to function reliably. Unfortunately, no practical model for a self-correcting quantum memory has been proposed to date, despite considerable effort [@BrownMemory16]. The models that come closest to this goal involve topological protection in the presence of physically imposed symmetries [@KitaevWire01; @KarzigScalable17], but even these are not expected to reduce error rates sufficiently for large computations. Therefore active protocols that require measuring the check operators of an error correcting code are probably necessary to realize scalable quantum computing.

There are three general approaches of fault-tolerant error correction (FTEC) applicable to a wide range of stabilizer codes due to Shor [@Shor96], Steane [@Steane97], and Knill [@KL2005]. There are also a number of promising code-specific FTEC schemes, most notably the surface code with a minimum weight matching error correction scheme [@BK98; @DKLP02; @FMMC12]. This approach gives the best fault-tolerant thresholds to date and only requires geometrically local measurements. A high threshold [@Shor96; @AB97; @Preskill98; @KLZ98] implies that relatively imperfect hardware could be used to reliably implement long quantum computations. Despite this, the hardware and overhead requirements for the surface code are sufficiently demanding that it remains extremely challenging to implement in the lab.

Fortunately, there are reasons to believe that there could be 