#  Model-Driven Engineering for Quantum Programming:
A Case Study on Ground State Energy Calculation

Furkan Polat  Electrical and Electronics Engineering, Koç University, &
Department of Computer Science, University of Antwerp, Antwerp, Belgium
<EMAIL_ADDRESS>Hasan Tuncer  Department of Physics and Astronomy, Purdue
University, IN, USA
<EMAIL_ADDRESS>Armin Moin  Department of Computer Science, University of
Colorado, Colorado Springs, CO, USA
<EMAIL_ADDRESS>Moharram Challenger  Department of Computer Science,
University of Antwerp, & Flanders Make Strategic Research Center, Antwerp,
Belgium
<EMAIL_ADDRESS>
###### Abstract

Abstract:

This study introduces a novel framework that brings together two main Quantum
Programming methodologies, gate-based Quantum Computing and Quantum Annealing,
by applying the Model-Driven Engineering principles. This aims to enhance the
adaptability, design, and scalability of quantum programs, facilitating their
design and operation across diverse computing platforms. A notable achievement
of this research is the development of a mapping method for programs between
gate-based quantum computers and quantum annealers, which can lead to the
automatic transformation of these programs. Specifically, this method is
applied to the Variational Quantum Eigensolver Algorithm and Quantum Annealing
Ising Model, targeting ground state solutions. Finding ground-state solutions
is crucial for a wide range of scientific applications, ranging from
simulating chemistry lab experiments to medical applications, such as vaccine
development. The success of this application demonstrates Model-Driven
Engineering for Quantum Programming frameworks’ practical viability and sets a
clear path for Quantum Computing’s broader use in solving intricate problems.

Keywords: Model-Driven Engineering, Quantum Computing Programming, Model
Transformation, Variational Quantum Eigensolver

##  I Introduction

Various models of computation exist for Quantum Computing (QC), such as
quantum circuit models (i.e., Quantum gates) and adiabatic (i.e., quantum
annealing), which are well-established ones[1]. However, it is already
theoretically proven that any quantum circuit algorithm can be transformed
into a quantum annealing algorithm with the same time complexity and vice-
versa, hence showing they are polynomial equivalent [2].

Furthermore, the key technologies for the actual physical implementation of
quantum processors comprise superconducting, ultra-cold atoms (e.g., trapped
ions), quantum optics, and spin-based. For instance, Linke et al. [3]
conducted experiments on two 5-qubit quantum computers with the quantum
circuit model, which had two different technologies and architectures, namely
superconducting and trapped ions. They illustrated that the superconducting
system offered faster gate clock speeds and a solid-state platform, whereas
the other one featured superior qubits and reconfigurable connections. They
concluded that the performance of those systems reflected the topology of
connections in the base hardware, thus supporting the idea that quantum
software and hardware should ideally be co-designed [4].

The Model-Driven Engineering (MDE) paradigm has already been proposed as a
reasonable choice to cope with the heterogeneity and complexity observed in
the QC domain and its programming [4, 5, 6].

In this paper, the vision of applying MDE techniques for Quantum Programming
(QP)[4, 7] is extended by proposing a new framework and applying it in the
context of the ground state energy calculation problem, which has concrete use
cases in various domains. This includes material science for developing new
materials [8], chemistry for creating efficient catalysts [9], pharmaceuticals
for advancing drug and vaccine development [10], and physics for exploring
phenomena like superconductivity and quantum magnetism [11].

This paper is structured as follows: Section II provides a review of related
work and presents an overview of QC, the MDE paradigm, and the ground-state
energy calculation problem. In Section III, it is introduced that MDE
framework for quantum programs, focusing on the strategy for developing
Platform Independent Model for Ground State Calculation. Section IV outlines
our proof of concept work for Model-Driven Engineering for Quantum Programs
(MDE4QP). After implementing a ground state solution separately in gate-based
QPUs and Quantum Annealers, it demonstrates the results from abstract mapping.
Finally, the paper concludes with a summary of our contributions and future
research directions in Section V.

##  II Related Work and Background

Previous research in QC has explored various models of computation and
technologies for physical implementation. Aharonov et al. [2] theoretically
demonstrated the equivalence between quantum circuit algorithms and quantum
annealing algorithms. This foundational work laid the groundwork for
understanding the relationship between different QC models.

In terms of physical implementations, Linke et al. [3] conducted experiments
using two different QC technologies: superconducting qubits and trapped ions.
They highlighted the difference in performance and architecture between these
systems, emphasizing the importance of co-designing quantum software and
hardware.

The MDE paradigm has emerged as a promising approach for managing the
complexity and heterogeneity in quantum programming [4, 5, 6]. Moin et al. [4]
proposed using MDE techniques for quantum programming, laying the foundation
for our work.

###  II.1 Quantum Computing

QC is a transformative approach to computing that integrates quantum mechanics
principles into information processing This field’s theoretical foundation was
pioneered by Richard Feynman, who proposed the concept of quantum simulation
in 1982. Feynman suggested that quantum systems could simulate physical
phenomena more efficiently than classical computers, addressing complex
quantum mechanical problems [12].

