Manuscript Em03-C@1 · Em03-C-v1.tex · section · 47dd71ec

A declared rotation family

C2: information-sensitive finite-register measurements › A declared rotation family · lines 1713–1757

Let $P_{B} = {I, X, Y, Z}^{\otimes B_{reg}}$ . For each nonidentity Pauli string $P$ , choose a product basis rotation $R_{P}$ using

U_{X} = e^{+ i π Y /4}, U_{Y} = e^{- i π X /4}, U_{Z} = I,

and use $I$ on positions where $P$ is the identity. These obey

U_{X}^{†} Z U_{X} = X, U_{Y}^{†} Z U_{Y} = Y .

The rotations are generated by bounded single-qubit Hamiltonians with pulse area of magnitude $π /2$ . Their phase convention is fixed by the apparatus reference.

The ideal selective maps are

M_{P, j} = (⟨ j ∣_{Q} R_{P}) \otimes I_{S} .

Their effects on $Q$ are

E_{P, j} = R_{P}^{†} ∣ j ⟩ ⟨ j ∣ R_{P} .

Thus

P_{P}^{S} (X) = j \sum ∣ y_{j} ⟩ ⟨ y_{j} ∣ \otimes M_{P, j} X M_{P, j}^{†}

is complete, including unused raw inputs. The setting label is also retained.

For a selected setting $P$ , let its implemented rotation have bounded generator deviation $Δ H_{P} (t)$ on $QS$ , and set

γ_{P} = min {1, \int ∥Δ H_{P} (t) ∥ d t} .

The pointer budget $δ_{P}$ is (eq:readbudget) for that setting.

Displayed formulas in this passage

unlabeled · line 1718
$U_{X} = e^{+ i π Y /4}, U_{Y} = e^{- i π X /4}, U_{Z} = I,$
unlabeled · line 1724
$U_{X}^{†} Z U_{X} = X, U_{Y}^{†} Z U_{Y} = Y .$
unlabeled · line 1732
$M_{P, j} = (⟨ j ∣_{Q} R_{P}) \otimes I_{S} .$
unlabeled · line 1737
$E_{P, j} = R_{P}^{†} ∣ j ⟩ ⟨ j ∣ R_{P} .$
unlabeled · line 1742
$P_{P}^{S} (X) = j \sum ∣ y_{j} ⟩ ⟨ y_{j} ∣ \otimes M_{P, j} X M_{P, j}^{†}$
unlabeled · line 1752
$γ_{P} = min {1, \int ∥Δ H_{P} (t) ∥ d t} .$

A declared rotation family

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