Manuscript En09-A@1 · En09-A.tex · corollary · eafec9de

Corollary: Circuit-history extraction

Quantum-computing analogs and selective extensions › Circuit history states · lines 811–839 · proof included or linked

En09-A@1#cor:circuit-history, En09-A@1#eq:circuit-extraction

Let $V_{0}, \dots, V_{d - 1}$ be unitaries on $H_{S}$ , and define

W_{V} = \frac{1}{d} k = 0 \sum^{d - 1} ∣ k ⟩_{C} \otimes V_{k} .

Then $W_{V}$ is an isometry, and computational-basis Enter on the clock realizes the channel

X ⟼ \frac{1}{d} k = 0 \sum^{d - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes V_{k} X V_{k}^{†} .

The clock names are consumed and the system names survive.

Proof. Orthogonality of the clock basis gives

W_{V}^{†} W_{V} = \frac{1}{d} k \sum V_{k}^{†} V_{k} = I_{S} .

The $k$ -branch destructive component is

(⟨ k ∣ \otimes I_{S}) W_{V} = d^{- 1/2} V_{k} .

Equation (eq:circuit-extraction) follows from Theorem thm:main. Its resource assertion is the same typing calculation.

Displayed formulas in this passage

En09-A@1#eq:circuit-extraction · line 819
$X ⟼ \frac{1}{d} k = 0 \sum^{d - 1} ∣ k ⟩ ⟨ k ∣_{R} \otimes V_{k} X V_{k}^{†} .$
unlabeled · line 814
$W_{V} = \frac{1}{d} k = 0 \sum^{d - 1} ∣ k ⟩_{C} \otimes V_{k} .$
unlabeled · line 829
$W_{V}^{†} W_{V} = \frac{1}{d} k \sum V_{k}^{†} V_{k} = I_{S} .$
unlabeled · line 833
$(⟨ k ∣ \otimes I_{S}) W_{V} = d^{- 1/2} V_{k} .$