Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices, where quantum approximation optimization algorithms (QAOAs) constitute promising candidates for demonstrating tangible quantum advantages based on NISQ devices. In this paper, we consider the maximum likelihood (ML) detection problem of binary symbols transmitted over a multiple-input and multiple-output (MIMO) channel, where finding the optimal solution is exponentially hard using classical computers. Here, we apply the QAOA for the ML detection by encoding the problem of interest into a level-p QAOA circuit having 2p variational parameters, which can be optimized by classical optimizers. This level-p QAOA circuit is constructed by applying the prepared Hamiltonian to our problem and the initial Hamiltonian alternately in p consecutive rounds. More explicitly, we first encode the optimal solution of the ML detection problem into the ground state of a problem Hamiltonian. Using the quantum adiabatic evolution technique, we provide both analytical and numerical results for characterizing the evolution of the eigenvalues of the quantum system used for ML detection. Then, for level-1 QAOA circuits, we derive the analytical expressions of the expectation values of the QAOA and discuss the complexity of the QAOA based ML detector. Explicitly, we evaluate the computational complexity of the classical optimizer used and the storage requirement of simulating the QAOA. Finally, we evaluate the bit error rate (BER) of the QAOA based ML detector and compare it both to the classical ML detector and to the classical MMSE detector, demonstrating that the QAOA based ML detector is capable of approaching the performance of the classical ML detector. This paves the way for a host of large-scale classical optimization problems to be solved by NISQ computers.