Significant progress in QC was marked by the development of Shor’s algorithm.
This algorithm, aimed at prime factorization, runs much faster on quantum
computers than the fastest algorithms on classical computers, highlighting the
advanced computational capabilities of quantum systems for certain tasks [13].

Following these early steps, quantum computing (QC) has undergone significant
advancements, demonstrating its capability for speed improvements in various
applications, a phenomenon known as quantum speedup. For instance, quantum
algorithms can search an unsorted database with $N$ entries in time
proportional to $\sqrt{N}$, as shown by Grover [14], unlike classical
computers, which take time proportional to $N$. Moreover, quantum computers
can perform Fourier transforms, invert sparse matrices, and find their
eigenvalues and eigenvectors in time that scales polynomially in $\log_{2}N$,
as demonstrated by Biamonte et al. [15]. This contrasts with classical
algorithms that scale as $N\log_{2}N$.

The Sycamore processor by Google demonstrated the quantum advantage by
performing a specific computational task exponentially faster than the current
state-of-the-art classical supercomputers, highlighting the potential of
quantum processors to achieve significant speed-ups for certain tasks [16].

The impact of QC spans various domains, from cryptography, where it could
break current encryption schemes [17], to optimization challenges in logistics
and manufacturing [18], and to detailed simulations in chemistry and physics
[19]. The potential for QC to accelerate drug discovery and materials science
highlights its implications across several fields [20].

###  II.2 Different Approaches in Quantum Computing

QC has a variety of computational paradigms, each distinguished by unique
principles, methodologies, and quantum phenomena exploitation. There are
various models of computation for QC, such as quantum circuit/logic gate,
adiabatic/annealing, topological, quantum walks, one clean qubit and
measurement-based. Since our work focuses on quantum circuit gate model and
Annealing model mapping, this subsection provides an overview of the two
primary approaches to them.

####  II.2.1 Quantum Gate-based Model

In this model, quantum operations are represented as sequences of quantum
gates, each performing a specific transformation on the qubits. These gates
include single-qubit gates, which act on individual qubits, and multi-qubit
gates, which entangle multiple qubits. By applying sequences of these gates to
the qubits in a quantum circuit, complex quantum computations can be
performed. Notably, the gate-based model forms the foundation for various
quantum algorithms, including Shor’s algorithm [13] for prime factorization,
Grover’s algorithm for database searching [14] and Variational Quantum
Algorithms (VQA) for optimization [21], eigenvalue estimation [22] and machine
learning [23].

####  II.2.2  Quantum Annealing

This model originates from the adiabatic theorem to solve optimization
problems by evolving an initial quantum state towards the ground state, which
represents the problem’s optimal solution. This evolution represents the
problem’s solution without manipulating the qubits through direct operators,
marking a difference from the step-by-step operations seen in gate-based
computing. AQC’s method is inherently suited for various optimization
challenges, encoding solutions in the system’s lowest energy state [24].
Quantum Annealers efficiently addresses optimization problems such as
logistical planning, financial portfolio optimization, and complex scheduling.
Quantum annealing can often reach satisfactory solutions faster than classical
computational methods by enabling the system to escape local minima and
explore more of the solution landscape. This makes it a valuable tool in
scenarios where speed is prioritized over absolute precision in finding the
lowest possible energy state [25, 26].

###  II.3 Challenges in Quantum Computing

One of the main challenges in quantum computing is maintaining the coherence
of quantum states. Qubits are extremely sensitive to their environment. Even
small temperature fluctuations, electromagnetic waves, or other interactions
with the environment can cause qubits to lose their quantum properties
(decoherence) and the information they contain to become corrupted. Quantum
noise can also interfere with the operation of qubits, causing errors in
calculations [27].

Therefore, robust error correction methods are very important. Quantum error
correction is more complex than classical error correction and requires
significant additional qubits and computational resources to encode
information in such a way that errors can be detected and corrected without
disrupting the quantum state [28].

Scalability is another important challenge. Building a quantum computer with
enough qubits to solve practical problems is a major challenge. Increasing the
number of qubits not only requires maintaining consistency across a larger
system but also exponentially increases the complexity of error correction.
The physical systems used to realize qubits (superconducting circuits, trapped
ions, etc.) currently face significant engineering and manufacturing
challenges to scale up while maintaining fidelity [29, 30].

Quantum computers will need to work together with classical computers to
perform practical tasks. This integration poses challenges in terms of
creating efficient interfaces and communication protocols between quantum and
classical systems [31].

Additionally, since quantum platforms take different approaches, their
efficiency on different problems may differ. In this case, it will be
important to develop a system that can run these platforms together. Our aim
is to eliminate heterogeneity through Model-Driven Engineering (MDE).

Quantum computers will need to work together with classical computers to
perform practical tasks. This integration poses challenges in terms of
creating efficient interfaces and communication protocols between quantum and
classical systems [31].

###  II.4 Model-Driven Engineering (MDE)

MDE is a software development approach that prioritizes creating and using
DSMLs to represent the core functionality of a system. DSMLs capture the
desired behaviour and properties at a higher level of abstraction instead of
focusing on the final code. This focuses on the problem domain, rather than
coding specifics, which leads to better communication and understanding among
involved developers[32]. MDE offers several benefits, including enabling early
validation and verification through techniques like model analysis and
simulation, ultimately leading to a more robust and efficient software
solution. Additionally, MDE can enhance development productivity by automating
tasks such as code generation [33].

