Channels and the observer interface

The class ${\mathfrak{P}}_N$ consists of finite deterministic protocols satisfying this assumption and $Q_{\mathrm{peak}}\le N$. Lower bounds permit arbitrary instruments. $Q_{\mathrm{peak}}$ counts delivered inputs, work, quantum records, retained data, and output. $Q_{\mathrm{stored},\mathrm{total}}$ additionally counts supplier buffers. The inaccessible core, reflected environment, and correctness reference are not observer storage.

A retained output is live memory. The no-early-export rule can make operational error nonmonotone in the mandatory delivery horizon.

The exact benchmark library contains $H,S,S^{†},X,Y,Z,\mathrm{CX},\mathrm{CZ}$, zero allocation, destructive computational readout, classical control, and return. A classically selected Pauli costs at most one gate. Parallel register readout takes one instruction round. Allocation, transfer, and return are instantaneous in this convention.

Preparing $I_2/2$ uses one recyclable slot and at most two gates: generate a random bit by $H$ and destructive readout, then reallocate and conditionally apply $X$.