the search for optimal
parameterized circuits for VQAs. QAS also finds application in determining
non-parameterized circuits as an approach for quantum program synthesis [6]
and multi-qubit maximally entangled state preparation [7].

One of the most prominent methods to tackle QAS problems is to use
Reinforcement Learning (RL) (RLQAS) [8, 7, 9, 10, 11, 12]. In this approach,
quantum circuits are defined as a sequence of actions generated by a trainable
policy. The cost function’s value, optimized independently by a classical
optimizer, serves as a signal for the reward function. This reward function
guides the policy updates to maximize expected returns and select optimal
actions for subsequent steps. Maximization of the expected return is achieved
by training neural networks (NNs). Recently, NNs have shown potential in
quantum information tasks [13, 14]. RLQAS with deep NNs can overcome the
trainability issues of VQAs [15, 16] and has demonstrated promising outcomes
in NISQ hardware [10, 11].

In a recent work [17], researchers have introduced Kolmogorov-Arnold Networks
(KANs) as a novel neural network architecture typically poised to replace
traditional Multi-Layer Perceptrons (MLPs). Contrary to the universal
approximation theorem-based MLPs, KAN utilizes the Kolmogorov-Arnold
representation theorem to approximate complicated functions. KAN has linear
weights replaced by spline-based univariate functions along the network edges
and is structured as learnable activation functions. Such a design enhances
the accuracy and interpretability of the networks, enabling them to achieve
comparable or superior results with smaller network sizes across a range of
tasks, including data fitting and solving partial differential equations.
Recently, different variants of KAN have been introduced [18, 19, 20, 21, 22,
23]. KAN has applications in time series analysis [24, 25, 26], satellite
image classification [27], and improving the robustness of neural network
architectures [28]. To understand the full potential and limitations of KANs
there is a need for further investigation towards robustness and compatibility
with other deep learning architectures.

In this article, we evaluate the practicality of KANs in quantum circuit
construction, analyzing their efficiency in terms of the probability of
success, frequency of optimal solutions and their dependencies on various
degrees of freedom of the network.

Figure 1: The schematic for KANQAS shows how the Kolmogorov Arnold network
replaces the multi-layer perception structure in the RL agent. The
environment, which is the quantum circuit, interacts with the RL agent
(containing the KAN) and gets updates after each step after the agent decides
on an action following a policy.

To the best of our knowledge, the application of Kolmogorov-Arnold Networks in
Quantum Architecture Search is still lacking in standard literature. We
propose the application of KAN in QAS by replacing the MLP of Double Deep
Q-Network (DDQN) in RLQAS with KAN to generate the desired quantum state,
introducing KANQAS111Link to KANQAS code repository. As shown in Figure 1, the
proposed framework of KANQAS features an RL agent, which contains the KAN,
interacting with a quantum simulator. The agent sequentially generates output
actions, which are candidates for quantum gates on the circuit. The fidelity
of the state from the constructed circuit is compared to the quantum state
fidelity of the circuit and is evaluated to determine how far it is from the
desired goal. The reward, based on fidelity, is sent back to the RL agent.
This process is repeated iteratively to train the RL agent. We show that in a
noiseless scenario when constructing Bell and GHZ states, the probability of
success and the number of optimal quantum circuit configurations to generate
the states are significantly higher than MLPs. In a noisy scenario, we show
that KAN can achieve a better fidelity in approximating GHZ state than MLPs,
where the performance of the MLP significantly depends on the depth of the
network and choice of activation function. We also address quantum chemistry
problems, where we find the ground state of 4-qubit $\texttt{H}_{2}$ and LiH
molecules using recently proposed Curriculum Reinforcement Learning for
Quantum Architecture Search (CRLQAS) [11]. In our version of CRLQAS, we
replace the MLP structure with KAN in the curriculum reinforcement learning
subroutine. Our results show that KAN demonstrates the ability to provide a
more compact parameterized quantum circuit, in terms of less number of 2 qubit
gates and depth for finding the ground state of the chemical Hamiltonian using
VQE, with a significantly reduced number of learnable parameters.

The structure of the paper is as follows. We present the problem statement in
Sec. II. In Sec. III, we provide the methods used in this work. Specifically,
we introduce the Kolmogorov-Arnold Q Network in Sec. III-A and discuss the RL
state, action and reward function in Sec. III-B. We present our results in
Sec. IV. Specifically, we discuss the results of our noiseless simulations in
Sec. IV-A1, noisy simulations in Sec. IV-A2 and the resource requirements for
the simulations in Sec. V. We provide concluding remarks and discuss open
problems in Sec. VI.

##  II Problem statement

###  II-A Quantum state preparation

To evaluate the efficiency of RL-assisted KANQAS in quantum circuit
construction we check if, in a noiseless and noisy scenario, multi-qubit
entanglement can be generated as expected. To this end, we target the
generation of two kinds of quantum states: Bell and
Greenberger–Horne–Zeilinger (GHZ) states. A Bell state is a maximal 2-qubit
entangled state and is given by

$|\Phi^{+}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right).$ (1)

The optimal circuit to produce a Bell state is given by

$\quantikz&\gate{H}\ctrl{1}\\\ \targ{}.$ (2)

Meanwhile, a GHZ state is a 3-qubit maximally entangled state given by

$|GHZ\rangle=\frac{1}{\sqrt{2}}\left(|000\rangle+|111\rangle\right)$ (3)

. and the optimal circuit produces a GHZ state given by

$\quantikz&\gate{H}\ctrl{1}\\\ \targ{}\ctrl{1}\\\ \targ{}.$ (4)

As a measure of the efficiency of the network, we define the probability of
success and the probability of optimal success. The probability of success is
given by

$\small\frac{\text{Number of times the network finds an admissible
circuit}}{\text{Total number of admissible circuits}},$ (5)

As the problem statement clarifies, any circuit that generates a Bell state
and a GHZ state is considered an admissible circuit. Meanwhile, Figure 2 and 4
are optimal admissible circuits to generate $2$\- and $3$-qubit maximally
entangled state.

###  II-B Quantum chemistry

Quantum chemistry [29] utilizes quantum mechanics to understand the behavior
of atoms and molecules. Its main goal is to understand the electronic
structure of chemical systems and dynamics, enabling predictions about their
chemical and physical properties. This involves examining electronic ground
and excited states, reaction pathways, and transition states, which are
crucial for explaining reactivity and interactions within chemical systems.

Molecule | Basis | Fermion to qubit mapping | Geometry | Number of qubits
---|---|---|---|---
$\texttt{H}_{2}$ | sto3g | Jordan-Wigner |  H ($0,0,-0.35$); H ($0,0,0.35$) | 4
LiH | sto3g | Parity |  Li ($0,0,0$); H ($0,0,3.4$) | 4
TABLE I: List of molecules included in our simulations, with configuration
coordinates provided in angstroms.

