Manuscript Hawk07-B@1 · Hawk07-B.tex · corollary · b9785538

Corollary: Noise-budgeted threshold

Noise, approximation, and reserved error headroom · lines 1620–1641 · proof included or linked

Hawk07-B@1#cor:noisethreshold, Hawk07-B@1#eq:noisethreshold

Fix full target $0 < ε < 1$ , and reserve $η + ι < ε /2$ . Put

u = ε /2 - η - ι > 0, ζ = u^{2} /2.

A sufficient Haar threshold for the same receiver to meet the implemented full-diamond target is

r_{*} \geq \frac{d ^{2} f}{u ^{2} p} .

Proof. The threshold gives ideal half-diamond error $2 ζ = u$ . Equation (eq:noise) then gives implemented half-diamond error at most $ε /2$ .

Displayed formulas in this passage

Hawk07-B@1#eq:noisethreshold · line 1631
$r_{*} \geq \frac{d ^{2} f}{u ^{2} p} .$
unlabeled · line 1625
$u = ε /2 - η - ι > 0, ζ = u^{2} /2.$