###  II.5 Ground State Calculation

The Ground State Calculation Problem is fundamental to understanding the
physical and chemical properties of systems at the quantum level. Calculating
the ground state of quantum systems enables scientists to predict behaviours
and interactions, crucial for fields such as material science and drug
discovery.

One area where ground-state energy calculations play a pivotal role is in the
development of new pharmaceuticals. Quantum chemistry simulations, facilitated
by Variational Quantum Algorithms, are essential for discovering Hamiltonian
spectra [22]. These algorithms are key to analyzing reaction rates and
understanding catalytic processes, which are vital for creating new drugs.
This method allows for the analysis of molecular energies, critical for
predicting chemical reactions and interactions, and showcases the potential of
near-term quantum hardware to enhance calculations in chemistry and drug
development.

####  II.5.1 Classical Algorithms for Ground State Calculation

Classical algorithms for calculating the ground states of the quantum states
are essential in understanding how atoms and molecules behave. These methods
help scientists find the lowest energy state of quantum systems, which is
important for predicting chemical reactions and material properties. Key
techniques include the Variational Principle, Hartree-Fock Method [34], Monte
Carlo Simulations [35]. Each traditional method has its own way of approaching
the problem, but all aim to provide accurate predictions about the quantum.

##### Density Matrix Renormalization Group (DMRG)

The development of the Density Matrix Renormalization Group (DMRG) algorithm
marked a significant advancement in computational physics, addressing some of
the limitations faced by classical algorithms. DMRG, primarily applied to one-
dimensional quantum systems, optimizes the calculation of low-energy
eigenstates by iteratively refining the system’s state space, focusing on the
most relevant degrees of freedom associated with entanglement. This process is
significantly more efficient than traditional methods, offering a glimpse into
the potential of specialized algorithms in overcoming computational challenges
[36].

##### Tensor Networks

Building upon the success of DMRG, tensor networks extend the concept of
efficient quantum state representation to higher dimensions. These networks
express the quantum state as a product of interconnected tensors, simplifying
the representation of complex entanglements and correlations within the
system:

$\displaystyle\Psi_{i_{1},i_{2},\ldots,i_{N}}=\sum_{\alpha_{1},\alpha_{2},\ldots,\alpha_{N}}A^{[1]}_{i_{1};\alpha_{1}}A^{[2]}_{i_{2};\alpha_{1}\alpha_{2}}\ldots
A^{[N-1]}_{i_{N-1};\alpha_{N-2}\alpha_{N-1}}A^{[N]}_{i_{N};\alpha_{N}\alpha_{N-1}}$
(1)

where $i_{k};\alpha_{k}$ represents indices corresponding to local properties,
and the bond indices labeled with $\alpha_{k}$ can encode correlations such as
entanglement. The decomposition is a special case known as the matrix product
state (MPS). The $\alpha_{k}$ indices’ maximum number of values, known as the
bond size, serves as a quantitative measure of entanglement.

##### Artificial Neural Networks in Quantum Physics

The intersection of quantum physics and computational science has further
expanded with the application of artificial neural networks (ANNs) to model
quantum systems. ANNs, particularly through structures like the Restricted
Boltzmann Machine (RBM), have shown promise in capturing the complex multi-
particle wave functions of quantum states [37, 38]. The RBM employs a layered
approach, where connections between visible and hidden units encode the
quantum system’s properties:

$\psi_{\text{RBM}}(\mathbf{j})=\exp\left(\sum_{i=1}^{L}a_{i}j_{i}\right)\prod_{i=1}^{M}\cosh\left(b_{i}+\sum_{k}W_{ik}j_{k}\right),$
(2)

This method,tensor networks, provides a useful tool for simulating quantum
systems, Through the adaptation of ANNs, researchers can model systems with a
level of efficiency and accuracy that opens new avenues for exploration and
understanding in quantum physics [39, 40, 41].

The progression from classical algorithms through artificial neural networks,
reflects the dynamic and interdisciplinary nature of computational research in
quantum physics. Notably, NetKet emerges as a pivotal framework in this
domain, offering a library for studying many-body quantum systems with machine
learning approaches, highlighting the potential of neural network quantum
states for quantum state tomography, supervised learning, and ground state
searches [37]. Further, the explicit demonstration of initial state
construction in ANNs using NetKet and validation with IBM Q Experience
Platform underscore the practical applicability and accuracy of these models
in simulating quantum computations [38]. With the release of NetKet 3, built
on JAX, the toolbox for many-body quantum systems now supports advanced neural
network ansätze, GPU/TPU accelerators, and discrete symmetry groups, marking
significant advancements in the field [39].

Those classical methods encounter challenges as the system’s complexity
escalates with the addition of more particles. Real-world problems often
involve large systems with many interacting particles, and the intricate
interactions within such systems result in an exponential growth in
computational workload. This presents a significant challenge that classical
computers, with their inherent limitations, struggle to manage effectively.

Quantum computers, on the other hand, offer an advantage, and Qubits play a
key role there. Adding a qubit to the system doubles the computational
capacity. So, with n qubits, the computational possibilities expand to
$2^{n}$. This scalability provides quantum computers with an edge, offering a
promising solution for handling the complexities of calculating the ground
states of the molecules [42, 43].

