Manuscript Hawk07-A@1 · Hawk07-A.tex · proposition · f9c0f5da

Proposition: A finite batch realization

One specified decoder core › Workspace and permissible operations · lines 1016–1050 · proof included or linked

Hawk07-A@1#prop:workspace, Hawk07-A@1#eq:workspace

The decoder Choi rank obeys

g \leq d q_{R} .

It has a Stinespring isometry

W_{D} : R_{k_{*}} \to L_{out} G, dim G = g .

With qubit spaces, a conservative realization at cut $k \geq k_{*}$ uses peak live quantum occupancy at most

lo g_{2} dim R_{k} + lo g_{2} d + ⌈ lo g_{2} g ⌉ \leq lo g_{2} dim R_{k} + lo g_{2} q_{R} + 2 lo g_{2} d .

This count includes early memory, all received cells, output, and allocated decoder environment. It excludes inaccessible $B_{k}$ and the untouched external reference.

Proof. Diagonalize the decoder Choi matrix to obtain at most $d q_{R}$ Kraus operators and stack them into $W_{D}$ . Allocate output and environment registers, then extend the isometry from its prepared subspace to a unitary. The displayed register count prices a nonoptimal implementation.

Arbitrary finite unitaries admit elementary decompositions when continuous one-qubit gates and CNOT are permitted Barenco. No efficient gate count or executed compilation is asserted.

Displayed formulas in this passage

Hawk07-A@1#eq:workspace · line 1028
$lo g_{2} dim R_{k} + lo g_{2} d + ⌈ lo g_{2} g ⌉ \leq lo g_{2} dim R_{k} + lo g_{2} q_{R} + 2 lo g_{2} d .$
unlabeled · line 1019
$g \leq d q_{R} .$
unlabeled · line 1023
$W_{D} : R_{k_{*}} \to L_{out} G, dim G = g .$

Nearby in this section

section: Workspace and permissible operations