# A QUBO Model for the Traveling Salesman Problem with Time Windows
## 1 Introduction  
Quantum computing promotes the exploit of quantum-mechanical principles for computation purposes. [PERSON] in 1982 suggested that computation could be done more efficiently by taking advantage of the power of quantum \"parallelism\" [1]. Since then, quantum computing has gained a lot of momentum and today appears to be one of the most prominent candidates to partially replace classical, silicon-based systems. There exist a number of quantum algorithms that run more efficiently than the best known classical counterparts; arguably, the most famous ones are [PERSON]'s [2] and [PERSON]'s [3] algorithms. For an in-depth study of quantum computing and quantum information, the reader is referred to [4]. Adiabatic quantum computation was proposed by [PERSON] et al. [5; 6] in the early 2000s [7]. It is based on an important theorem of quantum mechanics, the adiabatic theorem [8; 9]. This theorem asserts that if a quantum system is driven by a slowly changing Hamiltonian that evolves from \(H_{init}\) to \(H_{fin}\), then if the systems starts in the ground state of \(H_{init}\), the system will end up in the ground state of \(H_{fin}\). It has been shown that adiabatic quantum computation is equivalent to standard quantum computation [10]. Quantum annealing is based on the quantum adiabatic computation paradigm. Quantum annealing was initially proposed by [PERSON] and [PERSON] [11; 12] and has since then been used for tackling combinatorial optimization problems. The way that a quantum annealer tries to solve problems is very similar to the way optimization problem are solved using (classical) simulated annealing [13]. An energy landscape is constructed via a multivariate function, such that the ground state corresponds to the solution of the problem. The quantum annealing process is iterated until an optimal solution is found with a sufficiently high probability. The \"quantum\" in \"quantum annealer\" refers to the use of multi qubit tunneling [11; 12]. The high degree of parallelism is the advantage of quantum annealing over classical code execution. A quantum annealer explores all possible inputs in parallel to find the optimal solution to a problem, which might prove crucial when it comes to solvingNP-complete problems. We caution the reader however that, at present, these techniques should be considered more as an automatic heuristic-finding program than as a formal solver [14]. The prospect of employing quantum computing principles for solving optimization problems is particularly promising [15; 16; 17; 18; 19]. In this approach, which can be considered as an \"analog\" method to tackle optimization problems, the ground state of a Hamiltonian represents an optimal solution of the optimization problem at hand. Typically, the quantum annealing process starts with the system being in an equal superposition of all states. Then, an appropriate Hamiltonian is applied and the system evolves in a time-dependent manner according to the Schrodinger equation. The state of the system keeps changing according to strength of the local transverse field, which varies with respect to time. Finally, the transverse field is smoothly turned off and the system finds itself in the ground state of a properly chosen Hamiltonian that encodes an optimal solution of the optimization problem. The D-Wave platform, the first actual commercial system based upon quantum annealing, has demonstrated its ability to solve certain quadratic unconstrained (mainly binary) optimization problems. The core of D-Wave's machine that applies the quantum annealing principles for complex combinatorial optimization problems is the quantum processing unit (QPU). In the D-Wave computer, the quantum bits (which we refer to as qubits from now on) are the lowest energy states of superconducting loops [7; 20; 21]. In these states, there is a circulating current and a corresponding magnetic field. Since a qubit is a quantum object, its state can be a superposition of the 0 state and the 1 state at the same time.