Manuscript Em03-B@2 · Em03-B-v2.tex · proposition · c1a0977d

Proposition: Exact pulse compilation in the declared control set

B4 extraction: explicit swaps and control certificates › A declared control set and an exact pulse sequence · lines 1273–1298 · proof included or linked

Em03-B@2#prop:compile

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The $K$ simultaneous pulses (eq:swapPulse) generate $U_{N, K}$ exactly, with no relative swap phases. If the envelope limits are $Λ_{E} > 0$ and $B_{E} > 0$ , it suffices to take

T_{E} \geq max ⎩ ⎨ ⎧ \frac{2 π}{Λ _{E}}, \frac{2 π ^{2}}{B _{E}} ⎭ ⎬ ⎫ .

For input supported at occupation at most $K$ , the ideal sequence never exceeds occupation $K$ .

Proof. The projections commute and are mutually orthogonal. On each pair,

e^{- i π R_{n}} = I - 2 R_{n}

is the swap matrix, including its phase. Outside the pairs the evolution is the identity. The bounds follow by differentiating (eq:swapPulse). Its invariant subspace contains only memory occupations $0, \dots, K$ .

Displayed formulas in this passage

unlabeled · line 1279
$T_{E} \geq max ⎩ ⎨ ⎧ \frac{2 π}{Λ _{E}}, \frac{2 π ^{2}}{B _{E}} ⎭ ⎬ ⎫ .$
unlabeled · line 1291
$e^{- i π R_{n}} = I - 2 R_{n}$

Proposition: Exact pulse compilation in the declared control set

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