Circuit-to-open-graph resource encoding with normalized transport rounds

Let $C$ be a finite J/CZ circuit and let $\mathcal{T}(C)$ have rounds $(M_s,({\alpha }_{s1},\dots ,{\alpha }_{sn}){)}_{s=1}^R$ and recorded scalar ${\sigma }_C$ as in Normalized transport schedule for a J/CZ circuit. The schedule index $j\in \{1,\dots ,n\}$ denotes source circuit wire $j-1$; thus a schedule edge $\{j,k\}$ corresponds to the source-wire pair $\{j-1,k-1\}$. Coordinate $j$ of every frame vector denotes this same wire. The encoded graph $G_C$ has vertices $v_{j,s}$ for $1\le j\le n$ and $0\le s\le R$, horizontal edges $\{v_{j,s-1},v_{j,s}\}$, and vertical edges $\{v_{j,s-1},v_{k,s-1}\}$ for $\{j,k\}\in M_s$. It is a finite simple graph by Normalized transport encoding is a finite simple graph of bounded size. Its input vertices are $I=\{v_{j,0}:1\le j\le n\}$, its outputs are $O=\{v_{j,R}:1\le j\le n\}$, and its ordered vertex register follows wire index first and then column index. In particular, the inherited orders on $I$ and $O$ are $v_{1,0},\dots ,v_{n,0}$ and $v_{1,R},\dots ,v_{n,R}$, matching source circuit wire order $0,\dots ,n-1$. The pattern’s measurement order on the vertices $v_{j,s-1}$ is round-major, wire-minor, for $s=1,\dots ,R$ and $j=1,\dots ,n$.

The adaptive basis angles and terminal correction are defined from the outcome blocks $b^s=(b_1^s,\dots ,b_n^s)$ as follows. Let $A_s$ be the binary adjacency matrix of $M_s$ with vertex order $1,\dots ,n$ (vertex $j$ is position $j-1$ in the zero-based order of Binary adjacency matrix of an ordered finite simple graph), so $(A_s{)}_{jk}=1$ exactly when $\{j,k\}\in M_s$. Matrix-vector products and all frame vectors use these coordinates; binary dot products mean $u\cdot v={\sum }_{j=1}^n{u}_j{v}_j\in {\mathbb{F}}_2$. The frame functions are defined recursively by $x^0=z^0=0\in {\mathbb{F}}_2^n$ and the following formulas. Before round $s$, define

$z^{s,\ast }:=z^{s-1}\oplus A_s{x}^{s-1},{\theta }_{sj}:=(-1{)}^{x_j^{s-1}}{\alpha }_{sj}\mathrm{.}$

For an outcome block $b^s$, the post-round frame is defined by $x^s:=b^s\oplus z^{s,\ast }$ and $z^s:=x^{s-1}$. The function of earlier outcomes given by these recurrences selects the equatorial measurement basis at each vertex. For a complete transcript $\mathbf{b}$, the terminal frame and correction are the following functions. For $n\ge 1$, the zero-scalar Pauli frame is $P(x,z):=X^x{Z}^z=P(0,x,z)$ from Phase aware n qubit pauli group, and the terminal correction is $C(\mathbf{b}):=P(x^R,z^R{)}^{†}$. For $n=0$, the empty frame is $P((),()):=I_{\mathbb{C}}$ and the correction is the identity on the one-dimensional register.

For $n\ge 1$, define $\gamma (x,z,\alpha )=1$ if $x=0$ and $\gamma (x,z,\alpha )=(-1{)}^z{e}^{-i\alpha }$ if $x=1$, as in Adaptive J angle and Pauli-frame update, and put

${\lambda }_{\mathbf{b}}:={\sigma }_C{\prod }_{s=1}^R[(-1{)}^{{\sum }_{\{j,k\}\in M_s}{x}_j^{s-1}{x}_k^{s-1}}{\prod }_{j=1}^n\gamma (x_j^{s-1},z_j^{s,\ast },{\alpha }_{sj})]\mathrm{.}$

For $n=0$, the empty transcript has scalar $\lambda =1$. The encoding retains these exact scalars; no global phase is discarded. Its graph has $n(R+1)$ vertices and $R\le 3d$ for $d$ original layers. The item Normalized transport rounds preserve the circuit up to a recorded phase proves that the normalized round circuit is ${\sigma }_C{U}_C$; the branch theorem proves the resulting pattern's relation to $U_C$.