Manuscript Hawk02-C@2 · Hawk02-C.tex · section · bb332d66

A prescribed Hawk02-B endpoint

Finite-interface and predictive bridges › A prescribed Hawk02-B endpoint · lines 1444–1473

Fix a Hawk02-B schedule before comparison. Write its prepared endpoint as

W_{B} ∣ a ⟩ = s_{B} ∣ a ⟩_{M} ∣ vac ⟩ + ∣ 0 ⟩_{M} j = 1 \sum^{N} β_{j} ∣ a; j ⟩, ∣ s_{B} ∣^{2} + j \sum ∣ β_{j} ∣^{2} = 1.

For unmodified exchange pulses,

s_{B} = j \prod cos θ_{j}, β_{j} = - i sin θ_{j} k < j \prod cos θ_{k} .

Declared fixed phase gates may modify their phases.

Identify vacuum and the reached one-particle vectors by

∣ a; j ⟩ ⟼ ∣ ϕ_{j - 1}, a ⟩ .

Append vacant missed modes. This identifies the reached complete output supports. Unused toy vectors may be embedded in unused physical vectors; no energy-preserving identification of the full toy and continuum Hamiltonians is asserted.

Let $b_{n} (T) = ⟨ ϕ_{n}, f_{T} ⟩$ .

Displayed formulas in this passage

unlabeled · line 1448
$W_{B} ∣ a ⟩ = s_{B} ∣ a ⟩_{M} ∣ vac ⟩ + ∣ 0 ⟩_{M} j = 1 \sum^{N} β_{j} ∣ a; j ⟩, ∣ s_{B} ∣^{2} + j \sum ∣ β_{j} ∣^{2} = 1.$
unlabeled · line 1456
$s_{B} = j \prod cos θ_{j}, β_{j} = - i sin θ_{j} k < j \prod cos θ_{k} .$
unlabeled · line 1463
$∣ a; j ⟩ ⟼ ∣ ϕ_{j - 1}, a ⟩ .$