The Variational Quantum Eigensolver (VQE) is a quantum-classical algorithm
that is designed to find the ground state energy of quantum systems, making it
particularly useful for studying molecular systems where traditional
computational methods may struggle due to the complexity and the exponentially
growing size of the systems.

In VQE, the objective is to find the ground state energy of a chemical
Hamiltonian $H$ by minimizing the energy

$C(\vec{\theta})=\min_{\vec{\theta}}\left(\langle\psi(\vec{\theta})|H|\psi(\vec{\theta})\rangle\right).$
(6)

The trial state $|\psi(\vec{\theta})\rangle$ is prepared by applying a
Parameterized Quantum Circuit (PQC), $U(\vec{\theta})$, to the initial state
$|\psi_{\text{initial}}\rangle$, where $\vec{\theta}$ specify the rotation
angles of the local unitary operators in the circuit.

The structure of the PQC is one of the crucial factors affecting the
performance of the VQE. Among the recently proposed RLQAS methods for
automating the search for PQC in VQE, Curriculum Reinforcement Learning (CRL)
for QAS, i.e., CRLQAS [11] demonstrates promising outcomes to find the ground
state of various chemical Hamiltonian in the noiseless scenario and under
hardware error with short PQCs.

Hence, to evaluate the efficiency of KAN in PQC construction for quantum
chemistry problems we replace the MLP structure utilized in CRLQAS with KAN.
Utilizing the CRL-assisted KANQAS we find the ground state of 4-qubit
$\texttt{H}_{2}$ and LiH molecules. The exact molecular structures of these
molecules are provided in Table I.

##  III Methods

###  III-A Kolmogorov-Arnold Q Networks (KAQN)

In a recent work [17], Kolmogorov-Arnold Networks (KAN) was proposed as a
promising alternative to the Multi-Layer Perceptrons (MLP). KAN is based on
the Kolmogorov-Arnold representation theorem [30] instead of the Universal
Approximation Theorem [31] used in MLP. The Kolmogorov-Arnold representation
theorem states that a real-valued, smooth and continuous multivariate function
$g(\textbf{t}):[0,1]^{n}\to\mathbbm{R}$ can be represented by a superposition
of univariate functions [30]

$\displaystyle
g(\mathbf{t})=\sum_{j=1}^{2n+1}\Psi_{i}\bigg{(}\sum_{k=1}^{n}\psi_{jk}\bigg{)},$
(7)

where $\Psi_{j}:\mathbbm{R}\to{\mathbbm{R}}$ and
$\psi_{ij}:[0,1]\to{\mathbbm{R}}$. In other words, any multivariate continuous
function on a bounded domain can be expressed as a composition of continuous
functions of one variable. This reduces the task of learning complex multi-
variable functions to learning a polynomial number of single-variable
functions. It was noted by the authors of [17] that the representation of the
function in Eq. (7) has two layers of nonlinearity with $2n+1$ terms in the
middle layer, and we need to find functions $\Psi_{i}$ and $\psi_{ij}$ to
approximate $g(\textbf{t})$. The function $\psi_{ij}$ may be approximated
using B-splines [32].

A spline is a piecewise smooth polynomial function defined by a set of control
points. It is defined by the order $l$ of the polynomial used to interpolate
the curve between the control points. We denote the number of segments between
adjacent control points as $G$. The data points are connected by the segments
to form a smooth curve having $G+1$ grid points. It is observed that Eq. (7)
can be represented as a two-layer network having activation functions at the
edges and nodes performing the summation operation. However, such a network is
too restrictive to approximate any arbitrary function. A way to overcome this
was proposed in [17], where the authors propose a general architecture with
wider and deeper KANs.

The authors of [17] define a KAN layer by a matrix $\mathbf{\Psi}$ of
trainable univariate spline functions $\\{\psi_{jk}(.)\\}$ with
$j=1,...,n_{i}$ and $k=1,...,n_{o}$, where $n_{i}$ and $n_{o}$ denotes the
number of inputs and outputs respectively. The Kolmogorov-Arnold
representation theorem can be expressed as a two-layer KAN. The inner
functions constitute a KAN layer with $n_{i}=n$, $n_{o}=2n+1$ while the outer
function is another KAN with $n_{i}=2n+1$, $n_{o}=n$. We define the shape of a
KAN by $[n_{1},...,n_{k}]$ with $k$ denoting the number of layers of the KAN.
The Kolmogorov-Arnold representation theorem can be expressed as a KAN of
shape $[n,2n+1,1]$. A deeper KAN can be expressed by the composition $k$
layers:

$\displaystyle\textbf{z}=\text{KAN}(\textbf{t})=(\mathbf{\Psi}_{k}\circ\mathbf{\Psi}_{k-1}\circ...\circ\mathbf{\Psi}_{1})\mathbf{t}.$
(8)

Since all functions are differentiable, KAN can be trained using
backpropagation [33]. For the sake of simplicity, we describe a $2$-depth KAN
as $[n_{i},n,n_{o}]$, where the input layer is not included in the depth
count. The output and input layers will be comprised of $n_{o}$, and $n_{i}$
nodes corresponding to the total amount of time steps while $n$ describes the
number of the hidden layers.

KAN can learn features and compositional structure due to their outer
structure resembling MLPs and optimize the learned features by approximating
the univariate functions with high accuracy due to their internal spline
structure. It should be noted that increasing the number of layers of the
dimension of the grid increases the number of parameters and, hence, the
complexity of the network.

Motivated by the developments in [34], we replace the Multi-Layer Perceptron
(MLP) component of Deep Q-Networks (DQN) with the KAN. Furthermore, we employ
the Double Deep Q-Network (DDQN) update rule to enhance stability and learning
efficiency. DDQN [35] is a Q-learning algorithm based on standard DQN [36],
which features two neural networks to increase the stability of the prediction
of Q-values for each state and action pair. For more details, see Appendix -A.

Throughout the paper, the configurations of KANs and MLPs are written using
the short notation: $\text{KAN}_{d,N_{l}}$ for a KAN and
$\text{MLP}_{d,N_{l}}$ for an MLP. Here, $d$ is the depth (excluding the input
layer) of the network, and $N_{l}$ is the number of neurons per hidden layer.
For example, a KAN with the structure $[n_{i},3,3,3,n_{o}]$ is written as
$\text{KAN}_{4,3}$, where the depth $d$ is 4 and each hidden layer has
$N_{l}=3$ neurons.

