## 2 Work Context
Advances in quantum computing offer a way forward for efficient solutions to many cases of substantial eigenvalue problems unsolvable in a traditional way [52]. Quantum approaches to finding eigenvalues previously relied on the Quantum Phase Estimation (QPE) algorithm. The QPE is one of the essential subroutines in quantum computation. It serves as a central building block for many quantum algorithms and offers exponential acceleration compared to classical methods, and it requires several quantum operations \(O\Big{(}\frac{1}{p}\Big{)}\) to obtain an estimate with precision \(p\)[52; 53]. Variational Quantum Eigensolver (VQE), proposed by [PERSON] [52], based on the variational principle and form, estimates the ground state energy of the Hamiltonian of the problem [54]. The VQE is a hybrid quantum/classical algorithm originally proposed to approximate the ground state of a quantum system (the state attaining the minimum energy). Quantum Approximate optimization Algorithms (QAOA), based on the principles of adiabatic quantum computation [44; 53], is used to solve QUBO problems. [PERSON] and [PERSON] showed the advantages of QAOA compared to classical approaches [46; 47]. While [55] debated the problems of constrained polynomial optimization using adiabatic quantum computation methods, other scientists, such as [PERSON] and [PERSON] [56], worked on how to apply restrictions in QUBO systems to avoid the use of large numbers of the coefficients. This resulted in more qubits from the use of quadratic penalties, they proposed a new combinatorial design which involved solving problems of linear programming of mixed integers to adapt applications restitution. [PERSON], [PERSON], [PERSON], and [PERSON] [57] investigated and solved the Hamiltonian cycle problem in computational frameworks, such as quantum circuits, quantum walks, and adiabatic quantum computing. All of these advances in quantum computing have been applied to routing and scheduling techniques. Reference [58] contributed an expansive vision and discussions on Ising formulations for various NP-complete and NP-hard optimization problems, emphasizing using as the fewest possible as possible qubits. In the same way, there have been many works of literature on the VRP [59] and its variants. Amazon Braket [60] is a cloud-based (Figures 2 and 3), fully managed quantum computing service that helps researchers and developers get started into quantum world technology to accelerate research and discovery. Amazon Braket provides a development environment to explore, create, test, and run quantum algorithms, quantum circuit simulators, and different quantum hardware technologies. We will take advantage of all these related works to define an appropriate strategy for our proposal in this NISQ era. Studying and comparing different optimization methods of warehouse's challenge, like picking and batching, ref. [17] proposes three options: analytical models, simulation experiments, and mathematical programming. In our approach, we consider the latter. We use a set of mathematical expressions that describe the problem, represented by an objective mathematical function and constraints within the classical context and translate it to the quantum domain. While reviewing state of the art research, Reference [61] was found. The integrated order routing and the batch problem is modeled in such systems as an extended multi-tank vehicle routing problem with network flow formulations of three indices and two commodities. Such a variable neighborhood search algorithm provides close to optimal solutions within a computational time acceptable for classical but not quantum computing. Figure 3: Technical Specifications of Quantum Simulators systems where we can see the state vector simulator (34 qubits) and tensor network simulator (50 qubits). Figure 2: Technical specifications of the Quantum Hardware Technologies (Gate-based superconducting qubits, Gate-based ion traps, and Quantum annealing) available in Amazon Braket.