This work focuses on expressing the TSP with Time Windows (TSPTW for short) as a quadratic unconstrained binary optimization (QUBO) problem. The time windows impose time constraints that a feasible solution must satisfy. These take the form of inequality constraints, which are known to be particularly difficult to articulate within the QUBO framework. This is, we believe, the first time this major obstacle is overcome and the TSPTW is cast in the QUBO formulation. We have every reason to anticipate that this development will lead to the actual execution of small scale TSPTW instances on the D-Wave platform.

Game theory and its quantum extension apply in numerous fields that affect people’s social, political, and economical life. Physical limits imposed by the current technology used in computing architectures (e.g., circuit size) give rise to the need for novel mechanisms, such as quantum inspired computation. Elements from quantum computation and mechanics combined with game-theoretic aspects of computing could open new pathways towards the future technological era. This paper associates dominant strategies of repeated quantum games with quantum automata that recognize infinite periodic inputs. As a reference, we used the PQ-PENNY quantum game where the quantum strategy outplays the choice of pure or mixed strategy with probability 1 and therefore the associated quantum automaton accepts with probability 1. We also propose a novel game played on the evolution of an automaton, where players’ actions and strategies are also associated with periodic quantum automata.

Classical game theory is an important field with a long tradition of useful results. Recently, the quantum versions of classical games, such as the prisoner’s dilemma (PD), have attracted a lot of attention. This game variant can be considered as a specific type of game where the player’s actions and strategies are formed using notions from quantum computation. Similarly, state machines, and specifically finite automata, have also been under constant and thorough study for plenty of reasons. The quantum analogues of these abstract machines, like the quantum finite automata, have been studied extensively. In this work, we examine well-known conditional strategies that have been studied within the framework of the classical repeated PD game. Then, we try to associate these strategies to proper quantum finite automata that receive them as inputs and recognize them with a probability of 1, achieving some interesting results. We also study the quantum version of PD under the Eisert–Wilkens–Lewenstein scheme, proposing a novel conditional strategy for the repeated version of this game.

This paper studies the Hamiltonian cycle problem (HCP) and the traveling salesman problem (TSP) on D-Wave quantum systems. Motivated by the fact that most libraries present their benchmark instances in terms of adjacency matrices, we develop a novel matrix formulation for the HCP and TSP Hamiltonians, which enables the seamless and automatic integration of benchmark instances in quantum platforms. We also present a thorough mathematical analysis of the precise number of constraints required to express the HCP and TSP Hamiltonians. This analysis explains quantitatively why, almost always, running incomplete graph instances requires more qubits than complete instances. It turns out that QUBO models for incomplete graphs require more quadratic constraints than complete graphs, a fact that has been corroborated by a series of experiments. Moreover, we introduce a technique for the min-max normalization for the coefficients of the TSP Hamiltonian to address the problem of invalid solutions produced by the quantum annealer, a trend often observed. Our extensive experimental tests have demonstrated that the D-Wave Advantage_system4.1 is more efficient than the Advantage_system1.1, both in terms of qubit utilization and the quality of solutions. Finally, we experimentally establish that the D-Wave hybrid solvers always provide valid solutions, without violating the given constraints, even for arbitrarily big problems up to 120 nodes.

This paper presents a General Variable Search heuristic, trying to discover initial, feasible solution for the Travelling Salesman Problem with Time Windows. More specifically, we provide all relevant information regarding TSP-TW, Variable Neighborhood Search (VNS) and initialization methods, then we quote all related work in the direction on solving this NP-HARD problem, and at last we propose our new implementation for generating initial, feasible solution for the TSP-TW. We are thoroughly focused on the behavior of our main algorithm to different random-based or sorting-based initialization procedures of our main implemented algorithm. At last, we show experimentally that in some cases, the use of a sorting function as an initialization function in this algorithm did not work and never returned a feasible solution in some cases.