Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1

Transfer attempts with vacuum branches

Internally generated key and sequential logical transfer › Transfer attempts with vacuum branches · lines 744–780

Hawk01-B@1#eq:attemptmap, Hawk01-B@1#eq:completedmap, Hawk01-B@1#eq:weights

Choose fixed probabilities $λ_{t} \in [0, 1]$ , $2 \leq t \leq N$ . For a pending residual input define

V_{t} ∣ f, r; ψ ⟩ = λ_{t} ∣ t, r ⟩_{B_{t}} \otimes J_{t} ∣ ψ ⟩_{R_{t}} + 1 - λ_{t} ∣ f, r; ψ ⟩_{B_{t}} \otimes ∣ vac ⟩_{R_{t}} .

For an already completed branch $j < t$ ,

V_{t} ∣ j, r ⟩_{B_{t - 1}} = ∣ j, r ⟩_{B_{t}} \otimes ∣ vac ⟩_{R_{t}} .

Completed residual states have energy $M - e_{K} - e$ .

Define

s_{1} = 1, s_{t} = k = 2 \prod^{t} (1 - λ_{k}), w_{j} = s_{j - 1} λ_{j}, p_{t} = 1 - s_{t} = j = 2 \sum^{t} w_{j} .

The label $s_{t}$ denotes pending-transfer probability. A zero-probability branch may be retained in the ambient ledger.

Displayed formulas in this passage

Hawk01-B@1#eq:attemptmap · line 749
$V_{t} ∣ f, r; ψ ⟩ = λ_{t} ∣ t, r ⟩_{B_{t}} \otimes J_{t} ∣ ψ ⟩_{R_{t}} + 1 - λ_{t} ∣ f, r; ψ ⟩_{B_{t}} \otimes ∣ vac ⟩_{R_{t}} .$
Hawk01-B@1#eq:completedmap · line 761
$V_{t} ∣ j, r ⟩_{B_{t - 1}} = ∣ j, r ⟩_{B_{t}} \otimes ∣ vac ⟩_{R_{t}} .$
Hawk01-B@1#eq:weights · line 771
$s_{1} = 1, s_{t} = k = 2 \prod^{t} (1 - λ_{k}), w_{j} = s_{j - 1} λ_{j}, p_{t} = 1 - s_{t} = j = 2 \sum^{t} w_{j} .$