The role of entropy in topological quantum error correction

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 π/4. This seemingly innocuous change increases the distance of the code
by a factor of √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.