Quantum technologies in the second quantum revolution utilize quantum effects as a resource to go beyond technologies based on classical physics. In quantum computation entanglement and superposition are used to develop algorithms that can be superior to classical algorithms. To be useful such algorithms must be implemented on quantum computers that can handle sufficiently large problems. There are mainly two hardware paradigms that are available on the market today: quantum annealers and limited gate-based quantum computers of the NISQ-type (noisy-intermediate-scalequantum). On these, algorithms for solving optimization problems are of particular interest. Most of these algorithms are heuristic and hence there is no guarantee of success, but some studies indicate that there may be a quantum advantage in some cases. Implementation of algorithms based on fully error-corrected qubits lies in the far future. Current state of the art research on hardware is to implement a few qubits with rudimentary error-correction. In this report we first give an introduction to quantum computation and related concepts. Then quantum algorithms and quantum computers are introduced on a general level before the limited hardware that is available on the market today is discussed. Thereafter we describe quantum algorithms for optimization problems and their potential future applications in defense. Finally benchmarks that may be used to quantify how quantum and classical algorithms compare are discussed. The aim of this report is to give an introduction to why and how quantum computation may become an alternative route for solving certain military relevant optimization problems.