  The performance of a quantum error-correction process is determined by the
likelihood that a random configuration of errors introduced to the system will
lead to the corruption of encoded logical information. In this work we compare
two different variants of the surface code with a comparable number of qubits:
the surface code defined on a square lattice and the same model on a lattice
that is rotated by $\pi / 4$. This seemingly innocuous change increases the
distance of the code by a factor of $\sqrt{2}$.However, as we show, this gain
can come at the expense of significantly increasing the number of different
failure mechanisms that are likely to occur. We use a number of different
methods to explore this tradeoff over a large range of parameter space under an
independent and identically distributed noise model. We rigorously analyze the
leading order performance for low error rates, where the larger distance code
performs best for all system sizes. Using an analytical model and Monte Carlo
sampling, we find that this improvement persists for fixed sub-threshold error
rates for large system size, but that the improvement vanishes close to
threshold. Remarkably, intensive numerics uncover a region of system sizes and
sub-threshold error rates where the square lattice surface code marginally
outperforms the rotated model.
