Fixed-degree Clifford-intertwined Gaussian circuits and their boundary interpretation

For every natural number $d$, every intertwined family $F$, and every collection of normalized single-qubit vectors $\phi_{s,k}\in\mathbb C^2$, indexed by finite Boolean strings $s$ and wires $k<n(s)$ of its base width, assume that $F$’s structural code is polynomial-time computable, its numerical data is effectively approximable, the input-factor data is effectively approximable, and every stored observable word has length at most $d$. The structural code is the tuple of the base code $\operatorname{tuple}[1^{n(s)},1^{L(s)}]$, the unary measured-wire index, and the nested tuple of unary generator indices in each stored observable word. The numerical data lists base gate-matrix entries, then the four entries of each conjugated-generator factor, then observable coefficients; the input data lists each $\phi_{s,k}(\mathrm{false})$ followed by $\phi_{s,k}(\mathrm{true})$. Effective approximability of either numerical list $D(s)$ means that there exists a polynomial-time string function $a$ such that, for every position $j<|D(s)|$, precision $m\in\mathbb N$, and Boolean $b$,

Tuple encoding prefixes each constituent string with its unary length and a false bit; $\operatorname{dyad}_m$ interprets the first output bit as a negative-sign flag and the remaining bits as a binary magnitude divided by $2^m$, with an empty output representing zero. For each $s$, put $n=n(s)$, let $q<n$ be its measured wire, and define $c_i$, $i<2n$, as tensor products of $Z=\operatorname{diag}(1,-1)$ on wires preceding $\lfloor i/2\rfloor$, $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ or $Y=\begin{pmatrix}0&-\mathrm i\\\mathrm i&0\end{pmatrix}$ on that wire according as $i$ is even or odd, and identity afterward. The family supplies a unitary $T_s$ normalizing the subgroup generated by all $\mathrm i^a\bigotimes_{k<n}P_k$, with $a\in\{0,1,2,3\}$ and $P_k\in\{I,X,Y,Z\}$. Its conjugated generators $c'_i=T_s^\dagger c_iT_s$ have the supplied tensor factorizations, and its distinct increasing observable words satisfy $\sum_{(w,\alpha)}\alpha c'_{w_1}\cdots c'_{w_{|w|}}=Z_q$, where $Z_q$ applies $Z$ on wire $q$ and identity elsewhere. For any generator collection $a=(a_i)$, define

using the stored real skew-symmetric gate matrices $h_1,\ldots,h_L$. Then there exists a polynomial-time string function $f$ such that, for every $s$, every $m\in\mathbb N$, and both Booleans $b$,

Moreover, for every $s$, the following equality of complex inner products holds:

Here the inner product is conjugate-linear in its first argument. The statement includes $d=0$, empty observable words, empty gate lists, and $m=0$; empty products are identities, and the measured wire requires $n(s)>0$.