The Local Noise Model Composes With Circuit Semantics

Fix a fault pattern $F$. At each $\mathrm{\ell }\in F$, insert its declared local CPTP replacement; at each $\mathrm{\ell }\mathrm{\notin }F$, use the ideal CPTP location map. Each is a full-register channel by the schedule definition [F3]. Applying [F2] successively in chronological order shows the finite composition ${\mathrm{\Phi }}_F$ is CPTP.

Let $w_F$ be the probability of pattern $F$. The mutually disjoint pattern events cover the finite sample space, so their weights are nonnegative and sum to one by [F1].

Define $\bar{\mathrm{\Phi }}={\sum }_F{w}_F{\mathrm{\Phi }}_F$. It is completely positive: for every finite reference $R$ and positive $X$, $(\bar{\mathrm{\Phi }}\otimes {\mathrm{id}}_R)(X)={\sum }_F{w}_F({\mathrm{\Phi }}_F\otimes {\mathrm{id}}_R)(X)$ is a nonnegative weighted sum of positive operators. It is trace-preserving because $\mathrm{tr}\bar{\mathrm{\Phi }}(X)={\sum }_F{w}_F\mathrm{tr}{\mathrm{\Phi }}_F(X)={\sum }_F{w}_F\mathrm{tr}(X)=\mathrm{tr}(X)$ by steps 1.1 and 1.2.

For each input operator, averaging the conditional schedule outputs is exactly application of $\bar{\mathrm{\Phi }}$. Thus the finite stochastic schedule has a well-defined CPTP denotation.