Manuscript Em03-H@8 · Em03-H-v8.tex · section · 91e9e3bd

Amplitude, phase, and verdict

Continuum defect certification and the next accuracy target › Amplitude, phase, and verdict · lines 1862–1901

If a validated certificate gives

∣ q - q ∣ \leq e_{q}, ∣ q ∣ \in [m_{-}, m_{+}],

then

η_{-} = max (0, m_{-} - e_{q})^{2}, η_{+} = min (1, (m_{+} + e_{q})^{2}),

e_{specified} \in [1 - η_{+}, 1 - η_{-}] .

Thus:

η_{-} \geq 3/4 ⟹ specified decoder meets quarter error,

η_{+} < 3/4 ⟹ specified decoder fails quarter error .

Otherwise the certificate is inconclusive.

If an estimated phase $α$ is used, the decoded channel is

η Ad_{diag (1, e^{2 i (α - α)})} + (1 - η) R_{∣ 0 ⟩ ⟨ 0 ∣} .

Its error is at most

min (1, 1 - η + η ∣ sin (α - α) ∣) .

For $m_{-} > e_{q}$ , use $s_{phase} \leq e_{q} / m_{-}$ ; otherwise use one. A sufficient upper bound is $1 - η_{-} (1 - s_{phase})$ . Implementation error is separate.

Displayed formulas in this passage

unlabeled · line 1865
$∣ q - q ∣ \leq e_{q}, ∣ q ∣ \in [m_{-}, m_{+}],$
unlabeled · line 1869
$η_{-} = max (0, m_{-} - e_{q})^{2}, η_{+} = min (1, (m_{+} + e_{q})^{2}),$
unlabeled · line 1873
$e_{specified} \in [1 - η_{+}, 1 - η_{-}] .$
unlabeled · line 1877
$η_{-} \geq 3/4 ⟹ specified decoder meets quarter error,$
unlabeled · line 1881
$η_{+} < 3/4 ⟹ specified decoder fails quarter error .$
unlabeled · line 1888
$η Ad_{diag (1, e^{2 i (α - α)})} + (1 - η) R_{∣ 0 ⟩ ⟨ 0 ∣} .$
unlabeled · line 1893
$min (1, 1 - η + η ∣ sin (α - α) ∣) .$

Amplitude, phase, and verdict

Displayed formulas in this passage

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