Quantum Process Tomography (QPT) methods aim at identifying, i.e. estimating, a quantum process. QPT is a major quantum information processing tool, since it especially allows one to experimentally characterize the actual behavior of quantum gates, that may be used as the building blocks of quantum computers. We here consider unitary, possibly dense (i.e. without sparsity constraints) processes, which corresponds to isolated systems. Moreover, we aim at developing QPT methods that are applicable to a significant number of qubits and hence to a high state space dimension, which allows one to tackle more complex problems. Using the unitarity of the process allows us to develop methods that first achieve part of QPT by performing an eigenanalysis of the estimated density matrix of a process output. Building upon this idea, we first develop a class of complete algorithms that are single-stage, i.e. that use only one eigendecomposition. We then extend them to multiple-stage algorithms (i.e. with several eigendecompositions), in order to address high-dimensional state spaces while being less limited by the estimation errors made when using an arbitrary given Quantum State Tomography (QST) algorithm as a building block of our overall methods. We first propose two-stage methods and we then extend them to dichotomic methods, whose number of stages increases with the considered state space dimension. The relevance of our methods is validated by means of simulations. Single-stage and two-stage methods first yield the following results. Just running them with standard PC and software already makes it possible to evaluate their performance for up to 13 qubits, i.e. with state space dimensions up to a few thousands. This shows their attractiveness in terms of accuracy and proves that they solve the core of the dense QPT problem in a very limited time frame. For other test results, see the paper.