Error and loss tolerances of surface codes with general lattice structures

We propose a family of surface codes with general lattice structures, where the error-tolerances against bit and phase errors can be controlled asymmetrically by changing the underlying lattice geometries. The surface codes on various lattices are found to be efficient in the sense that their threshold values universally approach the quantum Gilbert-Varshamov bound. We find that the error-tolerance of surface codes depends on the connectivity of underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss-tolerance of surface codes exhibits an opposite behavior; the logical information on a lattice of higher connectivity has more robustness against qubit loss. As a result, we come upon a fundamental trade-off between error- and loss-tolerances in the family of the surface codes with different lattice geometries.