Finite open-graph adaptive measurement pattern

An open-graph adaptive measurement pattern consists of an ordered finite simple graph $G=(V,E)$, input and output subsets $I,O\subseteq V$, an ordering $m_1,\dots ,m_q$ of $M:=V\setminus O$, basis-choice functions ${\alpha }_j:\{0,1{\}}^{j-1}\to \mathbb{R}$, and a terminal correction function $C:\{0,1{\}}^q\to U({\mathcal{H}}_O)$. For each subset $S\subseteq V$, write ${\mathcal{H}}_S:={⨂}_{v\in S}{\mathbb{C}}^2$ in inherited vertex order, with ${\mathcal{H}}_{\mathrm{\varnothing }}=\mathbb{C}$. The tensor register ${\mathcal{H}}_V$ and all wire permutations use the fixed vertex order. Let ${\mathrm{\Pi }}_{I,V\setminus I}:{\mathcal{H}}_V\to {\mathcal{H}}_I\otimes {\mathcal{H}}_{V\setminus I}$ be the wire permutation listing $I$ first and then $V\setminus I$, each in inherited order. Let $a_{+}:\mathbb{C}\to {\mathbb{C}}^2$ be the linear map $c\mapsto c\mathrm{\mid }+⟩$, and let $a_{+,V\setminus I}:={⨂}_{v\in V\setminus I}{a}_{+}$ in inherited order. Using the empty-tensor identifications, define the ordered input map ${\iota }_I:={\mathrm{\Pi }}_{I,V\setminus I}^{-1}(I_{{\mathcal{H}}_I}\otimes a_{+,V\setminus I}):{\mathcal{H}}_I\to {\mathcal{H}}_V\mathrm{.}$ Thus the input is unchanged and each complementary vertex receives $\mathrm{\mid }+⟩$. Apply $U_G:={\prod }_{\{u,v\}\in E}^{\mathrm{l}\mathrm{e}\mathrm{x}}C{Z}_{uv}$. At step $j$, after transcript prefix $b_1,\dots ,b_{j-1}$, measure vertex $m_j$ in the ordered basis $\{\mathrm{\mid }{b}_j{⟩}_{{\alpha }_j(b_{<j})}:b_j\in \{0,1\}\}$ from J gate and ordered equatorial measurement vectors. The measured systems are removed and the vertices in $O$ are retained in their inherited order.

Let ${\mathrm{\Pi }}_{M,O}$ be the tensor permutation listing $m_1,\dots ,m_q$ first and the vertices of $O$ in inherited order. For a full transcript $\mathbf{b}=(b_1,\dots ,b_q)$, define the linear measurement contraction $B_{\mathbf{b}}:=({⨂}_{j=1}^q{}_{{\alpha }_j(b_{<j})}\text{\hspace{0.17em}⁣}⟨b_j{\mathrm{\mid }}_{m_j})\otimes I_{{\mathcal{H}}_O},$ where the bras are the linear functionals from Dirac Notation Conversion, their tensor map is formed in $m_1,\dots ,m_q$ order, so $B_{\mathbf{b}}:{\mathcal{H}}_{m_1}\otimes \cdots \otimes {\mathcal{H}}_{m_q}\otimes {\mathcal{H}}_O\to {\mathcal{H}}_O$. The scalar output ${\mathbb{C}}^{\otimes q}$ is identified with $\mathbb{C}$ by the tensor-unit convention. This tensor map is supplied by Tensor products of finitely many linear maps; for $q=0$ its measured-register factor is the identity on the empty tensor product. The branch map is the ordered composition

$K_{\mathbf{b}}=B_{\mathbf{b}}{\mathrm{\Pi }}_{M,O}{U}_G{\iota }_I:{\mathcal{H}}_I\to {\mathcal{H}}_O\mathrm{.}$

The measured systems are removed and the vertices in $O$ are retained in their inherited order. This pattern uses $\mathrm{\mid }V\mathrm{\mid }$ qubit registers, its width in the convention of Finite circuit resource accounting. Its projective measurement count is $q$, counting one occurrence for each listed vertex $m_j$. On a normalized pure input its probability is $\mathrm{\parallel }{K}_{\mathbf{b}}\psi {\mathrm{\parallel }}^2$; for a density operator $\rho$ it is $\mathrm{tr}(K_{\mathbf{b}}\rho K_{\mathbf{b}}^{†})$. The conditional output is normalized only when that probability is positive. The corrected branch is $C(\mathbf{b})K_{\mathbf{b}}$; the correction does not change the branch probability. This is a finite-dimensional mathematical circuit and instrument semantics, with no physical postulate or implementation claim.

If $I=\mathrm{\varnothing }$, the inputless resource is the graph state $\mathrm{\mid }G⟩$. If $M=\mathrm{\varnothing }$, the transcript is empty and no conditional normalization occurs. A zero-probability transcript has no normalized conditional output.