Manuscript Oai069-A@1 · Oai069-A.tex · theorem · a4913aab

Theorem: Simultaneous reference-preserving certification

Complete coverage and simultaneous finite-confidence certification › The simultaneous certificate · lines 1181–1243 · proof included or linked

Oai069-A@1#thm:simultaneous

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Under Assumption ass:validation,

Pr {L_{ν, k} \leq d_{⋄} (Φ^{⋆}_{ν, k}, T_{k}) \leq U_{ν, k} for every ν, k} \geq 1 - α_{s} - β .

The same event gives

L_{ν} \leq d_{⋄}_{\oplus} (Φ^{⋆}_{ν}, T) \leq U_{ν}

for every task.

If certification is declared only when $U_{ν} \leq ε_{ν}$ , then

Pr {some certified task has actual error > ε_{ν}} \leq α_{s} + β .

Proof. Conditional Hoeffding bounds give

Pr {∣ t_{ν k P} - m_{ν k P} ∣ > r_{ν k P} ∣ Z} \leq \frac{α _{s}}{M} .

A union bound gives simultaneous score coverage. On its intersection with $G$ , orthogonality of the Pauli basis implies

∥ J_{ν, k} - J (Φ_{ν, k}) ∥_{2}^{2} \leq \frac{1}{D} P \neq = I \sum a_{ν k P}^{2} .

Since $∥ A ∥_{1} \leq D ∥ A ∥_{2}$ ,

∥ J_{ν, k} - J (Φ_{ν, k}) ∥_{1} \leq R_{ν, k} .

The reverse triangle inequality gives

∣ δ_{ν, k} - δ (Φ_{ν, k}, T_{k}) ∣ \leq \frac{1}{2} R_{ν, k} .

Lemma lem:choi and the deployment triangle inequality give (eq:intervals). Equation (eq:blockerror) gives the direct-sum intervals.

Integrating the conditional statistical failure probability and adding $Pr (G^{c})$ proves the probability bound. Every invalid acceptance is outside the simultaneous coverage event.

Displayed formulas in this passage

unlabeled · line 1184
$Pr {L_{ν, k} \leq d_{⋄} (Φ^{⋆}_{ν, k}, T_{k}) \leq U_{ν, k} for every ν, k} \geq 1 - α_{s} - β .$
unlabeled · line 1194
$L_{ν} \leq d_{⋄}_{\oplus} (Φ^{⋆}_{ν}, T) \leq U_{ν}$
unlabeled · line 1203
$Pr {some certified task has actual error > ε_{ν}} \leq α_{s} + β .$
unlabeled · line 1212
$Pr {∣ t_{ν k P} - m_{ν k P} ∣ > r_{ν k P} ∣ Z} \leq \frac{α _{s}}{M} .$
unlabeled · line 1221
$∥ J_{ν, k} - J (Φ_{ν, k}) ∥_{2}^{2} \leq \frac{1}{D} P \neq = I \sum a_{ν k P}^{2} .$
unlabeled · line 1226
$∥ J_{ν, k} - J (Φ_{ν, k}) ∥_{1} \leq R_{ν, k} .$
unlabeled · line 1231
$∣ δ_{ν, k} - δ (Φ_{ν, k}, T_{k}) ∣ \leq \frac{1}{2} R_{ν, k} .$

Theorem: Simultaneous reference-preserving certification

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