Accurate simulations of planar topological codes cannot use cyclic boundaries

Cyclic boundaries are used in many branches of physics and mathematics, typically to assist the approximation of a large space. We show that when determining the performance of planar, fault-tolerant, topological quantum error correction, using cyclic boundaries leads to a significant underestimate of the logical error rate. We present cyclic and non-cyclic surface code simulations exhibiting this discrepancy, and analytic formulae precisely reproducing the observed behavior in the limit of low physical error. These asymptotic formulae are then used to prove that the underestimate is exponentially large in the code distance d at any fixed physical error rate p below the threshold error rate p_th.