Manuscript Hawk04-A@1 · Hawk04-A.tex · proposition · 526092ba

Proposition: A fixed-decoder finite-time interval

What a positive finite-time certificate must contain › One certified point can give a finite interval · lines 1700–1740 · proof included or linked

Hawk04-A@1#prop:interval

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Let $D$ be a fixed finite CPTP decoder. For the finite proxy, define

L_{m} = M_{N} + m Λ + 2 G m + 1 .

Suppose a validated computation at $t_{0} \in (0, T)$ proves $r_{dec} (t_{0}) \leq r_{0}$ . Then for the same decoder,

d_{⋄} (D N_{t}, id_{C}) \leq d r_{0} + d L_{m} ∣ t - t_{0} ∣ + b_{reg}

throughout $[0, T]$ , capped at one.

d r_{0} + b_{reg} < ε,

every

∣ t - t_{0} ∣ < \frac{ε - d r _{0} - b _{reg}}{d L _{m}}

inside $[0, T]$ belongs to a nonempty certified recovery interval.

Proof. The finite Hamiltonian has operator norm at most $L_{m}$ . Therefore its prepared isometries differ by at most $L_{m} ∣ t - t_{0} ∣$ . Lemma lem:isometry, decoder contraction, and the Bell-input test give

r_{dec} (t) \leq r_{0} + L_{m} ∣ t - t_{0} ∣.

Apply (eq:direct-certificate).

Displayed formulas in this passage

unlabeled · line 1704
$L_{m} = M_{N} + m Λ + 2 G m + 1 .$
unlabeled · line 1710
$d_{⋄} (D N_{t}, id_{C}) \leq d r_{0} + d L_{m} ∣ t - t_{0} ∣ + b_{reg}$
unlabeled · line 1718
$d r_{0} + b_{reg} < ε,$
unlabeled · line 1722
$∣ t - t_{0} ∣ < \frac{ε - d r _{0} - b _{reg}}{d L _{m}}$
unlabeled · line 1735
$r_{dec} (t) \leq r_{0} + L_{m} ∣ t - t_{0} ∣.$

Nearby in this section

section: One certified point can give a finite interval