Proposition: Explicit path-and-computation realization

Suppose a supplied path has $L$ edges and $n$-qubit memories at its $L+1$ vertices, admits adjacent SWAP operations, and ends at a laboratory supporting the declared circuit operations.

Then a finite protocol sends the payload along the path by $nL$ SWAP gates and executes $A^{O_f}$ at the receiver. A ledger retaining all blank memories has peak at most

qubits in the oracle model, $3nL$ routing CNOT gates, $g$ ordinary computation gates, and $q$ oracle calls.

Using Proposition prop:lookup instead of a primitive oracle adds $O(m+1)$ scratch and gives ordinary-gate count

The output is not available until routing and computation finish.

Proof. At each edge, SWAP the $n$ message qubits into the next blank memory. Induction transfers the entire message-reference operator and leaves previous memories blank. Each qubit SWAP is three CNOTs. The path memories number $n(L+1)$; the receiver uses another $w-n$ circuit workspace qubits. This gives $w+nL$. Apply the circuit or its explicit oracle realization.