Manuscript Hawk05-A@1 · Hawk05-A.tex · section · 449f85bb

Memory, cells, and transfer unitaries

An explicit conserving transfer schedule › Memory, cells, and transfer unitaries · lines 914–946

Hawk05-A@1#eq:rotation1, Hawk05-A@1#eq:rotation2, Hawk05-A@1#eq:energy

Let

H_{M} = C ∣ \emptyset ⟩ \oplus H_{C}, H_{R_{j}} = C ∣ \emptyset ⟩ \oplus H_{C} .

Initially the code occupies $M$ , and every cell is vacant. For $0 \leq θ_{j} \leq π /2$ , define a unitary $U_{j}$ on $M R_{j}$ by the rotations

U_{j} ∣ i, \emptyset ⟩ U_{j} ∣ \emptyset, i ⟩ = cos θ_{j} ∣ i, \emptyset ⟩ + sin θ_{j} ∣ \emptyset, i ⟩, = - sin θ_{j} ∣ i, \emptyset ⟩ + cos θ_{j} ∣ \emptyset, i ⟩ .

It is identity on the orthogonal complement of these mutually orthogonal two-dimensional subspaces.

Let $P_{M}$ and $P_{R_{j}}$ project onto the occupied logical sectors. For $ω > 0$ , declare

H_{0} = ω (P_{M} + j \sum P_{R_{j}}) .

Displayed formulas in this passage

Hawk05-A@1#eq:rotation1 · line 925
$U_{j} ∣ i, \emptyset ⟩ U_{j} ∣ \emptyset, i ⟩ = cos θ_{j} ∣ i, \emptyset ⟩ + sin θ_{j} ∣ \emptyset, i ⟩, = - sin θ_{j} ∣ i, \emptyset ⟩ + cos θ_{j} ∣ \emptyset, i ⟩ .$
Hawk05-A@1#eq:rotation2 · line 925
$U_{j} ∣ i, \emptyset ⟩ U_{j} ∣ \emptyset, i ⟩ = cos θ_{j} ∣ i, \emptyset ⟩ + sin θ_{j} ∣ \emptyset, i ⟩, = - sin θ_{j} ∣ i, \emptyset ⟩ + cos θ_{j} ∣ \emptyset, i ⟩ .$
Hawk05-A@1#eq:energy · line 942
$H_{0} = ω (P_{M} + j \sum P_{R_{j}}) .$
unlabeled · line 917
$H_{M} = C ∣ \emptyset ⟩ \oplus H_{C}, H_{R_{j}} = C ∣ \emptyset ⟩ \oplus H_{C} .$