##### Quantum Phase Estimation

Recent advancements in quantum computing have introduced new methods for
addressing eigenvalue problems that are challenging for classical
computational approaches. A key development in these quantum solutions is the
Quantum Phase Estimation (QPE) algorithm, notable for its efficiency advantage
over conventional computational methods. The QPE algorithm necessitates
$O(p^{-1})$ quantum operations to achieve a specified level of precision, $p$,
in estimating the eigenvalue linked to a known eigenvector of a Hermitian
operator $H$.

The precision $p$ highlights the accuracy in the eigenvalue estimation
process, with the algorithm assuming initial access to the eigenvector in
question. The aim is to uncover the associated eigenvalue. The quantum
operations required for this process, directly related to $O(p^{-1})$, mandate
sequential execution of the operation $e^{-iHt}$. In practical applications,
this could require to use of millions or even billions of quantum gates—far
exceeding the operational capacity of current quantum technologies, which are
predominantly capable of managing only tens to hundreds of gates [44, 45, 10].

##### Variational Quantum Eigensolver (VQE)

Figure 1: The schematic diagram illustrates a variational quantum algorithm
(VQA). [46]

In hybrid VQE solvers, the crucial quantum component involves the utilization
of Ising model solvers. Quantum computers work on the Ising model solution
based on a given ansatz as illustrated in Fig. 1. The inputs include a cost
function $C(\bm{\theta})$, where $\bm{\theta}$ represents parameters encoding
the solution, an ansatz whose parameters are trained to minimize the cost, and
potentially a set of training data $\\{\rho_{k}\\}$ used in optimization.
Typically, the cost function can be expressed as a set of functions
$\\{f_{k}\\}$.

The ansatz is depicted as a parameterized quantum circuit on the left, next to
a neural network shown on the right. During each iteration, a quantum computer
efficiently estimates the cost (or its gradients), with this information fed
into a classical computer utilizing optimizers to navigate the cost landscape
$\bm{\theta}$ and solve the optimization problem. Upon meeting termination
conditions, the VQE provides an estimate of the solution, with the form of the
output dependent on the specific task. The red-shaded box highlights common
types of outputs. is a hybrid algorithm designed to estimate the ground state
energy of a molecule by blending variational techniques with quantum
mechanics. It operates iteratively, fine-tuning wave function guesses to
minimize energy until it reaches the system’s most stable state. This process
involves both quantum and classical components: the quantum part computes the
system’s energy for a given Hamiltonian, while the classical part optimizes
variational parameters [22, 46].

The fundamental equation at the heart of VQE for ground sate calculation is
given by:

$H\ket{\psi}=E\ket{\psi},$ (3)

where $H$ represents the Hamiltonian acting on the wave function $\psi$, and
$E$ denotes the associated energy.

The Hamiltonian $H=\hat{T}+\hat{V}$ encompasses kinetic and potential
energies, incorporating terms that describe electron-electron, nucleus-
nucleus, and electron-nucleus interactions for electrons and nuclei. The
representation of the wave function evolves from classical Slater determinants
to second quantization. This methods expresses the states using fermionic
creation and annihilation operators. This formalism facilitates the expression
of the Hamiltonian for computation on quantum computers [47].

VQE employs techniques like the Jordan-Wigner transformation to map fermionic
operators to qubit operators that enables encoding of molecular systems into
qubits [48]. The goal is to minimize the expectation value of the Hamiltonian:

$E_{\theta}=\min_{\theta}\frac{\langle\Psi(\theta)|\hat{H}|\Psi(\theta)\rangle}{\langle\Psi(\theta)|\Psi(\theta)\rangle},$
(4)

where $\theta$ represents the parameters iteratively optimized through quantum
gates. This iterative process aims to closely approximate the ground state of
the molecular system, leveraging the precision and efficiency of quantum
computation for energy calculations

##### Quantum Annealing

is also employed for ground state calculations, as it uses the principles of
adiabatic quantum mechanics to solve optimization problems. It progresses from
an initial Hamiltonian, representing the problem at hand, to a final
Hamiltonian that corresponds to the ground state solution.

##### Ising Model

describes interacting spins in a system, where each spin interacts with its
neighboring spins on a lattice grid. The interactions are defined by coupling
constants, and there may also be local magnetic fields acting on each
spin.[49] In VQE, the Ising model is used to map the terms of the molecular
Hamiltonian onto a set of qubits, enabling quantum computers to simulate
molecular systems and calculate their properties efficiently.[46]

Additionally, the Ising model finds widespread application in other contexts,
such as in quantum annealers, where optimization problems are initially
formulated as Ising models. The ground energy of these Ising models represents
the optimal solution for various problems, including the traveling salesman
problem and financial optimization.

##  III Proposed Approach

A well-established MDE could provide a robust framework for developing quantum
algorithms because it has shown its ability for automated code generation,
model checking, and analysis in other domains. This paper presents a proof of
concept for applying MDE to a quantum application to its hybrid/quantum
equivalent or vice versa. As a first step, the paper is particularly focusing
on ground-state calculations.

