Independent Local Quantum Fault Model

Let $C$ be a finite channel-location schedule as in Finite ordered channel-location schedules, with finite ordered location set $L(C)$. A location is a supplied gate, preparation, measurement, reset, or explicitly scheduled idle channel with declared support $S_{\mathrm{\ell }}$. The model does not create idle locations that the schedule did not declare.

Fix $\eta \in [0,1]$. A fault pattern is a subset $F\subseteq L(C)$. Its probability is

$$\mathbb{P}(F)={\eta }^{\mathrm{\mid }F\mathrm{\mid }}(1-\eta {)}^{\mathrm{\mid }L(C)\setminus F\mathrm{\mid }},$$

where integer powers use the empty-product convention $x^0=1$, including at $x=0$. Thus each location faults with probability $\eta$, and the event classes generated by the fault/no-fault indicator at each location form an independent family in the sense of Independent families of event classes. Conditional on $F$, each $\mathrm{\ell }\in F$ replaces only the ideal operation at $\mathrm{\ell }$ by an arbitrary CPTP map supported on $S_{\mathrm{\ell }}$; all other locations apply their ideal channels. A replacement map may depend on the full pattern, but it acts only at its own location. Faults at measurement and reset locations are represented by channels on their declared quantum and classical registers.

This definition is the independent, local fault model: location indicators are independent, and each fault map acts at its own scheduled location.