  Various realizations of Kitaev's surface code perform surprisingly well for
biased Pauli noise. Attracted by these potential gains, we study the
performance of Clifford-deformed surface codes (CDSCs) obtained from the
surface code by applying single-qubit Clifford operators. We first analyze
CDSCs on the $3\times 3$ square lattice and find that, depending on the noise
bias, their logical error rates can differ by orders of magnitude. To explain
the observed behavior, we introduce the effective distance $d'$, which reduces
to the standard distance for unbiased noise. To study CDSC performance in the
thermodynamic limit, we focus on random CDSCs. Using the statistical mechanical
mapping for quantum codes, we uncover a phase diagram that describes random
CDSC families with $50\%$ threshold at infinite bias. In the high-threshold
region, we further demonstrate that typical code realizations outperform the
thresholds and subthreshold logical error rates, at finite bias, of the
best-known translationally invariant codes. We demonstrate the practical
relevance of these random CDSC families by constructing a translation-invariant
CDSC belonging to a high-performance random CDSC family. We also show that our
translation-invariant CDSC outperforms well-known translation-invariant CDSCs
such as the XZZX and XY codes.