This paper proposes a new method that combines MDE with QC to tackle the
complexities of systems with multiple particles. This approach aims to improve
upon traditional methods and provide a new perspective for conducting ground-
state calculations, where the complexity of systems involving multiple
particles often makes traditional computational approaches insufficient[50].
With the MDE paradigms that resolve heterogeneity in both classical and QC,
this research contributes to advancing QC applications for solving practical
problems [51, 4, 5, 6].

The future of Classical and Quantum Computing will likely involve a mix of
processors with different architectures and technologies. This will make
software development more complex and require more abstraction. Modeling
languages and Domain-Specific Modeling Languages (DSMLs) will be essential for
providing this abstraction. These methods allow developers to model their
applications without concerning about the underlying hardware or software. MDE
and DSMLs will also be important for analyzing and verifying these
applications, as the platforms will be complex and distributed. This will
require formalizing the semantics of QC and the specific target platforms for
QC. In other words, the knowledge of QC experts needs to be transferred to the
modeling languages and tools.

MDE4QP builds upon the existing MDE tools in classical platforms. Developers
can focus on their application’s core functionality without getting involved
with the intricacies of the underlying hardware or software. MDE4QP aspires to
bring this same level of abstraction to QC platforms, particularly
applications involving Quantum Machine Learning (QML).

Similar to how Domain- Specific Modeling (DSM) methodology offers a layer of
abstraction, MDE4QP promises to do the same for Quantum program development.
This approach is illustrated in Fig. 2. for the proof of concept case in
ground state calculation.

The proposed approach prioritizes flexibility and adaptability for hardware
platforms. Developers might not know the specific hardware (CPUs, GPUs). Still
application will run at runtime once they use DSMLs. MDE4QP aspires to extend
this flexibility to the world of QC by covering various Quantum Processor
Units (QPUs) with their diverse architectures and capabilities.

Additionally providing abstraction, MDE4QP aims to streamline the development
process further through automated code generation. This will be achieved by
implementing model-to-code transformations, essentially translating the high-
level models created by developers into code that can run on the target
platform. But the potential benefits go beyond mere code generation. The paper
explores the intriguing possibility of using model-to-model transformations to
achieve various functionalities at the modeling level. These functionalities
includes the ability to:

  * •

Convert between classical and quantum applications: This feature enables the
seamless transition of applications between classical and QC environments,
providing flexibility in the choice of computational model.

  * •

Translate between different models of quantum computation: Such as quantum
circuits and quantum annealing. This capability is essential for optimizing
applications for specific problem types and uses the strengths of different QC
approaches.

  * •

Fine-tune applications for specific QPU architectures: This feature allows
developers to optimize of applications to maximize performance and efficiency
on a particular QPU with its unique characteristics. Developers can design
applications with tailored optimizations for the target QPU, enhancing overall
computational performance.

Also, MDE4QP recognizes the importance of early-stage analysis and
verification. The approach aims to enable these activities at the modeling
level, allowing developers to catch errors and optimize performance before the
actual code is generated. This can significantly reduce development time and
effort while ensuring the robustness and efficiency of the final application.

Figure 2:  The Proposed Vision of MDE4QP in the context of Model Driven
Architecture for the ground state calculations in different quantum platforms

The first step towards the MDE4QP paradigm focuses on establishing a unified
abstract mapping to bridge the gap between theoretical quantum algorithms and
their practical implementation across various platforms. The work for this
framework begins with the conceptual mapping of quantum computations for
ground-state calculations. This initial step lays the groundwork for designing
and executing algorithms that can seamlessly operate across different
platforms.

There are two paths for calculating he ground states solution of an Ising
model: Adiabatic and VQE, as illustrated in 3. And Quantum Annealers uses the
adiabatic path to evolve the Ising model.On the other side, VQE path includes
the gate-based QPU processing of the model. The core objective of the
methodology is to implement a Platform Independent Model (PIM) of the Ising
model solution, which includes quantum gate-based models and quantum annealer
models. By conceptualizing the solutions to the Ising model in a platform-
agnostic manner, the flawless transition of quantum algorithms from one
computational model to another is enabled

This work explores the relationship between quantum annealing ising model and
Quantum Gate-Based VQE through the framework of abstract mapping. This
analysis reveals the potential for cross-platform compatibility between
different QC paradigms. Such compatibility facilitates the emergence of a new
generation of versatile quantum applications. These applications would bridge
the gap between diverse hardware platforms and enable the development of
adaptive solutions.

#####  Model-to-model Transformation

The next step in this paradigm shift involves the design of domain-specific
models (DSMs) that facilitate seamless transfer between diverse quantum and
classical computing platforms. A system for model-to-model (M2M)
transformation across various computation models is envisioned. This system
would leverage PIMs (as illustrated in Fig. 2) to translate algorithms for
execution on specific quantum hardware.

The ultimate goal lies in establishing a comprehensive framework that can
dynamically adapt to the operational paradigms of both quantum annealing and
gate-based architectures. This framework should ensure efficient algorithm
translation and execution across platforms, minimizing the need for extensive
re-engineering.

