Surface code design for asymmetric error channels

Surface codes are quantum error correcting codes typically defined on a 2D array of qubits. A [d(x), d(z)] surface code design is being introduced, where d(x)(d(z)) represents the distance of the code for bit (phase) error correction, motivated by the fact that the severity of bit flip and phase flip errors in the physical quantum system is asymmetric. We present pseudo-threshold and threshold values for the proposed surface code design for asymmetric error channels in the presence of various degrees of asymmetry of Pauli (X) over cap, (Y) over cap, and (Z) over cap errors in a depolarisation channel. We demonstrate 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. Our results show that for low degree of asymmetry, it is advantageous to increase d(x) along with d(z). However, as the asymmetry of the channel increases, higher pseudo-threshold is obtained with increasing d(z) when d(x) is kept constant at a low value. Additionally, we also show that the advantage in the pseudo-threshold rates begins to saturate for any possible degree of asymmetry in the error channel as the surface code asymmetry is continued to be increased.