Manuscript En08-B@1 · En08-B.tex · section · 23cac52a

Preparations, storage, waits, and decoding

Implementation errors and their boundaries › Preparations, storage, waits, and decoding · lines 1137–1177

En08-B@1#eq:preparation-ledger

The full estimate (eq:main-noise) charges complete tick maps, not merely active SWAP gates. This distinction is necessary when idle storage or the terminal wait is imperfect.

For example, suppose the $q - 1$ auxiliary preparations are independent states $σ_{v}$ with

d_{tr} (σ_{v}, σ) \leq ξ_{j} .

Product-state telescoping gives

d_{tr} v \neq = 0 ⨂ σ_{v}, v \neq = 0 ⨂ σ \leq (q - 1) ξ_{j} .

The corresponding encoding-channel half-diamond error has the same bound: its difference tensors the input with this fixed state difference.

Consequently, if each complete tick map has error at most $e_{j}$ and the final output map has error at most $β_{j}$ , then

d_{⋄} (Φ_{j} (T, x), id_{D}) \leq min {1, (q - 1) ξ_{j} + n_{j} e_{j} + β_{j}} .

Uniform conditions

j ξ_{j} \to 0, j e_{j} \to 0, β_{j} \to 0

are sufficient.

No independence between payload and reference is imposed. Independence is imposed on auxiliary preparation. Other preparation models may instead use a separately certified complete encoding allowance.

Displayed formulas in this passage

En08-B@1#eq:preparation-ledger · line 1161
$d_{⋄} (Φ_{j} (T, x), id_{D}) \leq min {1, (q - 1) ξ_{j} + n_{j} e_{j} + β_{j}} .$
unlabeled · line 1145
$d_{tr} (σ_{v}, σ) \leq ξ_{j} .$
unlabeled · line 1149
$d_{tr} v \neq = 0 ⨂ σ_{v}, v \neq = 0 ⨂ σ \leq (q - 1) ξ_{j} .$
unlabeled · line 1168
$j ξ_{j} \to 0, j e_{j} \to 0, β_{j} \to 0$