An integrity measure to benchmark quantum error correcting memories

Rapidly developing experiments acrossmultiple platformsnowaim to realise small quantumcodes, and so demonstrate amemorywithinwhich a logical qubit canbeprotected fromnoise. There is aneed to benchmark the achievements in these diverse systems, and to compare the inherent power of the codes they rely upon.Wedescribe a recently introducedperformancemeasure called integrity, which relates to the probability that an ideal agentwill successfully ‘guess’ the state of a logical qubit after aperiodof storage in thememory. Integrity is straightforward to evaluate experimentallywithout state tomography and it can be related to various establishedmetrics such as the logicalfidelity and thepseudo-threshold.Weoffer a set of experimentalmilestones that are steps towards demonstratingunconditionally superior encoded memories.Using intensive numerical simulationswe comparememories basedon thefive-qubit code, the seven-qubit Steane code, and anine-qubit codewhich is the smallest instanceof a surface code;we assess both the simple and fault-tolerant implementationsof each.While the ‘best’ codeuponwhich tobase a memorydoes vary according to thenature and severity of thenoise, nevertheless certain trends emerge.