We investigate the anisotropic quantum orbital compass model on an infinite square lattice by means of the infinite projected entangled-pair state algorithm. For varying values of the Jx and Jz coupling constants of the model, we approximate the ground state and evaluate quantities such as its expected energy and local order parameters. We also compute adiabatic time evolutions of the ground state, and show that several ground states with different local properties coexist at Jx = Jz. All our calculations are fully consistent with a first order quantum phase transition at this point, thus corroborating previous numerical evidence. Our results also suggest that tensor network algorithms are particularly fitted to characterize first order quantum phase transitions. Introduction.-When quantum many-body systems are cooled down close to zero temperature, important collective phenomena may occur [1]. A good example is provided by transition-metal oxides, whose physical properties have become of increasing interest in the last few years [2]. In these compounds the orbital degrees of freedom of the atomic electrons play a key role in determining properties such as metal-insulator transitions, high-temperature superconductivity and colossal magnetoresistance.The paradigmatic approach to these systems is based on the so-called orbital compass models [3,4], which have been the subject of many studies in the past both in the classical and quantum regimes. For these systems, JahnTeller effects produce an anisotropy of the pseudospin couplings which is intertwined with the orientation of the interaction bonds. The properties of these systems have attracted considerable attention since they are endowed with symmetries that effectively reduce the dimensionality of the system (the so-called dimensional reduction) [5,6]. Despite of their apparent simplicity, orbital compass models are relevant in a variety of contexts, such as in determining the physics of Mott insulators with orbital degrees of freedom [3] and the implementation of protected qubits for quantum computation in Josephson junction arrays [7]. These systems are also candidates to exhibit topological quantum order [8]. Furthermore, it was recently shown how to simulate these models using polar molecules in optical lattices and systems of trapped ions with state-of-the-art technology [9,10].Generally speaking, the symmetries in these systems involve large degeneracies in their energy spectra, which make their numerical simulation difficult [11]. This fact, together with the lack of exact solutions, makes it hard to elucidate their phase diagrams. In this paper we use a tensor product state (TPS) [12,13] or projected entangled-pair state (PEPS) [14] to study the twodimensional anisotropic quantum orbital compass model (AQOCM) and, in particular, to investigate whether its phase transition is of first order [15,16] or second order [17]. More specifically, we use the infinite PEPS (iPEPS)