Manuscript Hawk07-B@1 · Hawk07-B.tex · theorem · b9785538

Theorem: Explicit multi-cut design for any qubit payload

Payload-resource tradeoffs before a positive cutoff › Fixed residual dimension · lines 704–744 · proof included or linked

Hawk07-B@1#thm:design, Hawk07-B@1#eq:rwindow, Hawk07-B@1#eq:receivercost

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Let $d = 2^{ℓ}$ , $ℓ \geq 1$ , take $f = 1$ , choose $0 < ε < 1$ , $0 < p < 1$ , and integers $q \geq 1$ , $w \geq 0$ . Set

b_{*} = 2^{q + w}, H_{*} = \frac{4 b _{*} d}{ε ^{2} p}, h = 2^{m}, m = ⌈ lo g_{2} H_{*} ⌉ .

Let the vault have $n = ℓ + m$ qubits and declare

k_{*} = n - (q + w), K = n - q .

Then $0 \leq k_{*} \leq K < n$ , there are $w + 1$ retained cuts, and with Haar probability at least $1 - p$ all those cuts have full-code recovery error at most $ε$ . The cutoff retains $q$ physical vault qubits.

At the first retained cut,

\frac{4 d ^{2}}{ε ^{2} p} dim R_{*} = \frac{d h ^{2}}{b _{*}} \leq r_{*} < \frac{8 d ^{2}}{ε ^{2} p}, < \frac{64 b _{*} d ^{3}}{ε ^{4} p} .

Proof. The choice of $h$ gives $H_{*} \leq h < 2 H_{*}$ . In particular $d h / b_{*} \geq 4 d^{2} / (ε^{2} p)$ ; hence $k_{*}$ is a valid nonnegative integer and Theorem thm:threshold applies. Nested release reaches residual dimension $2^{q}$ at $K$ . Substitution proves (eq:rwindow)–(eq:receivercost).

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Hawk07-B@1#eq:rwindow · line 725
$\frac{4 d ^{2}}{ε ^{2} p} dim R_{*} = \frac{d h ^{2}}{b _{*}} \leq r_{*} < \frac{8 d ^{2}}{ε ^{2} p}, < \frac{64 b _{*} d ^{3}}{ε ^{4} p} .$
Hawk07-B@1#eq:receivercost · line 725
$\frac{4 d ^{2}}{ε ^{2} p} dim R_{*} = \frac{d h ^{2}}{b _{*}} \leq r_{*} < \frac{8 d ^{2}}{ε ^{2} p}, < \frac{64 b _{*} d ^{3}}{ε ^{4} p} .$
unlabeled · line 709
$b_{*} = 2^{q + w}, H_{*} = \frac{4 b _{*} d}{ε ^{2} p}, h = 2^{m}, m = ⌈ lo g_{2} H_{*} ⌉ .$
unlabeled · line 716
$k_{*} = n - (q + w), K = n - q .$

Theorem: Explicit multi-cut design for any qubit payload

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