Quantum and classical ports

A finite network is a directed acyclic circuit of channels with input and output ports, fixed initial ancillas, and declared discarded outputs. Its vertices include:

• quantum and classical register states;

• local gates and routed wire segments;

• ancilla preparations;

• destructive measurements and their outcome registers;

• resets, discards, guards, and classical feed-forward;

• every classical variable read to select a later operation.

For a conservative dependency graph, connect every input port of a gate to every output port of that gate. A tighter graph is allowed only with a proved joint-output support certificate for that gate. Identity wires have their ordinary directed edges.

Classical registers use a distinguished basis. Controller maps first read the declared classical blocks and implement fixed branch maps. They are channels independent of an unknown input density operator. A canonical pinching extension allows the finite matrix-channel arguments to apply on a common ambient matrix space.