  To date, a great deal of attention has focused on characterizing the
performance of quantum error correcting codes via their thresholds, the maximum
correctable physical error rate for a given noise model and decoding strategy.
Practical quantum computers will necessarily operate below these thresholds
meaning that other performance indicators become important. In this work we
consider the scaling of the logical error rate of the toric code and
demonstrate how, in turn, this may be used to calculate a key performance
indicator. We use a perfect matching decoding algorithm to find the scaling of
the logical error rate and find two distinct operating regimes. The first
regime admits a universal scaling analysis due to a mapping to a statistical
physics model. The second regime characterizes the behavior in the limit of
small physical error rate and can be understood by counting the error
configurations leading to the failure of the decoder. We present a conjecture
for the ranges of validity of these two regimes and use them to quantify the
overhead -- the total number of physical qubits required to perform error
correction.
