  Surface codes are quantum error correcting codes normally defined on 2D
arrays of qubits. In this paper, we introduce a surface code design based on
the fact that the severity of bit flip and phase flip errors in the physical
quantum systems is asymmetric. For our proposed surface code design for
asymmetric error channels, we present pseudo-threshold and threshold values in
the presence of various degrees of asymmetry of Pauli $\hat{X}$, $\hat{Y}$, and
$\hat{Z}$ errors in a depolarization channel. We show that, compared to
symmetric surface codes, our asymmetric surface codes can provide almost double
the pseudo-threshold rates while requiring less than half the number of
physical qubits in the presence of increasing asymmetry in the error channel.
We also demonstrate that as the asymmetry of the surface code increases, the
advantage in the pseudo-threshold rates begins to saturate for any degree of
asymmetry in the channel.
