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

Exact extraction and global control error

Joint extraction and complete correlated readout › Exact extraction and global control error · lines 1363–1400

Initialize two binary blocks in their joint zero state. For each memory $i$ and $1 \leq n \leq K_{i}$ , swap the disjoint basis pairs

∣ n ⟩_{c_{i}} ∣ 0 ⟩_{Q_{i}} ⟷ {∣ 0 ⟩_{c_{i}} ∣ n ⟩_{Q_{i}}, ∣ n ⟩_{c_{i}} ∣ a_{i} ⟩_{Q_{i}}, n \leq a_{i}, a_{i} < n \leq K_{i} .

All other states are fixed. Denote the resulting product unitary by $U_{a, K}$ .

For each pair of normalized vectors $A_{i, n}, B_{i, n}$ , set

R_{i, n} = \frac{1}{2} ∣ A_{i, n} - B_{i, n} ⟩ ⟨ A_{i, n} - B_{i, n} ∣ .

An exact interaction-picture pulse is

H_{E} (t) = i, n \sum u_{i} (t) R_{i, n}, u_{i} (t) = \frac{2 π}{T _{E}} sin^{2} (π t / T_{E}), 0 \leq t \leq T_{E} .

Within each block the projections are orthogonal; between blocks they commute. Since each pulse has area $π$ , its evolution is the stated swap with no relative swap phase.

These number-selective matrix elements are a declared effective control set. They are not inferred from a generic oscillator controllability result.

Let $E_{a, K}$ be the register channel obtained by discarding the memories.

Displayed formulas in this passage

unlabeled · line 1368
$∣ n ⟩_{c_{i}} ∣ 0 ⟩_{Q_{i}} ⟷ {∣ 0 ⟩_{c_{i}} ∣ n ⟩_{Q_{i}}, ∣ n ⟩_{c_{i}} ∣ a_{i} ⟩_{Q_{i}}, n \leq a_{i}, a_{i} < n \leq K_{i} .$
unlabeled · line 1380
$R_{i, n} = \frac{1}{2} ∣ A_{i, n} - B_{i, n} ⟩ ⟨ A_{i, n} - B_{i, n} ∣ .$
unlabeled · line 1385
$H_{E} (t) = i, n \sum u_{i} (t) R_{i, n}, u_{i} (t) = \frac{2 π}{T _{E}} sin^{2} (π t / T_{E}), 0 \leq t \leq T_{E} .$

Exact extraction and global control error

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