## 1 Introduction  
As a relatively new and rapidly evolving field in both science and technology, the full potential of quantum computing remains largely uncharted. Researchers are still exploring its practical usefulness across various domains. While significant advancements have been made, such as [PERSON]'s algorithm [1] for factoring integers and [PERSON]'s algorithm [2] for unstructured search problems, which demonstrate the theoretical advantages of quantum computing, its real-world applications are still being uncovered. Among the different branches of quantum computing, one area has demonstrated significant practical potential: quantum annealers. Quantum annealing, a method specifically designed for solving optimization problems, has shown promising results and near-practical utility. Quantum annealers, such as those developed by D-Wave Systems, are being increasingly utilized for their ability to find approximate solutions to complex optimization problems more efficiently than any other quantum architecture paradigm [3]. However, the downside is that quantum annealers are not universal quantum computers. Despite the fact that the vast majority of combinatorial optimization problems can be represented as quadratic unconstrained binary optimization (QUBO) formulations, researchers continue to explore and improve other approaches. With simultaneous advancements in the quality of universal quantum computers, the true leader in quantum optimization has yet to be determined. Most quantum optimization algorithms in the NISQ (noisy intermediate-scale quantum) era involve representing the optimization problem as a Hamiltonian [4]. In physics, a Hamiltonian represents the energy of a system and governs its time evolution. In quantum computing, it often serves as a crucial component for determining measurable quantities and finding the most favorable solutions by seeking the system's ground state. Although some general guidelines for constructing such Hamiltonians have been established [5], there is no single, universal method for doing so. The construction of a Hamiltonian for a given problem can vary based on the emphasis placed on different aspects of the problem or by considering different quantum paradigms. For example, quantum annealers are designed to handle optimization problems in QUBO form, which constrains the Hamiltonian to contain entanglements of at most second-degree, i.e., entangling at most two qubits with each other. Another aspect of encoding the Hamiltonian is to introduce interpretability to the energy values. In its simplest form, the Hamiltonian can be monotonic, meaning that better solutions have lower energy than worse ones. However, the Hamiltonian can also be encoded to provide more detailed information. For instance, the energy value might represent the number of unsatisfied constraints, giving a clear indication of how far a solution is from being feasible. Additionally, each digit or component of the Hamiltonian's energy value could correspond to different constraints or optimization criteria, allowing for a more granular analysis of the solution's quality. Choosing the encoding for a Hamiltonian should not be considered in isolation from the performance of the algorithm. Hamiltonians that encode more detailed information in their energy values tend to be more complex. This complexity often necessitates the algorithm to invest more time and computational effort to achieve convergence. Additionally, such Hamiltonians may require the construction of more intricate quantum circuits, which involve costly operations and are more susceptible to noise. This is particularly problematic for current quantum computers, which are limited by noise and error rates in the NISQ era. Therefore, a balance must be struck between the richness of the information provided by the Hamiltonian and the practical limitations of the quantum hardware. In this paper, we focus on a specific subset of gate-based quantum optimization algorithms: namely, variational algorithms.