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Sensitivity to noise makes most of the current quantum computing schemesprone to error and nonscalable, allowing only for small proof-of-principledevices. Topologically-protected quantum computing aims at solving this problemby encoding quantum bits and gates in topological properties of the hardwaremedium that are immune to noise that does not impact the entire system at once.There are different approaches to achieve topological stability or active errorcorrection, ranging from quasiparticle braidings to spin models and topologicalcolor codes. The stability of these proposals against noise can be quantifiedby their error threshold. This figure of merit can be computed by mapping theproblem onto complex statistical-mechanical spin-glass models with localdisorder on nontrival lattices that can have many-body interactions and aresometimes described by lattice gauge theories. The error threshold for a givensource of error then represents the point in the temperature-disorder phasediagram where a stable symmetry-broken phase vanishes. An overview of thetechniques used to estimate the error thresholds is given, as well as a summaryof recent results on the stability of different topologically-protected quantumcomputing schemes to different error sources.

Stability of topologically-protected quantum computing proposals as seen through spin glasses