Manuscript Oai280-B@1 · Oai280-B.tex · proposition · 6d92ab75

Proposition: Encoding circuit and isometry

Conventions and the explicit five-qubit encoding · lines 262–295 · proof included or linked

Oai280-B@1#prop:encoding, Oai280-B@1#eq:encircuit

The gate sequence

U_{enc} = U_{G} U_{par} (I_{1} \otimes H_{2} \otimes H_{3} \otimes H_{4} \otimes H_{5}) Z_{1}

satisfies

U_{enc} (∣ i ⟩_{1} ∣ 0000 ⟩_{2345}) = V ∣ i ⟩, V^{†} V = I_{2} .

It uses one $Z$ , four Hadamards, four CNOTs, and five controlled $Z$ gates, with the displayed leg ordering.

Proof. After $Z_{1}$ and the four Hadamards, the state is

\frac{( - 1 ) ^{i}}{4} y_{2}, \dots, y_{5} \sum ∣ i ⟩_{1} ∣ y_{2} y_{3} y_{4} y_{5} ⟩ .

The four CNOTs change the first bit to $x_{1} = i + y_{2} + y_{3} + y_{4} + y_{5}$ , so the final string has parity $i$ . The five controlled $Z$ gates multiply its amplitude by $(- 1)^{q (x)}$ . This gives (eq:Vmatrix).

Each column has exactly $16$ entries of magnitude $1/4$ , hence norm one. The columns have disjoint parity supports, hence are orthogonal. This also tests $V^{†} V = I_{2}$ directly.

Displayed formulas in this passage

Oai280-B@1#eq:encircuit · line 265
$U_{enc} = U_{G} U_{par} (I_{1} \otimes H_{2} \otimes H_{3} \otimes H_{4} \otimes H_{5}) Z_{1}$
unlabeled · line 272
$U_{enc} (∣ i ⟩_{1} ∣ 0000 ⟩_{2345}) = V ∣ i ⟩, V^{†} V = I_{2} .$
unlabeled · line 282
$\frac{( - 1 ) ^{i}}{4} y_{2}, \dots, y_{5} \sum ∣ i ⟩_{1} ∣ y_{2} y_{3} y_{4} y_{5} ⟩ .$

Proposition: Encoding circuit and isometry

Displayed formulas in this passage

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