###  III-B RL state, action space and reward function

The RL environment in KANQAS is encoded using the tensor-based one-hot
encoding method described in [11]. However, the tensor’s dimensions vary
depending on the size of the action space. The RL state encoding translates a
quantum circuit into a tensor of size $D\times N\times(N+N_{\text{rot}})$,
where $D$ is a hyperparameter and is defined as the maximum depth of the
circuit, $N$ is the number of qubits and $N_{\text{rot}}$ defines the number
of 1-qubit gates. The first $N\times N$ encodes the position of the 2-qubit
gate and the remaining $N\times N_{\text{rot}}$ encodes the position of the
1-qubit gate. Meanwhile, the definition of the reward function varies
depending on the problem under consideration. These aspects are further
elaborated in the following sections

##### Quantum state preparation

In this problem, we initialize with an empty quantum circuit (i.e. without any
gates). Depending on a fidelity-based reward function of the form

$R=\begin{cases}\mathcal{R},&\text{if }F(s_{t})\geq 0.98\\\
F(s_{t}),&\text{otherwise}\end{cases}$ (9)

and by following an $\epsilon$-greedy policy the RL agent sets the probability
of selecting a random action. Where $F(s_{t})$ is the fidelity of a state at
step $t$ generated by KANQAS and $\mathcal{R}>>F(s_{t})$, is a hyperparameter.

The random action is chosen from a predefined action space $(\mathbb{A})$
which contains non-parametrized $1$\- and $2$-qubit gates [7]

$\mathbb{A}=\\{CX,X,Y,Z,H,T\\}.$ (10)

Depending on the action the RL state, which is encoded into a tensor of
dimension $D\times N\times(N+5)$, is modified in the next step.

##### Quantum chemistry

Following the same principle as discussed in the previous section, in this
problem, we initialize with an empty quantum circuit (i.e. without any gates).
However, for a fair comparison with MLP bases CRLQAS, we utilize a reward
function of the form

$R=\begin{cases}5&\text{ if }C_{t}<\xi,\\\ -5&\text{ if }t\geq T_{s}^{e}\text{
and }C_{t}\geq\xi,\\\
\max\left(\frac{C_{t-1}-C_{t}}{C_{t-1}-C_{\min}},-1\right)&\text{ otherwise
}\end{cases}$ (11)

where $C_{t}$ is the VQE cost function (see Eq. 6) at RL step $t$ and $\xi$,
is a hyperparameter, but for VQE it is the chemical accuracy $0.0016$ Hartree.
The random action is chosen from a predefined action space $(\mathbb{A})$
which contains parametrized $1$\- and non-parameterized $2$-qubit gates [11,
9, 8]

$\mathbb{A}=\\{CX,RX,RY,RZ\\}.$ (12)

Depending on the action the RL state, which is encoded into a tensor of
dimension $D\times N\times(N+3)$, is modified in the next step.

##  IV Results

###  IV-A Quantum state preparation

####  IV-A1 Noiseless simulation

In the noiseless case, we consider the structure of the KAN and the MLP as
presented in Table V. In the case of MLP, the activation function is a
hyperparameter and is fixed throughout the training, but in KAN, it is
learnable. For the construction of Bell state, we run a total of $10000$
episodes where each episode is divided into $6$ steps. For a better intuition,
we divide the total number of episodes into 4 intervals where each interval
contains $2500$ episodes.

In Figure 2(a) investigate the probability of success in each interval which
is calculated as per Eq. 5. Meanwhile, in Figure 2(b), we see the variation in
the number of the optimal admissible circuits in each interval.

(a)  (b)

Figure 2: In (a) the probability of successful circuits and in (b) the
probability of optimal successful circuits in finding a 2-qubit maximally
entangled state is slightly higher with KAN than MLP. A total of 10000
episodes are divided into 4 separate intervals where each interval contains
2500 episodes. In (a), each point in an interval corresponds to the
probability of occurrence of a successful episode (see Eq. 5). Similarly, (b)
corresponds to the number of occurrences of an optimal successful episode. The
results are averaged over 20 random seeds (i.e. initialization) of the
networks.

In each interval, the probability is averaged over $20$ random seeds, where
each seed corresponds to the random initialization of the network. The
probability of success with an MLP and KAN is comparable in the first 3
intervals, but in the 4th interval, i.e., in the episodes, falls in the range
of $7500-10000$, the probability of success achieved by KAN is higher. With
MLP, the best probability of success achievable in the 4th interval is
$35.36\%$ whereas with KAN, we can achieve a success probability $36.31\%$ but
the number of optimal admissible ansatz achievable by both the networks is the
same. This indicates that KANs have the potential to generate more diverse
solutions to the same problem than MLPs.

(a)  (b)

Figure 3: In (a) the probability of successful circuits and in (b) the total
number of optimal successful circuits in finding a 3-qubit maximally entangled
state is noticeably higher with KAN than MLP. A total of 8000 episodes are
divided into 4 separate intervals, where each interval contains 2000 episodes.
In (a), each point in an interval corresponds to the probability of occurrence
of a successful episode (see Eq. 5). The results are averaged over 15 random
seeds (i.e. initialization) of the networks. The range defines the best
performance of each interval for both networks. The range defines the region
between the best and the worst performance in each interval for both networks.

Meanwhile, the difference in performance between KAN and MLP becomes
significant in the task of constructing the GHZ state. Here we consider a
depth 2 MLP with $3$ neurons ($\textrm{MLP}_{2,3}$) and depth 4 MLP with $30$
neurons ($\textrm{MLP}_{4,30}$) and compare its performance with a depth 2 KAN
containing $3$ neurons ($\textrm{KAN}_{2,3}$). We run a total of $8000$
episodes where each episode is divided into $12$ steps. Just like the previous
$2$-qubit experiment, we divide the total number of episodes into 4 intervals
where each interval contains $2000$ episodes.

In Figure 3(a) we observe that the average probability of success with
($\textrm{KAN}_{2,3}$) is higher than both $\textrm{MLP}_{2,3}$ and
$\textrm{MLP}_{4,30}$ over $15$ random initialization of the networks. This
difference becomes truly significant when we consider the best performance at
the 4th interval. With $\textrm{KAN}_{2,3}$ we can achieve a probability of
success in the final interval $48.41\%$ whereas with $\textrm{MLP}_{2,3}$ and
$\textrm{MLP}_{4,30}$ we get $38.23\%$ and $38.61\%$ respectively, indicating
a $10.18-9.8\%$ performance enhancement with KAN.

In Figure 3(b) we observe that on average (over $15$ random initialization of
the networks) one can achieve a higher number of optimal admissible ansatz
using $\textrm{KAN}_{2,3}$ than the MLPs. Meanwhile, when we consider the best
performance in the 4th interval the number of optimal admissible ansatz
achievable by KAN is $1.84\times$ to $5.16\times$ higher than
$\textrm{MLP}_{2,3}$ and $\textrm{MLP}_{4,30}$ respectively.

