Monte Carlo studies of the properties of the Majorana quantum error correction code: is selfcorrection possible during braiding?
Abstract
The Majorana code is an example of a stabilizer code where the quantum information is stored in a system supporting wellseparated Majorana Bound States (MBSs). We focus on onedimensional realizations of the Majorana code, as well as networks of such structures, and investigate their lifetime when coupled to a paritypreserving thermal environment. We apply the Davies prescription, a standard method that describes the basic aspects of a thermal environment, and derive a master equation in the BornMarkov limit. We first focus on a single wire with immobile MBSs and perform error correction to annihilate thermal excitations. In the hightemperature limit, we show both analytically and numerically that the lifetime of the Majorana qubit grows logarithmically with the size of the wire. We then study a trijunction with four MBSs when braiding is executed. We study the occurrence of dangerous error processes that prevent the lifetime of the Majorana code from growing with the size of the trijunction. The origin of the dangerous processes is the braiding itself, which separates pairs of excitations and renders the noise nonlocal; these processes arise from the basic constraints of moving MBSs in 1D structures. We confirm our predictions with Monte Carlo simulations in the lowtemperature regime, i.e. the regime of practical relevance. Our results put a restriction on the degree of selfcorrection of this particular 1D topological quantum computing architecture.
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.00892
 Bibcode:
 2015arXiv150700892P
 Keywords:

 Condensed Matter  Mesoscale and Nanoscale Physics;
 Quantum Physics
 EPrint:
 Main text: 20 pages, Supplementary Material: 66 pages. Short version: arXiv:1505.03712