Topological quantum systems were introduced as an attempt to overcome the challenge of decoherence due to the self-correcting properties of their energy eigenstates. One of the simplest and most popular topologically ordered systems is known as “Kitaev’s toric code.” We study the dynamics of the quasiparticle excitations that live on the two-dimensional surface of the toric code in the zero temperature limit. We also look at a modified version of the toric code in which the continuity of the system is retained in one direction and removed in the other, thus forming a cylindrical surface. We argue that the main process that can result in a logical error in this limit would be a random walker of quasiparticles that can move around the surface in a topologically non-trivial manner. Using a discrete Monte Carlo method, we find the probability of the occurrence of a logical error as a function of the number of steps taken by the walker and the system size. We find that the cylindrical code does indeed show a notable advantage in its passive fault-tolerance when compared to the toric code.