##### Model-to-code Transformation (Automated Code Generation)

Related to designing Model-to-Model Transformation, Automated Code Generation
through Model-to-code transformation is essential. This approach, as shown in
Fig. 2, would enable the automatic generation of code from platform-specific
models into desired application programming interfaces (APIs).

Automated code generation from high-level models into executable quantum code
would significantly accelerate the development of quantum algorithms. This
would, in turn, enhance the accessibility of QC for researchers and
practitioners who might lack in-depth expertise in quantum programming
languages.

To achieve this goal, the development of specialized tools and compilers is
necessary. These tools would be responsible for interpreting high-level
algorithmic descriptions and generating optimized quantum code tailored for
various platforms. These tools will require a level of intelligence to
navigate the complexities of quantum computation. This includes tasks like
qubit allocation, gate selection, and the incorporation of error correction
strategies. Ultimately, the aim is to produce efficient and reliable quantum
programs.

Figure 3:  The schematic diagram illustrates the abstract mapping of solutions
from different platforms to ground state calculation. The upper green side
represents the Quantum Annealing aproach, while the lower orange side depicts
the Hybrid VQE approach in a Quantum Gate-based model. This conceptually
simplifies the two methods for solving the same ground state calculation
problem.”

##  IV Case Study: Ground State Energy Calculation using Gate-based QC and
Quantum Annealer

This research explores a new method for applying the Ising model to both gate-
based quantum computers and quantum annealers. This method uses a special
mapping process to translate quantum algorithms between these two platforms.
This allows researchers to easily switch between different types of quantum
computers to solve problems, which can lead to faster discoveries and better
solutions.

This method is tested by applying it to three different versions of the Ising
model on both gate-based and quantum annealing machines. The goal is to find
the lowest possible energy state (ground state) for each version of the model.
Here’s the Ising model’s general Hamiltonian, which describes its energy:

$H=\sum_{i,j}J_{ij}\sigma_{i}^{z}\sigma_{j}^{z}+\sum_{i}h_{i}\sigma_{i}^{z}$
(5)

where $J_{ij}=1$ denotes uniform interaction strength between connected qubits
$i$ and $j$, $h_{i}=1$ represents a uniform external magnetic field applied to
each qubit, and $\sigma_{i}^{z}$ is the Pauli-Z operator acting on qubit $i$.
The choice of $J_{ij}=1$ is often made for simplicity and normalization
purposes. It allows easier comparison and analysis of different systems.
Additionally, this choice simplifies mathematical expressions and calculations
while still capturing the essential behavior of the system.

The specific entanglement patterns are described as follows:

  * •

Full-Entangled Ising Model: The model contains interactions among all pairs of
qubits, which can be mathematically described as:

$\sum_{i<j}\sigma_{i}^{z}\sigma_{j}^{z}\quad$ (6)

This notation sums over all unique pairs $i,j$, indicating all-to-all
connectivity.

  * •

Linear-Entangled Ising Model: This model features a sequential chain of
interactions, which can be expressed as:

$\sum_{i=0}^{N-2}\sigma_{i}^{z}\sigma_{i+1}^{z}$ (7)

where $N$ is the total number of qubits, illustrating a linear sequence of
connections from qubit $0$ to $N-1$.

  * •

Circular-Entangled Ising Model: To represent the circular connectivity where
the last qubit is also connected back to the first:

$\sum_{i=0}^{N-1}\sigma_{i}^{z}\sigma_{(i+1)\mod N}^{z}$ (8)

Here, $\mod N$ ensures that after the last qubit, the summation loops back to
the first qubit, forming a circular entanglement pattern.

For those interested in further exploration or contributions, the project’s
code is available on GitHub. 111https://github.com/micss-lab/MDE4QP/.

###  IV.1 VQE with Different Ising Models using Gate-Based Quantum Computing

The First step of the investigation highlights the VQE approach to tackle the
Ising model with gate-based QC. VQE, a hybrid quantum-classical algorithm,
uses the quantum mechanical strengths of superposition and entanglement,
combined with classical optimization routines, to efficiently approximate the
ground state of quantum systems.

The analysis uses a 1-D Ising model, articulated through Pauli spin matrices,
to explore the interplay between quantum spins under the influence of an
external magnetic field ($\lambda$), set to a constant value of 1 for this
study.

This Hamiltonian in 5 serves as the objective function for the VQE algorithm,
guiding the search for the system’s ground state.

(a) Full-entangled ansatz

(b) Linear-entangled ansatz

(c) Circular-entangled ansatz

Figure 4: Circuits of different entangled ansatzes. Each subfigure depicts a
unique quantum circuit layout corresponding to its entanglement strategy.

A 4-qubit system is used to explore various entanglement patterns within the
ansatz designs: full-entangled (Fig. 4(a)), linear-entangled (Fig. 4(b)), and
circular-entangled (Fig. 4(c)). These designs are instrumental in assessing
the impact of entanglement on the optimization process and the algorithm’s
ability to approximate the ground state energy of the corresponding
model.These ansatz circuits derived from [53] which were initially used in
[54]. The convergence towards the ground state energy is visually tracked
through energy plots for each ansatz, revealing the algorithm’s effectiveness
in approximating the Ising model’s ground state.

