Manuscript Oai212-A@1 · Oai212-A.tex · corollary · 79323841

Corollary: One explicit growing-resource choice

Bridge 2: an engineered entanglement-cost membrane › Finite-dimensional preparation and entropy errors · lines 1057–1080 · proof included or linked

Oai212-A@1#cor:quench-resources

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Suppose $M_{n}, N_{n}^{*} = O (n^{2})$ . For $n \geq 2$ , choose

K_{n} = \frac{6}{λ} lo g n, D_{n} = ⌈ \frac{8 lo g n}{- lo g tanh ( K _{n} /2 )} ⌉ .

Then the tail allowance is $O (n^{- 4})$ , the good-event preparation allowance is $O (n^{- 3})$ , and the optimization error is $o (n)$ . Thus these finite-dimensional cost optimizations have the same macroscopic norm as the exact exponential target.

Proof. The tail bound is $M_{n} n^{- 6} = O (n^{- 4})$ . The choice of $D_{n}$ ensures $z_{K_{n}}^{D_{n}} \leq n^{- 8}$ . Hence the preparation bound is $O (n) n^{- 4} = O (n^{- 3})$ . For large $n$ , $d_{D_{n}} (K_{n}) \leq 4 n^{- 8}$ , so the optimization error is $O (n^{- 6})$ .

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unlabeled · line 1061
$K_{n} = \frac{6}{λ} lo g n, D_{n} = ⌈ \frac{8 lo g n}{- lo g tanh ( K _{n} /2 )} ⌉ .$

Corollary: One explicit growing-resource choice

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