####  IV-A2 Noisy simulation

For the noisy simulation of Bell state preparation, we consider two forms of
gate errors depending on the number of qubits. The gate error refers to the
imperfection in any quantum operation we perform. For the $1$-qubit gate
error, we consider random X noise where with probability $p_{\text{x}}$ an X
gate is applied to the circuit and with $1-p_{\text{x}}$ it applies an
$\mathbb{I}$. Meanwhile, for $2$-qubit gate error, we apply depolarizing noise
which replaces the state of any qubit with a random state of probability
$p_{\text{dep}}$. The configuration of MLP and KAN utilized are summarized in
Table II.

Network | Configuration | Spline and grid | Activation func. | Fidelity
---|---|---|---|---
$\text{KAN}_{3,10}$ | $[84,10,10,12]$ | B-spline, $k=4,G=5$ | learnable | 0.7328
$\text{MLP}_{4,30}$ | $[84,30,30,30,12]$ | NA | ReLU | 0.5005
| NA | LeakyReLU | 0.7300
$\text{MLP}_{4,100}$ |  |  | ELU | 0.6830
$[84,100,100,100,12]$ | NA | ReLU | 0.7300
|  | LeakyReLU | 0.8500
TABLE II: KAN outperforms most of the configurations of MLPs except when
$\text{MLP}_{4,100}$ is operated with LeakyReLU activation function for noisy
Bell state preparation with $p_{\text{x}}=0.3$ and $p_{\text{dep}}=0.2$. This
helps us conclude that to achieve competitive/better performance with an MLP,
it is necessary to fine-tune not just the network’s depth and width but also
the activation function.

At the first stage, we consider noise levels $p_{\text{x}}=0.1$ and
$p_{\text{dep}}=0.01$, where the $\text{MLP}_{4,30}$ (i.e. an MLP of depth $4$
and $30$ neurons) we can achieve a fidelity $99.25\%$ whereas the same
fidelity can be achieved with KAN with just depth $2$ and $2$ neurons.

To elevate the hardness of the problem, in the second experiment, we consider
the following noise levels: $p_{\text{x}}=0.3$ and $p_{\text{dep}}=0.2$. With
KAN configuration presented in Table II we can achieve a fidelity of $73.28\%$
which $\text{MLP}_{4,30}$ with ReLU and LeakyReLU activation functions cannot
achieve. Even when the number of neurons is increased tenfold compared to KAN,
$\text{MLP}_{4,100}$ (i.e., MLP with depth 4 and 100 neurons) still fails to
surpass KAN’s fidelity using ELU and ReLU activation functions. However, with
LeakyReLU activation, it achieves a higher fidelity of $85\%$.

Figure 4: KAN outperforms MLP in finding a parameterized quantum circuit that
solves the 4-qubit LiH and $\texttt{H}_{2}$ molecule. We solve a molecule when
its energy goes below the chemical accuracy, i.e., $1.6\times 10^{-3}$
Hartree. In the plot, the average corresponds to the average performance of
the neural networks over three distinct initializations, and the minimum is
the best-performing seed among the three. We consider $\text{KAN}_{4,3}$ that
corresponds to a KAN of depth 4 and 3 neurons, whereas $\text{MLP}_{6,1000}$
and $\text{MLP}_{4,100}$ defines MLP of depth 6 with 1000 neurons and MLP of
depth 4 with 100 neurons respectively. It should be noted that an MLP of
configuration $\text{MLP}_{6,1000}$ outperforms KAN in the number of 2-qubit
gates, depth, and the total number of parameters, but the number of trainable
parameters required for the MLP is $5.15\times 10^{6}$ which is very large
compared to $5.52\times 10^{4}$ parameters of KAN.

Hence, to achieve competitive/better performance with an MLP, it is necessary
to fine-tune not just the network’s depth and width but also the activation
function. However, with KAN, this process becomes much simpler, as the
activation functions are learnable.

Figure 5:  The number of splines ($k$) has more impact in improving and
stabilizing the performance of the KAN than the grid size ($G$). Here we
measure the improvement of the KAN while constructing the GHZ state by
calculating the average number of gates and depth in each setting. Each point
is marked as $(a,b)$ where $a$ is the total number of successful episodes and
$b$ total optimal successful episodes.

One can argue that KAN has two additional parameters: splines ($k$) and grid
($G$) and tuning these hyperparameters can heavily affect the performance of
the KAN. In Fig. 5, for constructing GHZ state, we show that to achieve better
performance in terms of the depth and number of gates variation in the number
of splines (i.e. $k$) is more effective and stable for the network than
variation in $G$. Whence, in ref. [26], the authors show that achieving a
better minimization of the loss function grid size within the splines of KANs
has a notable impact. Meanwhile, loss function minimization is crucial for
Variational Quantum Algorithms (VQAs), so it is much needed to fine-tune the
grid size while optimizing parameterized gates within parameterized quantum
circuits of VQAs.

###  IV-B Quantum chemistry

In this section, we utilize the recently introduced CRLQAS [11] algorithm to
automate the search for a parameterized quantum circuit to find the ground
state of 4-qubit $\texttt{H}_{2}$ and LiH in the noiseless scenario. Replacing
the MLP in the curriculum reinforcement learning algorithm in CRLQAS by KAN,
we evaluate the performance of the neural networks. The molecular structure of
the $\texttt{H}_{2}$ and LiH are discussed in detail in Appendix -B in Tab. I.

In finding the ground state of $\texttt{H}_{2}$ molecule, we consider an MLP
of depth 6, containing 1000 neurons i.e. $\text{MLP}_{6,1000}$, which is
compared with a KAN of depth 4 with 3 neurons i.e. $\text{MLP}_{4,3}$.
Meanwhile, to find the ground state of LiH molecule we consider
$\text{MLP}_{4,100}$ and compare its performance with $\text{KAN}_{4,3}$. For
more details on the configuration of the neural networks see Tab. VI.

In Fig. 4, we benchmark using the total number of 2-qubit gates, the circuit
depth, and the total number of gates in the parameterized quantum circuit.
These parameters are labeled as 2-qubit gate, Depth, and Numb. gate on the
x-axis of Fig. 4. In both cases, we observe that $\text{KAN}_{4,3}$
outperforms various MLP configurations in terms of average (minimum) depth and
2-qubit gate count. This indicates that $\text{KAN}_{4,3}$ can find more
compact parameterized quantum circuits for solving the $\texttt{H}_{2}$ and
LiH molecules compared to $\text{MLP}_{6,1000}$ and $\text{MLP}_{4,100}$.
However, the total number of gates required to solve these problems is smaller
with MLP, suggesting that for a parameterized quantum circuit with fewer
1-qubit gates, it is preferable to choose MLP over KAN in CRLQAS.

