Manuscript Em03-B@2 · Em03-B-v2.tex · section · c1a0977d

The exact finite-domain interaction

B4 extraction: explicit swaps and control certificates › The exact finite-domain interaction · lines 1186–1200

Use the binary register $Q$ itself, initially in $∣ 0 ⟩_{Q}$ . For $1 \leq n \leq K$ , define the disjoint pairs

A_{n} = ∣ n ⟩_{c} ∣ 0 ⟩_{Q}, B_{n} = {∣ 0 ⟩_{c} ∣ n ⟩_{Q}, ∣ n ⟩_{c} ∣ N ⟩_{Q}, n \leq N, N < n \leq K .

These pairs are mutually disjoint. Let $U_{N, K}$ swap each $A_{n}, B_{n}$ and act as the identity on the remaining basis states.

Displayed formulas in this passage

unlabeled · line 1190
$A_{n} = ∣ n ⟩_{c} ∣ 0 ⟩_{Q}, B_{n} = {∣ 0 ⟩_{c} ∣ n ⟩_{Q}, ∣ n ⟩_{c} ∣ N ⟩_{Q}, n \leq N, N < n \leq K .$