Manuscript Hawk06-A@1 · Hawk06-A.tex · proposition · c59fe67c

Proposition: A priced batch implementation

A specified deterministic decoder › Finite coherent realization and its workspace · lines 896–934 · proof included or linked

Hawk06-A@1#prop:workspace, Hawk06-A@1#eq:workspace

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The decoder has Kraus rank

g \leq d a_{A} .

It has a Stinespring isometry

W_{D} : A \to L_{out} \otimes G, dim G = g .

For a qubit interface, a batch implementation using fresh output and environment registers has peak quantum occupancy at most

lo g_{2} a_{A} + lo g_{2} d + ⌈ lo g_{2} g ⌉ \leq 2 lo g_{2} a_{A} + 2 lo g_{2} d .

The received early and late registers are included in this count. The inaccessible reference and residual are outside the laboratory decoder's holdings.

Proof. Diagonalize the decoder Choi matrix to obtain at most $d a_{A}$ Kraus operators. Their column stack is a Stinespring isometry. Allocate the output and a padded environment, and extend

∣ x ⟩_{A} ∣ 0 ⟩_{L G} ⟼ ∣ 0 ⟩_{A} W_{D} ∣ x ⟩

from its orthonormal prepared subspace to a unitary. The displayed register count prices this deliberately nonoptimal construction. A finite decomposition into elementary gates exists with unrestricted continuous gate parameters Barenco. No efficient gate-count bound is asserted.

Displayed formulas in this passage

Hawk06-A@1#eq:workspace · line 909
$lo g_{2} a_{A} + lo g_{2} d + ⌈ lo g_{2} g ⌉ \leq 2 lo g_{2} a_{A} + 2 lo g_{2} d .$
unlabeled · line 899
$g \leq d a_{A} .$
unlabeled · line 903
$W_{D} : A \to L_{out} \otimes G, dim G = g .$
unlabeled · line 923
$∣ x ⟩_{A} ∣ 0 ⟩_{L G} ⟼ ∣ 0 ⟩_{A} W_{D} ∣ x ⟩$

Nearby in this section

section: Finite coherent realization and its workspace