In Appendix -C, we further investigate two different configurations of KAN and
MLP to find the ground state of the 4-qubit LiH. The results reveal that while
larger configurations, such as $\text{MLP}_{6,1000}$, outperform KAN, they
require $5$-$100\times$ more learnable parameters. However, when considering a
comparable number of learnable parameters, for instance, $\text{MLP}_{4,500}$
and $\text{KAN}_{4,50}$, KAN achieves the ground state with a parameterized
quantum circuit of smaller depth and fewer 2-qubit gates. Although MLP results
in a lower error in estimating the ground state, our objective in quantum
chemistry is to attain chemical accuracy, defined as $1.6\times 10^{-3}$
Hartree222The chemical accuracy is often defined as being within 1 kcal/mol,
which is an experimental value, as 1 Hartree = 627.5095 kcal/mol, therefore 1
kcal/mol is equal to approximately 0.0015934 $\approx$ 0.0016 Hartree [37].
Given that the noise in 2-qubit gates is significantly higher than in 1-qubit
gates [38], KAN demonstrates considerable promise for solving quantum
chemistry problems on real quantum hardware.

##  V Resource details

Here we quantify the resources required by KAN and MLP for the simulations
presented in the previous section. In this quantification, we consider (1) the
total number of learnable parameters and (2) the time of execution of each
episode with the two networks.

###  V-A Parameter count

Problem | Network | Configuration | Spline and grid | Parameter count
---|---|---|---|---
Bell state prep. (noiseless) | KAN | $\text{KAN}_{2,2}$ |  B-spline, $k=3,G=5$ | 1728
MLP | $\text{MLP}_{4,30}$ | NA | 4782
GHZ state prep. (noiseless) | KAN | $\text{KAN}_{2,3}$ |  B-spline, $k=3,G=5$ | 5562
MLP | $\text{MLP}_{4,30}$ | NA | $11181$
Bell state prep. (noisy) | KAN | $\text{KAN}_{3,10}$ |  B-spline, $k=4,G=5$ | 9540
MLP | $\text{MLP}_{4,100}$ | NA | $29912$
$\texttt{H}_{2}$ ground state | KAN | $\text{KAN}_{4,3}$ |  B-spline, $k=4,G=5$ | $\mathbf{5.525\times 10^{4}}$
MLP | $\text{MLP}_{6,1000}$ | NA | $5.150\times 10^{6}$
LiH ground state | KAN | $\text{KAN}_{4,3}$ |  B-spline, $k=4,G=5$ | $\mathbf{5.525\times 10^{4}}$
MLP | $\text{MLP}_{4,100}$ | NA | $1.348\times 10^{5}$
TABLE III: KAN requires $2$-$3\times$ less learnable parameters in the task of
quantum state preparation. Meanwhile, in finding the ground state of LiH and
$\texttt{H}_{2}$ KAN takes $2.4$-$93\times$ fewer parameters respectively and
still solves the problems with smaller parameterized quantum circuits than
MLP. To simplify, we denote the networks by their depth and number of neurons.
For instance, a KAN with configuration $[288,3,21]$ is abbreviated as
$\text{KAN}_{2,3}$, where 2 represents the network depth (excluding the input
layer) and 3 denotes the number of neurons in the hidden layer. The same
shorthand notation is applied to MLP networks.

With depth $L$ and width $N$ MLP only needs $O(N^{2}L)$ parameters whereas KAN
requires $O(N^{2}L(G+k))$ parameters. In Table III for the quantum state
generation problem and in Tab. VII for quantum chemistry problems, we
calculate the total number of learnable parameters required for KANs and MLPs
in noisy and noiseless scenarios. We observe that in all cases KAN requires a
lesser number of parameters than MLP.

###  V-B Time of executing each episode

Although KAN requires fewer learnable parameters than MLPs, the average time
of executing each episode for KAN is $120\times$ higher. A recently released
version of KAN, named MultKAN [39], executes an episode approximately
$3.46\times$ slower than MLPs.

Problem | Network | Configuration | Avg. time per episode
---|---|---|---
GHZ state prep. | KAN | $\text{KAN}_{2,3}$ | 0.2049
MLP | $\text{MLP}_{4,30}$ | 0.0592
$\texttt{H}_{2}$ ground state | KAN | $\text{KAN}_{4,3}$ | 2.2103
MLP | $\text{MLP}_{6,1000}$ | 0.7552
LiH ground state | KAN | $\text{KAN}_{4,3}$ | 3.3737
MLP | $\text{MLP}_{4,100}$ | 1.2102
TABLE IV: A new version of KAN [39] is approximately $3.46\times$ slower than
MLPs, which is far better than the older version, which was $120\times$ slower
for quantum state preparation. Meanwhile, KAN is $2.92\times$ and
$2.787\times$ slower than MLP in finding the ground state of 4-qubit
$\texttt{H}_{2}$ and LiH molecule respectively.

##  VI Discussion and future work

In this work, we analyse the practicality of using the recently introduced
Kolmogorov-Arnold Networks in constructing quantum circuits in terms of
probability of success, frequency of optimal solutions and the degrees of
freedom of the network. To this end, we propose KANQAS which replaces the
Multi-Layer Perception (MLP) in reinforcement learning-assisted quantum
architecture search in the double deep Q-network with the Kolmogorov Arnold
network (KAN).

Our experiments with KANQAS reveal that in constructing multi-qubit maximally
entangled states with non-parameterized gates KAN can increase the probability
of success of the RL agent in finding optimal quantum circuits as compared to
MLPs. Moreover, when noise is present in the quantum device, achieving similar
or better outcomes with an MLP necessitates a deeper and wider network
compared to KAN, as well as a careful selection of the appropriate activation
function. Since finding the optimal activation function for deep learning
remains an ongoing challenge [40, 41], KAN has an advantage, as its activation
functions are inherently learnable.

In addressing quantum chemistry problems, KAN demonstrates the ability to
provide a more compact parameterized quantum circuit, with less number of 2
qubit gate and depth for finding the ground state of the chemical Hamiltonian
using VQE, with a significantly reduced number of learnable parameters. It is
important to note that for larger configurations, such as
$\text{MLP}_{6,1000}$, MLP outperforms KAN configurations but requires
$5$-$100\times$ more learnable parameters. When considering a comparable
number of learnable parameters, for example, $\text{MLP}_{4,500}$ and
$\text{KAN}_{4,50}$, KAN achieves the ground state with a parameterized
quantum circuit of smaller depth and fewer 2-qubit gates. Although MLP results
in a lower error in estimating the ground state, our objective in quantum
chemistry is to attain chemical accuracy, defined as $1.6\times 10^{-3}$
Hartree. Given that the noise in 2-qubit gates is significantly higher than in
1-qubit gates [38], KAN demonstrates considerable promise for solving quantum
chemistry problems on real quantum hardware.

