Error and loss tolerances of surface codes with general lattice structures

Abstract

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 the surface codes depends on the connectivity of the underlying lattices; the error chains on a lattice of lower connectivity are easier to correct. On the other hand, the loss tolerance of the 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 surface codes with different lattice geometries. We also provide the physical aspects of the present results from the viewpoint of statistical physics, which leads to an equality that captures well both the error and loss tolerances of these surface codes.

Lifetime of topological quantum memories in thermal environment

“…By localizing anyons to certain regions on the lattice, they are less likely to travel far apart, leading to better error correction properties. However, deforming the surface code breaks the symmetry between the X and Z error correction properties, creating an asymmetry in the error threshold values and their corresponding lifetimes [6]. This is because stabilizer operators are no longer symmetric under the lattice duality transformation (i.e.…”

“…The error threshold of the toric code for the optimal decoding is related to the phase transition point of the random bond Ising model and found to be approximately 11% [5]. Dependence of the phase transition point on the lattice geometry has been studied for various lattices in [6,7].…”

Thresholds of surface codes on the general lattice structures suffering biased error and loss

“…1) [8]. Their constructions and error-correction procedures are basically the same as those of the original surface code.…”

Implications of ignorance for quantum-error-correction thresholds

“…Hence, setting p =p yieldsp X +p Z < 2p C , as compared to the non-transformed version which only successfully corrects if max(p X ,p Z ) < p C . The transformed version has more natural symmetry properties and negates the requirement of recent studies [12,13] to adjust the lattice geometry for each different asymmetry betweenp X andp Z . Fig.…”

“…Other authors [12,13] have recently concerned themselves with the idea that two different error types, X and Z, could occur at different rates. The standard version of the Toric code in 2D does not tolerate these well, with a threshold of the form max(p X , p Z ) ≤ p C , and so they have studied how one might alter the lattice geometry in order to better tolerate asymmetries between the parameters p X and p Z .…”