Manuscript Em03-E@1 · Em03-E-v1.tex · lemma · c4b07e6a

Lemma: Assignment and pointer certificate

Destructive verification and complete records › A complete pointer instrument · lines 1625–1658 · proof included or linked

Em03-E@1#lem:readout

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For ideal destructive readout $P_{0}^{S}$ ,

d_{⋄} (P_{W}^{S}, P_{0}^{S}) = j max [1 - W (y_{j} ∣ j)] .

If readiness fails with probability $r_{F}$ , the ready pointer preparation has error $e_{ptr}$ , and its joint bounded copying-generator deviation has integral $I_{ptr}$ , then a complete readout budget is

δ_{read} = min {1, r_{F} + (1 - r_{F}) (e_{ptr} + I_{ptr} + j max [1 - W (y_{j} ∣ j)])} .

Proof. Pinch a referenced input operator into its tested computational blocks $X_{j j}$ . Pinching is trace-norm contractive and $\sum_{j} ∥ X_{j j} ∥_{1} \leq ∥ X ∥_{1}$ . The complete assignment difference is at most

j max y \sum ∣ W (y ∣ j) - δ_{y, y_{j}} ∣ = 2 j max [1 - W (y_{j} ∣ j)] .

A maximizing basis input attains equality. Preparation replacement, bounded-generator stability, and assignment telescope on the ready branch. The visible failed branch costs at most one and has weight $r_{F}$ .

Displayed formulas in this passage

unlabeled · line 1628
$d_{⋄} (P_{W}^{S}, P_{0}^{S}) = j max [1 - W (y_{j} ∣ j)] .$
unlabeled · line 1636
$δ_{read} = min {1, r_{F} + (1 - r_{F}) (e_{ptr} + I_{ptr} + j max [1 - W (y_{j} ∣ j)])} .$
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$j max y \sum ∣ W (y ∣ j) - δ_{y, y_{j}} ∣ = 2 j max [1 - W (y_{j} ∣ j)] .$

Lemma: Assignment and pointer certificate

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