Although the number of learnable parameters in KAN is on average
$2$-$100\times$ lesser than in MLPs, one of the biggest disadvantages of KAN
is that it requires $2$-$3\times$ more execution time per episode as compared
to MLPs. However, due to their effectiveness and efficiency in finding
solutions in noiseless and noisy quantum devices, KAN thus appears to be a
reasonable alternative to traditional MLPs in solving quantum architecture
search problems. In the following, we discuss the future research directions
as a follow-up to our research.

  * •

KAN for VQAs A primary direction for future research is addressing the quantum
architecture search problem within variational quantum algorithms, in this
paper we just show the simulation of 4-qubit $\texttt{H}_{2}$ and LiH
molecules, this study should be expanded to bigger molecules such as
$\texttt{H}_{2}\texttt{O}$ 8-qubits. These KAN-assisted algorithms could have
significant applications in quantum chemistry and combinatorial optimization
problems.

  * •

Optimizing computational time of KAN Another important goal is to explore the
use of specialized accelerators, such as tensor processing units or digital
signal processors, to reduce the computation time of KAN.

  * •

Interpretability of KAN Focusing on the interpretability of KAN, future
research should investigate its applicability to similar but scalable problems
to enhance understanding. This can, for example, include the investigation of
subclasses of families of the activation function after training KAN. We
invite readers to explore our Git repository [42] for inspiration and further
motivation in this direction.

##  VII Acknowledgement

AK would like to thank Ziming Liu for fruitful discussions about KAN. AK wish
to thank the Finnish Computing Competence Infrastructure (FCCI) for supporting
this project with computational and data storage resources.

###  -A Double Deep Q-Network (DDQN)

Deep Reinforcement Learning (RL) techniques utilize Neural Networks (NNs) to
adapt the agent’s policy to optimize the return:

$\displaystyle G_{t}=\sum_{k=0}^{\infty}\gamma^{k}r_{t+k+1},$ (13)

where $\gamma\in[0,1)$ is the discount factor. For each state-action pair
$(s,a)$, an action value is assigned, quantifying the expected return from
state $s$ at step $t$ when taking action $a$ under policy $\pi$:

$\displaystyle q_{\pi}(s,a)=\mathbbm{E}_{\pi}[G_{t}|s_{t}=s,a_{t}=a].$ (14)

The goal is to determine the optimal policy that maximizes the expected
return, which can be derived from the optimal action-value function
$q_{\ast}$, defined by the Bellman optimality equation:

$\displaystyle
q_{\ast}(s,a)=\mathbbm{E}\bigg{[}r_{t+1}+\max_{a^{\prime}}q_{\ast}(s_{t+1},a^{\prime})|s_{t}=s,a_{t}=a\bigg{]}.$
(15)

Instead of solving the Bellman optimality equation directly, value-based RL
aims to learn the optimal action-value function from data samples. Q-learning
is a prominent value-based RL algorithm, where each state-action pair $(s,a)$
is assigned a Q-value $Q(s,a)$, which is updated to approximate $q_{\ast}$.
Starting from randomly initialized values, the Q-values are updated according
to the rule:

$\displaystyle Q(s_{t},a_{t})\leftarrow Q(s_{t},a_{t})$
$\displaystyle\qquad\qquad+\alpha\bigg{(}r_{t+1}+\gamma\max_{a^{\prime}}Q(s_{t+1},a^{\prime})-Q(s_{t},a_{t})\bigg{)},$

where $\alpha$ is the learning rate, $r_{t+1}$ is the reward at time $t+1$,
and $s_{t+1}$ is the state encountered after taking action $a_{t}$ in state
$s_{t}$. This update rule is proven to converge to the optimal Q-values in the
tabular case if all $(s,a)$ pairs are visited infinitely often [43]. To ensure
sufficient exploration in a Q-learning setting, an $\epsilon$-greedy policy is
used. Formally, the policy is defined as:

$\displaystyle\pi(a|s)\coloneqq\begin{cases}1-\epsilon_{t}&\text{if
$a=\max_{a^{\prime}}Q(s,a^{\prime})$},\\\
\epsilon_{t}&\text{otherwise}.\end{cases}$ (17)

Network | Problem | Configuration | Spline and grid | Activation func.
---|---|---|---|---
KAN | Bell state prep. | $\textrm{KAN}_{2,2}$ |  B-spline, $k=3,G=5$ | learnable
GHZ state prep. | $\textrm{KAN}_{2,3}$ |  B-spline, $k=4,G=5$ | learnable
MLP | Bell state prep. | $\textrm{MLP}_{4,30}$ |  |
GHZ state prep. | $\textrm{KAN}_{2,3}$ | NA | LeakyReLU
GHZ state prep. | $\textrm{KAN}_{4,30}$ |  |
TABLE V: Configuration for noiseless GHZ and Bell state preparation. In the
case of MLP, the activation function is fixed whereas it is learned during the
network training for KAN. For the sake of simplicity, we represent the
networks by their depth and number of neurons for example, for a KAN of
configuration $[288,3,21]$, we use the shorthand notation $\text{KAN}_{2,3}$
as the KAN is of depth 2 (excluding the input layer) and contains 3 neurons in
the hidden layer.

Network | Molecule | Network configuration | Spline and grid | Activation func.
---|---|---|---|---
KAN | $\texttt{H}_{2}$ | $\text{KAN}_{4,3}$ |  |
| $\text{KAN}_{2,3}$ |  B-spline, $k=10,G=5$ | learnable
LiH | $\text{KAN}_{4,30}$ |  |
| $\text{KAN}_{4,50}$ |  |
MLP | $\texttt{H}_{2}$ | $\text{MLP}_{6,1000}$ |  |
| $\text{MLP}_{4,100}$ |  |
LiH | $\text{MLP}_{4,500}$ | NA | LeakyReLU
| $\text{MLP}_{6,1000}$ |  |
TABLE VI: The configuration of KAN and MLP utilized in tackling quantum
chemistry problems specifically for finding the ground state of 4-qubit
$\texttt{H}_{2}$ and LiH molecule. A detailed configuration of these molecules
is in Tab. I. As discussed in the caption of Tab. V, we represent the networks
by their depth and number of neurons for example, for an MLP of configuration
$[1121,500,500,500,24]$, we use the shorthand notation $\text{MLP}_{4,500}$ as
the MLP of depth 4 (excluding the input layer) and of 500 neurons in the
hidden layer.

