Manuscript En02-C@1 · En02-C-v1.tex · section · 7549c6df

Registers and gates

An exact three-step emission benchmark › Registers and gates · lines 542–587

En02-C@1#eq:batchU, En02-C@1#eq:V

The message initially occupies position $2$ . The four internal qubits occupy $1, 3, 4, 5$ and begin in $∣ 0000 ⟩$ . Early radiation $E$ is one dimensional.

Write

∣ Φ_{2} ⟩_{ab} = \frac{∣ 00 ⟩ _{ab} + ∣ 11 ⟩ _{ab}}{2} .

Let $B$ prepare Bell pairs on $1, 4$ and $3, 5$ : chronologically,

H_{1}, CX_{1 \to 4}, H_{3}, CX_{3 \to 5} .

On positions $1, 2, 3$ , define

U = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3} .

The rightmost operation is applied first. This is a nine-gate Clifford circuit.

Define the full encoding unitary

J = (U^{†}_{123} \otimes I_{45}) B,

and its message isometry

V ∣ ψ ⟩ = J (∣ 0 ⟩_{1} ∣ ψ ⟩_{2} ∣ 0 ⟩_{3} ∣ 0 ⟩_{4} ∣ 0 ⟩_{5}) .

The authorized-triple channel uses the decoder normal form of En02-B. The remnant purification and the following sequential dynamics are specified here.

Displayed formulas in this passage

En02-C@1#eq:batchU · line 561
$U = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3} .$
En02-C@1#eq:V · line 577
$V ∣ ψ ⟩ = J (∣ 0 ⟩_{1} ∣ ψ ⟩_{2} ∣ 0 ⟩_{3} ∣ 0 ⟩_{4} ∣ 0 ⟩_{5}) .$
unlabeled · line 549
$∣ Φ_{2} ⟩_{ab} = \frac{∣ 00 ⟩ _{ab} + ∣ 11 ⟩ _{ab}}{2} .$
unlabeled · line 556
$H_{1}, CX_{1 \to 4}, H_{3}, CX_{3 \to 5} .$
unlabeled · line 573
$J = (U^{†}_{123} \otimes I_{45}) B,$