Fault Locations, Paths, and Rectangles

For a finite channel-location schedule $C$, its elementary fault locations are the elements of the finite set $L(C)$ in Independent Local Quantum Fault Model and Finite ordered channel-location schedules. A fault path is a subset $F\subseteq L(C)$; it records where faults occur, not the CPTP maps chosen at those locations.

Fix a recursively replaced gadget family. A level-zero rectangle is one elementary location. A level-$(r+1)$ rectangle is the complete set of elementary locations used by one level-$r$ operation gadget, including its declared preparation, correction, measurement, reset, classical-control, and idle locations. Its level-$r$ child rectangles partition its elementary location set; in the ordinary rectangle recursion these child sets are disjoint. Let $a$ be a uniform upper bound on the number of child rectangles in any such parent. The extended rectangle for a gate is the union of its rectangle and its adjacent correction rectangles; extended rectangles may overlap and are not covered by the disjoint-child convention.

Use a separate integer $b$ for a uniform upper bound on the number of elementary locations in a level-one extended rectangle. The constants $a$ and $b$ count different objects.