Molecule | Network configuration | Min. 2-qubit gate | Min. depth | Min. 1-qubit gate | error | Parameter count
---|---|---|---|---|---|---
4-qubit LiH | $\text{KAN}_{4,30}$ | 11 | 18 | 15 | $1.129\times 10^{-4}$ | $\mathbf{5.784\times 10^{5}}$
$\text{KAN}_{4,50}$ | 8 | 17 | 24 | $2.265\times 10^{-4}$ | $9.960\times 10^{5}$
$\text{MLP}_{4,500}$ | 13 | 18 | 13 | $1.100\times 10^{-4}$ | $1.074\times 10^{6}$
$\text{MLP}_{6,1000}$ | 6 | 11 | 13 | $\mathbf{3.606\times 10^{-5}}$ | $5.152\times 10^{6}$
TABLE VII: We observe that although $\text{MLP}_{6,1000}$ gives the best
results in all benchmarking variables, it requires $10\times$ more learnable
parameters than $\text{KAN}_{4,30}$, making it significantly more expensive in
terms of training time. For a fair comparison, $\text{KAN}_{4,50}$ and
$\text{MLP}_{4,500}$ show a tradeoff between the number of parameterized
1-qubit gates and 2-qubit gates, with $\text{MLP}_{4,500}$ having 1.078 times
more parameters than $\text{KAN}_{4,50}$. Despite having fewer parameters, KAN
provides a competitive approximation of the ground state with a shorter
parameterized quantum circuit, fewer 2-qubit gates, and smaller depth than
MLP.

The $\epsilon$-greedy policy introduces randomness to the actions during
training, but after training, a deterministic policy is used.

We employ NN ad function approximations to extend Q-learning to large state
and action spaces. NN training typically requires independently and
identically distributed data. This problem is circumvented by experience
replay. This method divides experiences into single-episode updates and
creates batches which are randomly sampled from memory. For stabilizing
training, two NNs are used: a policy network, which is continuously updated,
and a target network, which is an earlier copy of the policy network. The
current value is estimated by the policy network, while the target network
provides a more stable target value $Y$ given by :

$\displaystyle
Y_{\text{DQN}}=r_{t+1}+\gamma\max_{a^{\prime}}Q_{\text{target}}(s_{t+1},a^{\prime}).$
(18)

In the DDQN algorithm, we sample the action for the target value from the
policy network to reduce the overestimation bias present in standard DQN. The
corresponding target is defined as:

$\displaystyle Y_{\text{DDQN}}=r_{t+1}+\gamma
Q_{\text{target}}\bigg{(}s_{t+1},\arg\max_{a^{\prime}}Q_{\text{policy}}(s_{t+1},a^{\prime})\bigg{)}.$
(19)

This target value is approximated via a loss function. In this work, we
consider the loss function as the smooth L1-norm.

###  -B Configuration of neural networks

The detailed configurations of the KANs and the MLPs utilized in the main text
to tackle quantum state preparation and quantum chemistry problems are
provided in Tab. V and Tab. VI

###  -C Quantum chemistry with different configurations of KAN and MLP

In this section, we further explore the different configurations of KAN and
MLP in finding the ground state of the 4-qubit LiH molecule. We use the
following variables to benchmark the performance of the different neural
networks: the minimum number of 1-qubit gates (Min. 1-qubit gate), the minimum
number of 2-qubit gates (Min 2-qubit gate), the error in estimating the ground
state energy (error), and the number of learnable parameters in the network
(Parameter count). The results are presented in detail in the Tab. VII.

From the Tab. VII we observe although the $\text{MLP}_{6,1000}$ provides us
with the best result in terms of all the benchmarking variables, the number of
learnable parameters it requires to achieve this outcome is $10\times$ higher
than $\text{KAN}_{4,30}$, which is far more expensive in terms of training
time. Meanwhile, for the sake of fair comparison, if we observe the
performance of $\text{KAN}_{4,50}$ and $\text{MLP}_{4,500}$ where the number
of parameters in MLP is $1.078\times$ more than KAN, we notice a tradeoff
between the number of parameterized 1-qubit gates and the number of 2-qubit
gates. The KAN provides us with a competitive approximation of the ground
state with a shorter parameterized quantum circuit, containing less 2-qubit
gate and smaller depth than MLP despite having a smaller number of parameters.
Considering that the noise in 2-qubit gates in real quantum hardware is many
scales higher than 1-qubit gates, KAN with a comparable number of parameters,
shows a large potential in quantum architecture search in quantum hardware.

