Batch occupancy

Write

For qubit dimensions, a decoder of Choi rank $g\le d{A}_{\ast }$ has a Stinespring realization with at most

environment qubits.

Allocate a fresh output and environment, extend the Stinespring isometry on its prepared subspace to a unitary, and discard unused factors. This gives

The inaccessible active supplier register contains $n-k$ qubits. Keeping it separate, the corresponding upper bound on live named processing qubits is

Neither count includes an untouched test reference, retired physical record stores, or an unspecified thermal bath.

A useful ledger is therefore

with

If discarded decoder factors are implemented by Enter and their outcomes are retained, an elementary realization records at most $q_k+q_G$ such local bits. Its recovery need not send those bits between sites. Preparation distribution and quantum-cell transport are separate communication resources.

Arbitrary finite unitaries admit finite decompositions using continuous one-qubit gates and CNOT Barenco. We do not supply numerical values for $G_{\mathrm{gates}}$ or $D_{\mathrm{circuit}}$, nor an efficient synthesis algorithm. The matrix variable $C$ alone has $(d{A}_{\ast }{)}^2$ real Hermitian parameters.