Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

A bounded encoding

An exact finite response-table instrument › A bounded encoding · lines 1678–1725

II-B@1#eq:score, II-B@1#eq:probe-prep

Set $B_{*} = 768$ , and let $Q$ be a fresh qubit in $∣ 0 ⟩$ . For a real value $∣ v ∣ \leq B_{*}$ , define

a_{v} = \frac{1 + v / B _{*}}{2}, b_{v} = \frac{1 - v / B _{*}}{2},

V_{v} = (a_{v} b_{v} - b_{v} a_{v}), ∣ ψ_{v} ⟩ = V_{v} ∣ 0 ⟩ .

Then $V_{v}$ is unitary and, for $Z = diag (1, - 1)$ ,

⟨ ψ_{v} ∣ B_{*} Z ∣ ψ_{v} ⟩ = v .

Let $f$ be one setting in the finite list

F = {0, m, n_{1}, \dots, n_{15120}} .

Lemma lem:bound and (eq:target-value) ensure $∣ f (δ (k)) ∣ \leq B_{*}$ . After clock consumption, apply $V_{f (δ (k))}$ to $Q$ , controlled by the classical record at the sender.

The resulting complete preparation instrument is

P_{f} (ρ) = k = 0 \sum^{3} ∣ k ⟩ ⟨ k ∣_{R} \otimes σ_{k} \otimes ∣ ψ_{f (δ (k))} ⟩ ⟨ ψ_{f (δ (k))} ∣_{Q} .

For a reference, $σ_{k}$ is the joint $X E$ block.

This uses a finite gate library: there are only finitely many specified values. Exact gates are admitted mathematical primitives of the model. No compilation into an unspecified hardware gate set is claimed.

Displayed formulas in this passage

II-B@1#eq:score · line 1697
$⟨ ψ_{v} ∣ B_{*} Z ∣ ψ_{v} ⟩ = v .$
II-B@1#eq:probe-prep · line 1712
$P_{f} (ρ) = k = 0 \sum^{3} ∣ k ⟩ ⟨ k ∣_{R} \otimes σ_{k} \otimes ∣ ψ_{f (δ (k))} ⟩ ⟨ ψ_{f (δ (k))} ∣_{Q} .$
unlabeled · line 1682
$a_{v} = \frac{1 + v / B _{*}}{2}, b_{v} = \frac{1 - v / B _{*}}{2},$
unlabeled · line 1687
$V_{v} = (a_{v} b_{v} - b_{v} a_{v}), ∣ ψ_{v} ⟩ = V_{v} ∣ 0 ⟩ .$
unlabeled · line 1703
$F = {0, m, n_{1}, \dots, n_{15120}} .$