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

Circuit history states

Quantum-computing analogs and selective extensions › Circuit history states · lines 839–868

En09-A@1#eq:propagation-H

A preparation interface for $W_{V}$ consists of

∣ + ⟩^{\otimes m} \otimes ∣ ϕ ⟩ ⟼ (k \sum ∣ k ⟩ ⟨ k ∣_{C} \otimes V_{k}) (∣ + ⟩^{\otimes m} \otimes ∣ ϕ ⟩) .

The controlled unitaries are declared operations. Exact synthesis from a particular finite gate library, its approximation error, and its storage cost are separate implementation obligations.

If $V_{k + 1} = G_{k} V_{k}$ , define maps

L_{k} = \frac{1}{2} (⟨ k + 1 ∣ \otimes I_{S} - ⟨ k ∣ \otimes G_{k}), 0 \leq k < d - 1.

Then the positive propagation Hamiltonian

H_{prop} = k = 0 \sum^{d - 2} L_{k}^{†} L_{k}

annihilates $W_{V}$ , because $L_{k} W_{V} = 0$ . This is a legitimate history-state constraint, but generally not the noninteracting sum $H_{C} + H_{S}$ . A propagation constraint is not silently substituted for the frozen additive constraint.

Displayed formulas in this passage

En09-A@1#eq:propagation-H · line 860
$H_{prop} = k = 0 \sum^{d - 2} L_{k}^{†} L_{k}$
unlabeled · line 842
$∣ + ⟩^{\otimes m} \otimes ∣ ϕ ⟩ ⟼ (k \sum ∣ k ⟩ ⟨ k ∣_{C} \otimes V_{k}) (∣ + ⟩^{\otimes m} \otimes ∣ ϕ ⟩) .$
unlabeled · line 853
$L_{k} = \frac{1}{2} (⟨ k + 1 ∣ \otimes I_{S} - ⟨ k ∣ \otimes G_{k}), 0 \leq k < d - 1.$

Circuit history states

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