Manuscript En02-D@4 · En02-D-v4.tex · section · e19ec871

The Clifford benchmark and its inherited unlocking state

The Clifford benchmark and its inherited unlocking state · lines 304–330

The message occupies $Q_{2}$ among five qubits. Let

∣ Φ_{2} ⟩_{ab} = (∣ 00 ⟩_{ab} + ∣ 11 ⟩_{ab}) / 2,

and define, rightmost factor first,

U V ∣ ψ ⟩ = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3}, = (U^{†}_{123} \otimes I_{45}) (∣ ψ ⟩_{2} \otimes ∣ Φ_{2} ⟩_{14} \otimes ∣ Φ_{2} ⟩_{35}) .

There are thirteen preparation gates. Put $F = CZ_{1, 2}$ and $V_{L} = F^{L} V$ . Only $L$ 's parity matters.

For $A_{L} (A) = Tr_{45} (V_{L} A V_{L}^{†})$ ,

A_{0} (A) = U^{†} (I_{1} /2 \otimes A_{2} \otimes I_{3} /2) U .

Displayed formulas in this passage

unlabeled · line 313
$U V ∣ ψ ⟩ = Y_{2} CX_{3 \to 2} CX_{1 \to 2} (H_{1} S_{1}^{†}) (H_{3} S_{3}^{†}) CZ_{1, 2} CZ_{2, 3}, = (U^{†}_{123} \otimes I_{45}) (∣ ψ ⟩_{2} \otimes ∣ Φ_{2} ⟩_{14} \otimes ∣ Φ_{2} ⟩_{35}) .$
unlabeled · line 309
$∣ Φ_{2} ⟩_{ab} = (∣ 00 ⟩_{ab} + ∣ 11 ⟩_{ab}) / 2,$
unlabeled · line 327
$A_{0} (A) = U^{†} (I_{1} /2 \otimes A_{2} \otimes I_{3} /2) U .$