Incoherent dynamics in the toric code subject to disorder

Abstract

We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. Firstly, a new class of codes based on random lattices of stabilizer operators is presented, and shown to be superior to the standard square lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable CSS codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed onsite potentials. In the presence of repulsive long-range interaction between the anyons, a surprising increase with disorder strength of the lifetime of encoded states is reported and explained by an entirely incoherent mechanism. Finally, the coherent transport of the anyons in the presence of both forms of disorder is investigated, and a significant suppression of the anyon motion is found.

Duality analysis on random planar lattices

“…For instance, it has opened a way to analyze the theoretical limitation in the classical/quantum coding theory, where orderdisorder phase transition corresponds to the theoretical limitation of the error correction codes. Specifically, the optimal error thresholds of a class of quantum error correction codes, so-called surface codes [9], have been evaluated in terms of the duality with the real-space renormalization [10,11], On the other hand, newly developed quantum error correction codes [12] also motivate us to study the associated spin glass models, which have not been considered so far.…”

Implications of ignorance for quantum-error-correction thresholds

“…For a fixed ratiõ p X /p Z , the fraction of 500 different realisations of an error distribution giving a logical error was computed for varying error rates, enabling determination of the failure probability (the threshold at which a transition in logical error rate from 0 to 50% occurs). Similar numerics, for a perfectly identified error model, are present in [12].…”

“…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 .…”

Breakdown of surface-code error correction due to coupling to a bosonic bath

“…Ref. [18] shows that variations of the surface code tailored for stability against biased noise (p x = p z ) give thresholds that fall only a few percents short of the ones suggested by such entropic arguments -even with error correction performed by an efficient approximate algorithm. We will use two different algorithms for performing error correction for the above error model.…”