(a) Full-entangled ansatz

(b) Linear-entangled ansatz

(c) Circular-entangled ansatz

Figure 5: Step vs energy plots for different entangled ansatzes. Each
subfigure illustrates the energy convergence with the red line indicating the
exact ground state energy.

  * •

The full-entangled ansatz reached an energy state of $-2.0$, with the qubit
configuration $\ket{0,1,0,1}$.The iteration steps are shown in figure. 5(a).

  * •

The linear-entangled ansatz approached the ground state with an energy of
$-3.0$, with the qubit configuration $\ket{0,1,0,1}$.The iteration steps are
are shown in figure. 5(b).

  * •

The circular-entangled ansatz demonstrated significant promise by converging
to a ground state energy value of $-4.0$, with the qubit configuration
$\ket{0,1,0,1}$.The iteration steps shown in figure. LABEL:subfig:cir_vqe.

This VQE model investigates the 1-D Transverse Ising model using computational
techniques to determine -state energy properties. The investigation via VQE
with Qiskit demonstrates convergence to the target ground-state energies
through optimization. Across all ansatz types, convergence to the ground-state
energies (-2, -3, -4) for corresponding full linear and circular Ising models
is observed.

###  IV.2 Quantum Annealer Ground State Solution for Different Ising Models

After gate-based QC, this study also investigates the application of quantum
annealing to solve the Ising model using the D-Wave system. Quantum annealing
uses quantum mechanical phenomena such as superposition and quantum tunneling,
offering a specialized and efficient way to find the system’s ground state.
This process is inherently suited to solving optimization problems, like
finding the minimum energy configuration in the Ising model, by gradually
transitioning from a superposition of all possible states towards the lowest
energy state.

The D-Wave system facilitates this exploration through its Quantum Processing
Unit (QPU), designed around a lattice of qubits that can be programmatically
entangled and biased to represent the Ising model. The Ising model’s
Hamiltonian 5, representing the energy landscape of the system, is directly
mapped onto the QPU. This mapping involves defining the external magnetic
field (h) affecting individual qubits and the interaction strength (J) between
qubits as circular, linear, and full-entangled models.

The implementation of each Ising model variants on the D-Wave system has a
sequence of critical steps. The process begins with the initialization phase,
where the Ising model parameters, specifically the $h$ and $J$ values, are
configured to define the problem space accurately. Then, the
EmbeddingComposite and DWaveSampler are employed to sample the Ising model
multiple times. This step is essential for exploring the solution space
thoroughly to identify low-energy states that approximate the ground state of
the system. Furthermore, the following graphs for each Ising model variants—
full-entangled (Fig.6(c)), Linear- entangled (Fig. 6(b)), and circular-
entangled (Fig. 6(a))—provide a visual representation of the qubit
interactions and their implementation on D-Wave’s QPU within the Pegasus
topology. Finally, the culmination of this process involves the determination
of the ground state by extracting the sample with the lowest energy from the
result set, along with its corresponding energy value.

The ground state energy and corresponding qubit configurations for the
different Ising models are as follows:

  * •

Full-Entangled Model: Achieved a ground state energy of $-2.0$ with the qubit
configuration $\ket{0,1,0,1}$.

  * •

Linear Model: Achieved a ground state energy of $-3.0$ with the same qubit
configuration $\ket{0,1,0,1}$.

  * •

Circular Model: Achieved a ground state energy of $-4.0$ with the qubit
configuration $\ket{0,1,0,1}$.

These results highlight the D-Wave system’s capability to efficiently solve
the Ising model, with variations in the entanglement patterns affecting the
ground state energy and qubit configurations. The ability to visualize the
problem setup and solutions via the D-Wave Inspector helps in understanding
the quantum annealing process and the dynamics of the Ising model on a quantum
annealer.

(a) Full-entangled Ising Model

(b) Linear-entangled Ising Model

(c) Circular-entangled Ising Model

Figure 6: Schematics of different Ising Models on a Quantum Annealer,
constructed Leap SDK’s default inspector tool, depicted across three
entanglement configurations. Each subfigure shows the connections among qubits
on the left and the implementation on D-Wave’s QPU within Pegasus topology on
the right.

###  IV.3 Abstract Mapping For Platform Independent Model

The primary objective of this study is to establish a conceptual framework and
PIM that enables the integration of quantum annealing and gate-based quantum
devices for solving the ground state energy for multi-partite quantum systems.
PIM standardizes quantum computations for ground state calculation across
different platforms like D-Wave’s Leap SDK and IBM’s Qiskit.

The mapping approach and the implementation of the PIM facilitate a seamless
transition between computational paradigms and standardize methodology. Table
1 provides a simplified comparison between the quantum annealing approach and
the VQE method in gate-based QC for solving the Ising model:

Table 1:  Comparison of Quantum Annealing and Quantum Gate-Based VQE Approaches in Ground State Calculation Aspect | Quantum Annealer Approach | VQE Approach
---|---|---
Hamiltonian Mapping | Directly onto hardware qubits | As quantum circuit operators
State Preparation | Physical qubits initialized in superposition | Dynamic state preparation
Entanglement | Fixed by hardware design | Configurable via software
Parameter Optimization | Annealing controls transitions | Algorithmic parameters
Ground State Search | Natural evolution | Iterative optimization
Result Analysis | Immediate hardware output | Computed post-process

This comparative table illustrates the distinctive methodologies used in
quantum annealing and the VQE method for the specific case, The Ground State
Calculation problem. It highlights the shift from a more theoretical
discussion to a practical application within the PIM framework, which has been
fully realized and now supports these two QC approaches effectively for ground
state problem.

