Manuscript Em03-E@1 · Em03-E-v1.tex · section · c4b07e6a

Controlled Pauli randomization

E1: explicit internal and emission Hamiltonians › Controlled Pauli randomization · lines 458–477

Put $P_{ab} = X^{a} Z^{b}$ . Apply controlled $Z$ from $K_{b}$ to $Q$ , then controlled $X$ from $K_{a}$ to $Q$ . The resulting unitary is

U_{pad} = a, b \sum ∣ a, b ⟩ ⟨ a, b ∣_{K} \otimes P_{ab} .

Its generators are the two explicit projections

G_{Z} = ∣ 1 ⟩ ⟨ 1 ∣_{K_{b}} \otimes \frac{I - Z _{Q}}{2}, G_{X} = ∣ 1 ⟩ ⟨ 1 ∣_{K_{a}} \otimes \frac{I - X _{Q}}{2},

each with pulse area $π$ .

The ordering fixes all phases. In particular, $P_{11} = X Z$ is not replaced silently by a differently phased Pauli matrix. Its conjugation action is nevertheless that of $Y$ .

Displayed formulas in this passage

unlabeled · line 463
$U_{pad} = a, b \sum ∣ a, b ⟩ ⟨ a, b ∣_{K} \otimes P_{ab} .$
unlabeled · line 468
$G_{Z} = ∣ 1 ⟩ ⟨ 1 ∣_{K_{b}} \otimes \frac{I - Z _{Q}}{2}, G_{X} = ∣ 1 ⟩ ⟨ 1 ∣_{K_{a}} \otimes \frac{I - X _{Q}}{2},$