## References

  * [1] P. Ren, Y. Xiao, X. Chang, P.-Y. Huang, Z. Li, X. Chen, and X. Wang, “A comprehensive survey of neural architecture search: Challenges and solutions,” _ACM Computing Surveys (CSUR)_ , vol. 54, no. 4, pp. 1–34, 2021\.
  * [2] S.-X. Zhang, C.-Y. Hsieh, S. Zhang, and H. Yao, “Differentiable quantum architecture search,” _Quantum Science and Technology_ , vol. 7, no. 4, p. 045023, 2022.
  * [3] Z. Lu, P.-X. Shen, and D.-L. Deng, “Markovian quantum neuroevolution for machine learning,” _Physical Review Applied_ , vol. 16, no. 4, p. 044039, 2021.
  * [4] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, “The theory of variational hybrid quantum-classical algorithms,” _New Journal of Physics_ , vol. 18, no. 2, p. 023023, 2016.
  * [5] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio _et al._ , “Variational quantum algorithms,” _Nature Reviews Physics_ , vol. 3, no. 9, pp. 625–644, 2021\.
  * [6] A. Sarkar, “Automated quantum software engineering,” _Automated Software Engineering_ , vol. 31, no. 1, pp. 1–17, 2024.
  * [7] E.-J. Kuo, Y.-L. L. Fang, and S. Y.-C. Chen, “Quantum architecture search via deep reinforcement learning,” _arXiv preprint arXiv:2104.07715_ , 2021.
  * [8] M. Ostaszewski, L. M. Trenkwalder, W. Masarczyk, E. Scerri, and V. Dunjko, “Reinforcement learning for optimization of variational quantum circuit architectures,” _Advances in Neural Information Processing Systems_ , vol. 34, pp. 18 182–18 194, 2021.
  * [9] A. Kundu, P. Bedełek, M. Ostaszewski, O. Danaci, Y. J. Patel, V. Dunjko, and J. A. Miszczak, “Enhancing variational quantum state diagonalization using reinforcement learning techniques,” _New Journal of Physics_ , vol. 26, no. 1, p. 013034, 2024.
  * [10] Y. Du, T. Huang, S. You, M.-H. Hsieh, and D. Tao, “Quantum circuit architecture search for variational quantum algorithms,” _npj Quantum Information_ , vol. 8, no. 1, p. 62, 2022.
  * [11] Y. J. Patel, A. Kundu, M. Ostaszewski, X. Bonet-Monroig, V. Dunjko, and O. Danaci, “Curriculum reinforcement learning for quantum architecture search under hardware errors,” _arXiv preprint arXiv:2402.03500_ , 2024.
  * [12] A. Sadhu, A. Sarkar, and A. Kundu, “A quantum information theoretic analysis of reinforcement learning-assisted quantum architecture search,” _arXiv preprint arXiv:2404.06174_ , 2024.
  * [13] A. R. Morgillo, S. Mangini, M. Piastra, and C. Macchiavello, “Quantum state reconstruction in a noisy environment via deep learning,” _Quantum Machine Intelligence_ , vol. 6, no. 2, 2024. [Online]. Available: http://dx.doi.org/10.1007/s42484-024-00168-x
  * [14] A. Lumino, E. Polino, A. S. Rab, G. Milani, N. Spagnolo, N. Wiebe, and F. Sciarrino, “Experimental phase estimation enhanced by machine learning,” _Physical Review Applied_ , vol. 10, no. 4, p. 044033, 2018.
  * [15] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, “Noise-induced barren plateaus in variational quantum algorithms,” _Nature Communications_ , vol. 12, no. 1, p. 6961, 2021.
  * [16] L. Bittel and M. Kliesch, “Training variational quantum algorithms is np-hard,” _Physical Review Letters_ , vol. 127, p. 120502, Sep 2021. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.127.120502
  * [17] Z. Liu, Y. Wang, S. Vaidya, F. Ruehle, J. Halverson, M. Soljačić, T. Y. Hou, and M. Tegmark, “Kan: Kolmogorov-arnold networks,” _arXiv preprint arXiv:2404.19756_ , 2024.
  * [18] A. A. Aghaei, “fkan: Fractional kolmogorov-arnold networks with trainable jacobi basis functions,” _arXiv preprint arXiv:2406.07456_ , 2024.
  * [19] R. Genet and H. Inzirillo, “Tkan: Temporal kolmogorov-arnold networks,” _arXiv preprint arXiv:2405.07344_ , 2024.
  * [20] Z. Bozorgasl and H. Chen, “Wav-kan: Wavelet kolmogorov-arnold networks,” _arXiv preprint arXiv:2405.12832_ , 2024.
  * [21] D. W. Abueidda, P. Pantidis, and M. E. Mobasher, “Deepokan: Deep operator network based on kolmogorov arnold networks for mechanics problems,” _arXiv preprint arXiv:2405.19143_ , 2024.
  * [22] M. Kiamari, M. Kiamari, and B. Krishnamachari, “Gkan: Graph kolmogorov-arnold networks,” _arXiv preprint arXiv:2406.06470_ , 2024.
  * [23] J. Xu, Z. Chen, J. Li, S. Yang, W. Wang, X. Hu, and E. C.-H. Ngai, “Fourierkan-gcf: Fourier kolmogorov-arnold network–an effective and efficient feature transformation for graph collaborative filtering,” _arXiv preprint arXiv:2406.01034_ , 2024.
  * [24] R. Genet and H. Inzirillo, “A temporal kolmogorov-arnold transformer for time series forecasting,” _arXiv preprint arXiv:2406.02486_ , 2024.
  * [25] K. Xu, L. Chen, and S. Wang, “Kolmogorov-arnold networks for time series: Bridging predictive power and interpretability,” _arXiv preprint arXiv:2406.02496_ , 2024.
  * [26] C. J. Vaca-Rubio, L. Blanco, R. Pereira, and M. Caus, “Kolmogorov-arnold networks (kans) for time series analysis,” _arXiv preprint arXiv:2405.08790_ , 2024.
  * [27] M. Cheon, “Kolmogorov-arnold network for satellite image classification in remote sensing,” _arXiv preprint arXiv:2406.00600_ , 2024.
  * [28] Y. Wang, J. Sun, J. Bai, C. Anitescu, M. S. Eshaghi, X. Zhuang, T. Rabczuk, and Y. Liu, “Kolmogorov arnold informed neural network: A physics-informed deep learning framework for solving pdes based on kolmogorov arnold networks,” _arXiv preprint arXiv:2406.11045_ , 2024.
  * [29] I. N. Levine, D. H. Busch, and H. Shull, _Quantum chemistry_. Pearson Prentice Hall Upper Saddle River, NJ, 2009, vol. 6.
  * [30] A. N. Kolmogorov, “On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition,” in _Doklady Akademii Nauk_ , vol. 114, no. 5. Russian Academy of Sciences, 1957, pp. 953–956.
  * [31] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” _Neural networks_ , vol. 2, no. 5, pp. 359–366, 1989.
  * [32] G. D. Knott, _Interpolating cubic splines_. Springer Science & Business Media, 1999, vol. 18.
  * [33] D. E. Rumelhart, R. Durbin, R. Golden, and Y. Chauvin, “Backpropagation: The basic theory,” in _Backpropagation_. Psychology Press, 2013, pp. 1–34.
  * [34] R. Waris and Corentin, “Kolmogorov-Arnold Q-Network (KAQN)-KAN applied to Reinforcement learning, initial experiments,” https://github.com/riiswa/kanrl, 2024.
  * [35] H. Van Hasselt, A. Guez, and D. Silver, “Deep reinforcement learning with double q-learning,” in _Proceedings of the AAAI conference on artificial intelligence_ , vol. 30, no. 1, 2016.
  * [36] V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski _et al._ , “Human-level control through deep reinforcement learning,” _nature_ , vol. 518, no. 7540, pp. 529–533, 2015.
  * [37] Y. Chen, L. Zhang, H. Wang, and W. E, “Ground state energy functional with hartree–fock efficiency and chemical accuracy,” _The Journal of Physical Chemistry A_ , vol. 124, no. 35, pp. 7155–7165, 2020.
  * [38] E. Wilson, F. Mueller, L. Bassman Oftelie, and C. Iancu, “Empirical evaluation of circuit approximations on noisy quantum devices,” in _Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis_ , 2021, pp. 1–15.
  * [39] “MultKAN,” https://github.com/KindXiaoming/pykan/blob/master/kan/MultKAN.py, 2024.
  * [40] S. Hayou, A. Doucet, and J. Rousseau, “On the selection of initialization and activation function for deep neural networks,” _arXiv preprint arXiv:1805.08266_ , 2018.
  * [41] P. Ramachandran, B. Zoph, and Q. V. Le, “Searching for activation functions,” _arXiv preprint arXiv:1710.05941_ , 2017.
  * [42] A. Kundu, “KANQAS GitHub,” https://github.com/Aqasch/KANQAS_code, 2024\.
  * [43] F. S. Melo, “Convergence of q-learning: A simple proof,” _Institute Of Systems and Robotics, Tech. Rep_ , pp. 1–4, 2001.