Manuscript XVI-O1-ST@5 · XVI-O1-ST-v5.tex · section · dd45e0d1

Numerical reports and sharp complete error

Numerical reports and sharp complete error · lines 604–633

For real $r_{j}$ with $∣ r_{j} ∣ \leq B$ , define the binary report

p_{r} (b ∣ j) = \frac{1 + b r _{j} / B}{2}, b = \pm 1,

I_{r}^{B} (X) = j, b \sum p_{r} (b ∣ j) ∣ j, b ⟩ ⟨ j, b ∣ \otimes D_{j} X D_{j}^{†} .

The coefficient is a bounded expectation, not an exact rational value recovered from one shot.

For complex $v_{j}$ , $∣ v_{j} ∣ \leq B$ , define

τ_{B} (v_{j}) = \frac{1}{2} (I + \frac{Re v _{j}}{B} X_{P} + \frac{Im v _{j}}{B} Y_{P}),

Q_{v}^{B} (X) = j \sum ∣ j ⟩ ⟨ j ∣ \otimes τ_{B} (v_{j}) \otimes D_{j} X D_{j}^{†} .

The Bloch bound makes $τ_{B} (v_{j})$ a state. This interface returns a classical task label and a quantum report qubit, not a wholly classical coefficient word. Both maps are channels: they measure the clock and append valid classical or quantum states while retaining the branch block.

Displayed formulas in this passage

unlabeled · line 608
$p_{r} (b ∣ j) = \frac{1 + b r _{j} / B}{2}, b = \pm 1,$
unlabeled · line 611
$I_{r}^{B} (X) = j, b \sum p_{r} (b ∣ j) ∣ j, b ⟩ ⟨ j, b ∣ \otimes D_{j} X D_{j}^{†} .$
unlabeled · line 619
$τ_{B} (v_{j}) = \frac{1}{2} (I + \frac{Re v _{j}}{B} X_{P} + \frac{Im v _{j}}{B} Y_{P}),$
unlabeled · line 624
$Q_{v}^{B} (X) = j \sum ∣ j ⟩ ⟨ j ∣ \otimes τ_{B} (v_{j}) \otimes D_{j} X D_{j}^{†} .$

Numerical reports and sharp complete error

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