Manuscript Hawk05-C@1 · Hawk05-C.tex · theorem · ba0719f1

Theorem: A proved two-curve clocked proxy

Replacing the clock: compatibility, crossover, and cutoff › A compatible, rounded two-curve sketch · lines 1100–1146 · proof included or linked

Hawk05-C@1#thm:sketch, Hawk05-C@1#eq:sketchQ, Hawk05-C@1#eq:sketchL

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For this declared construction,

r (t) = 2^{k (t)}, b (t) = 2^{n - k (t)}, D = 2^{n},

and

∣ lo g_{2} r (t) - A (M (t)) ∣ \leq 1, ∣ lo g_{2} b (t) - B (M (t)) ∣ \leq 1.

Its exact coherent and record means are

Q (t) = P (2^{k (t)}, 2^{n - k (t)}), L (t) = L (2^{k (t)}, 2^{n - k (t)}) .

They obey

∣ Q (t) - F (t) ∣ ∣ L (t) - A (M (t)) ∣ \leq 1 + lo g_{2} (1 + 2^{- ∣ n - 2 k (t) ∣}), \leq 1 + \frac{2 ^{- (n - k (t))}}{ln 2} .

If $B (M_{cut}) \geq 1$ , at least one residual qubit remains at the cutoff. No terminal drain is included.

Proof. Before the cap in $k (t)$ , the radiation rounding error is less than one and

n - k (t) - B (M (t)) = n - B_{0} + A (M (t)) - ⌊ A (M (t))⌋

lies between $- 1$ and $1$ . After the cap, both target errors are at most one because $A + B = B_{0}$ and $A \geq n$ . Apply Proposition prop:deficits. If $B (M_{cut}) \geq 1$ , then $A (M_{cut}) \leq B_{0} - 1$ , so $k (t_{cut}) \leq n - 1$ .

Displayed formulas in this passage

Hawk05-C@1#eq:sketchQ · line 1118
$∣ Q (t) - F (t) ∣ ∣ L (t) - A (M (t)) ∣ \leq 1 + lo g_{2} (1 + 2^{- ∣ n - 2 k (t) ∣}), \leq 1 + \frac{2 ^{- (n - k (t))}}{ln 2} .$
Hawk05-C@1#eq:sketchL · line 1118
$∣ Q (t) - F (t) ∣ ∣ L (t) - A (M (t)) ∣ \leq 1 + lo g_{2} (1 + 2^{- ∣ n - 2 k (t) ∣}), \leq 1 + \frac{2 ^{- (n - k (t))}}{ln 2} .$
unlabeled · line 1103
$r (t) = 2^{k (t)}, b (t) = 2^{n - k (t)}, D = 2^{n},$
unlabeled · line 1107
$∣ lo g_{2} r (t) - A (M (t)) ∣ \leq 1, ∣ lo g_{2} b (t) - B (M (t)) ∣ \leq 1.$
unlabeled · line 1112
$Q (t) = P (2^{k (t)}, 2^{n - k (t)}), L (t) = L (2^{k (t)}, 2^{n - k (t)}) .$
unlabeled · line 1135
$n - k (t) - B (M (t)) = n - B_{0} + A (M (t)) - ⌊ A (M (t))⌋$

Theorem: A proved two-curve clocked proxy

Displayed formulas in this passage

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