Error correcting power of small topological codes

Many quantum technologies are now reaching a high level of maturity and control, and it is likely that the first demonstrations of suppression of naturally occurring quantum noise using small topological error correcting codes will soon be possible. In doing this it will be necessary to define clear achievable metrics to measure and compare the performance of real codes. Here we consider the smallest examples of several families of topological codes: surface codes, color codes, and the gauge color code, and determine the minimum requirements to demonstrate error suppression. We use an exact decoder to compare the performance of the codes under simple models of both physical error and measurement error, and determine regions of correctability in the physical parameters for each code.