###  IV.4 Platform Independent Model (PIM) for Ground State Calculations

This paper proposes a PIM for Ground state calculations to achieve
standardized representation of quantum computations across various platforms,
such as D-Wave’s Leap SDK and IBM’s Qiskit. The PIM allows users to define
quantum systems using high-level parameters, thus eliminating the need for
intricate details of the underlying hardware and SDK.

Unlike universal languages, quantum computers require specific instructions
tailored to their unique capabilities. These Platform Specific Models (PSMs)
act like custom blueprints, accounting for a machine’s strengths, like qubit
connections, and limitations, such as available operations. While PSMs unlock
a quantum computer’s full potential, they often demand in-depth knowledge of
the underlying hardware, creating a hurdle for programmers.

By contrasting these platform-specific approaches, the need for a PIM becomes
evident, as it abstracts these details into a unified model that can be
understood and utilized without deep platform-specific knowledge.

####  IV.4.1 Key PIM Parameters

The PIM simplifies the quantum system definition by standardizing the
following key parameters:

  * •

Number of Qubits: Defines the quantum system’s size.

  * •

External Magnetic Field Constant: Represents the average influence of external
magnetic fields across the system.

  * •

Coupler Constant: Captures the strength of interactions (couplings) between
qubits.

  * •

Platform Name: Identifies the targeted QC platform (e.g., ’dwave’, ’qiskit’).

  * •

Entanglement Type: Describes the interconnection pattern of qubits (e.g.,
linear, circular, full).

####  IV.4.2 Translation Functions between PSM and PIM

The system implements two primary function types:

  1. A.

PSM to PIM Conversion Functions: These functions (e.g., PSM_to_PMI_dwave and
PSM_to_PMI_qiskit) translate PSMs into the platform-independent PIM format.
This conversion is crucial for enabling uniform analysis and understanding of
quantum systems across different platforms.

  2. B.

PIM to PSM Conversion Function: The PIM_ground_state_calculator function takes
the platform-independent specifications and generates the necessary platform-
specific model. This allows direct execution of quantum simulations or
calculations on the designated hardware or simulation framework.

####  IV.4.3 Use Cases and Benefits of PIM

The proposed PIM system offers several practical advantages:

  * •

Platform-Agnostic Quantum System Design: Users can specify quantum systems
using PIM parameters without being limited to a specific platform. This is
ideal for theoretical and educational purposes where understanding core
concepts outweighs platform-specific details.

  * •

Cross-Platform Translation: Users working on one platform (e.g., D-Wave’s Leap
SDK) can effortlessly translate their models to another platform (e.g., IBM’s
Qiskit) using the conversion functions. This is particularly beneficial for
comparing results across platforms or leveraging specific features offered by
different platforms.

  * •

Simplified Interface for Quantum Developers: By abstracting the complexities
of each platform into a common set of parameters, PIM simplifies quantum
software development. This can accelerate development cycles and reduce the
learning curve for new quantum programmers.

##  V Conclusion and Future Work

###  V.1 Key Findings and Contributions

This study not only proposes a unified framework for integrating different
quantum computing (QC) platforms but also successfully implements it for a
specific use case of ground state calculation. A significant milestone in this
framework is the implementation of the PIM for ground state calculation, which
bridges the gap between quantum annealing and gate-based QC. It enables the
use of a single set of high-level parameters to define and manipulate quantum
systems across different platforms. Additionally, it allows transitions from
one PSM to another between the Quantum Annealer model and the Gate-Based
Quantum system.

###  V.2 Future Research and Potential Impact

The results outline possibilities for further research, particularly in
refining these mapping techniques for even more sophisticated and broader QC
challenges. Future work could explore extending the PIM to support emerging
quantum technologies and integrating Domain-Specific Languages (DSLs) that
could further simplify quantum algorithm development across platforms.

####  V.2.1 Model Enhancements

To further improve the PIM system, several aspects can be considered:

  * •

Validation and Error Handling: The conversion functions should be robust in
handling exceptional cases and provide informative error messages for
unsupported configurations or invalid inputs.

  * •

Support for Additional Platforms: As the field of QC evolves, incorporating
support for emerging platforms will ensure the continued relevance and utility
of the PIM system.

  * •

User-Friendly Documentation and Examples: Comprehensive documentation and
illustrative use cases will empower users to use the PIM system effectively.
Tutorials or interactive web applications could significantly enhance user
engagement and understanding.

  * •

Extending Compatibility: Currently, the framework is tailored only for Ising
model solutions for the ground state problem. To fully tap into the potential
of quantum technologies, it is essential to expand its applicability to
encompass all programming algorithms in quantum computing.

These improvements will solidify MDE frameworks as valuable tools for
facilitating standardized development and execution of quantum computations
across diverse platforms for quantum or hybrid applications.

The impact of using MDE principles in QC is profound, potentially impact how
quantum algorithms are developed and making QC more accessible to a wider
audience.

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