Finite ordered channel-location schedules

Fix a nonzero finite-dimensional ordered tensor register $H=H_0\otimes \cdots \otimes H_{n-1}$ and an integer $M\ge 0$. A finite channel-location schedule is an ordered list of slots ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_M$. Slot ${\mathrm{\ell }}_t$ has an ideal quantum channel ${\mathrm{\Phi }}_{{\mathrm{\ell }}_t}:\mathcal{L}(H)\to \mathcal{L}(H)$ and a declared support $S_{{\mathrm{\ell }}_t}\subseteq \{0,\dots ,n-1\}$. A channel is supported on $S_{{\mathrm{\ell }}_t}$ when it has a finite Kraus representation all of whose Kraus operators are supported there in the sense of Support and weight of an operator on a finite register. The order is chronological: slot ${\mathrm{\ell }}_1$ acts first. Its circuit denotation is the linear map

${\mathrm{\Phi }}_{{\mathrm{\ell }}_M}\circ \cdots \circ {\mathrm{\Phi }}_{{\mathrm{\ell }}_1},$

with the empty schedule denoting the identity map on $\mathcal{L}(H)$. The definition records the full-register channels at the slots; gate, preparation, measurement, reset, classical-control, and scheduled-idle operations are included only when their corresponding local channels and supports are supplied. Any ancillary or classical record registers they use are factors of $H$. No unlisted idle slot is implicit. A pattern-dependent replacement at slot $\mathrm{\ell }$ must be a quantum channel supported on the same $S_{\mathrm{\ell }}$; the chosen replacement may depend on the full fault pattern, but no extra joint operation across distinct slots is part of this schedule.