Every finite open-graph pattern is a measurement circuit

Construct the finite graph operation list by adjoining $\mathrm{\mid }+⟩$ on $V\setminus I$, applying one $CZ$ for each edge, and then applying the pattern's ordered projective measurements with the transcript-dependent bases. The finite edge-gate list denotes its chronological product by [F3], and the ordered measurement contractions compose after that product. The resulting branch maps are exactly the $K_{\mathbf{b}}$ of [F1], with input register ${\mathcal{H}}_I$, output register ${\mathcal{H}}_O$, and finite outcome set $\{0,1{\}}^q$. This finite specification has one wire per vertex, one $CZ$ occurrence per edge, and one measurement occurrence per listed measured vertex, so [F5] gives the stated resource coordinates. The set of exact angle and correction labels is finite because both the vertex set and transcript set are finite; no finite-bit encoding or synthesis-time claim follows.

For each transcript define ${\mathcal{I}}_{\mathbf{b}}(X):=K_{\mathbf{b}}X{K}_{\mathbf{b}}^{\ast }$. Each branch map is completely positive: for any finite reference register and positive $Y$, its extension sends $Y$ to $(I\otimes K_{\mathbf{b}})Y(I\otimes K_{\mathbf{b}}^{\ast })$, which is positive. Let $A:={\sum }_{\mathbf{b}}{K}_{\mathbf{b}}^{\ast }{K}_{\mathbf{b}}$. For every unit vector $u$, [F6] applied to $\mathrm{\mid }u⟩⟨u\mathrm{\mid }$ gives $⟨Au,u⟩={\sum }_{\mathbf{b}}\mathrm{\parallel }{K}_{\mathbf{b}}u{\mathrm{\parallel }}^2=1$. In an orthonormal basis, applying this identity to each basis vector shows the diagonal entries of $A-I$ vanish; applying it to $(e_j+e_k)\mathrm{/}\sqrt{2}$ and $(e_j+i{e}_k)\mathrm{/}\sqrt{2}$ shows the real and imaginary parts of each off-diagonal entry vanish. Hence $A=I$. By trace cyclicity [F8], ${\sum }_{\mathbf{b}}\mathrm{tr}({\mathcal{I}}_{\mathbf{b}}(X))=\mathrm{tr}(XA)=\mathrm{tr}(X)$ for every operator $X$. Thus these maps form a finite instrument from ${\mathcal{H}}_I$ to ${\mathcal{H}}_O$ by [F7].

For density input $\rho$, step 1.1 gives exactly the same raw branch state $K_{\mathbf{b}}\rho K_{\mathbf{b}}^{\ast }$ in the raw circuit and raw pattern. Its trace is $p_{\mathbf{b}}$, and division by that trace gives the same ${\rho }_{\mathbf{b}}$ when $p_{\mathbf{b}}>0$. The controller in [F4] stores the observed transcript and appends $C(\mathbf{b})$, giving the exact corrected branch operator $L_{\mathbf{b}}=C(\mathbf{b})K_{\mathbf{b}}$ in both corrected presentations. Unitarity gives $L_{\mathbf{b}}^{\ast }{L}_{\mathbf{b}}=K_{\mathbf{b}}^{\ast }{K}_{\mathbf{b}}$, so cyclicity [F8] gives the same probability. Normalizing gives $C(\mathbf{b}){\rho }_{\mathbf{b}}C(\mathbf{b}{)}^{\ast }$ on positive branches. There is no normalized output on zero-probability branches.

Retaining the transcript associates each $\mathbf{b}$ to the raw map $X\mapsto K_{\mathbf{b}}X{K}_{\mathbf{b}}^{\ast }$ or to the corrected map $X\mapsto L_{\mathbf{b}}X{L}_{\mathbf{b}}^{\ast }$, respectively. Corresponding maps agree exactly in the two raw or two corrected presentations. The corrected maps are completely positive by the same reference-register calculation as in step 1.2, and ${\sum }_{\mathbf{b}}{L}_{\mathbf{b}}^{\ast }{L}_{\mathbf{b}}=I$ by step 1.2 and step 2.1, so they too form an instrument. Summing the corrected maps therefore gives equal trace-preserving channels after discarding the transcript. These are exact operator identities before passing to branch states, so no scalar phase has been discarded. By [F4]–[F5], the declared cost of the selected suffix call, or every occurrence and extra wire in its supplied construction, must be charged in addition to the raw prefix coordinates. Finiteness alone provides no access or efficiency bound.