Exact encoded range-two matchgate simulation

There exists a positive natural number $C$ such that the following holds for every source family $F$. Such a family assigns to every finite Boolean string $s$ a width $n_s$ and a finite ordered list of gates, each either an arbitrary one-qubit unitary on a specified wire or a controlled-$Z$ gate on two specified wires whose indices satisfy $i+1=j$. Assume $n_s>0$ for every $s$, and assume the family is effective in this precise sense: write $u(r)$ for $r$ consecutive true bits and define $\tau([v_1,\ldots,v_\ell])$ by concatenating $u(|v_j|)$, one false bit, and $v_j$ for each $j$; the string $\tau([u(n_s),u(L_s)]\mathbin{+\!\!+}\text{gate descriptions})$, where $L_s$ is the gate-list length and a gate description is $\tau([[{\rm false}],u(i)])$ for a one-qubit gate or $\tau([[{\rm true}],u(i),u(j)])$ for controlled-$Z$, is computable from $s$ in polynomial time; moreover, the list of numerical data, obtained by listing each one-qubit matrix’s entries in the order $(00,01,10,11)$ and listing no entries for controlled-$Z$, has a polynomial-time approximation procedure. Specifically, on $\tau([s,u(j),u(r),[b]])$, for every valid data index $j$, precision $r\in\mathbb N$, and Boolean $b$, that procedure outputs a string interpreted as a signed binary integer divided by $2^r$, with error at most $2^{-r}$ from the real part of entry $j$ when $b$ is false and its imaginary part when $b$ is true; the first output bit is the sign, true meaning negative, and the remaining bits encode the magnitude in binary, with missing bits defaulting to false. Then there exists a family $G_s$ of ordered gate lists on $4n_s$ wires. Each gate specifies an injection of two local wires into these wires and two Boolean-indexed $2\times2$ unitary matrices $A,B$; its local matrix has entry $0$ between strings of different parity, entry $A_{x_0,y_0}$ when both strings have even parity, and entry $B_{x_0,y_0}$ when both have odd parity. Every such gate in $G_s$ satisfies $\det A=\det B$; its two wire indices differ by at most $2$; and $|G_s|\le C L_s$. This new family is effective: its exact code is $\tau([u(4n_s),u(|G_s|)]\mathbin{+\!\!+}[\tau([u(i_g),u(j_g)])]_g)$, and its numerical data list concatenates the four entries of $A_g$ followed by the four entries of $B_g$ for every gate, with polynomial-time approximation in the same sense just specified. For every $s$, every source wire $k$ with index $0$, every target wire $q$ with index $0$, every unit vector $\psi$ on length-$n_s$ bit strings, and every Boolean $b$, the conclusion is

Here $V_s$ and $W_s$ apply the source and target gates, respectively, in their listed order, each gate acting identically on untouched wires, and

The assertion covers every finite string $s$, including the empty string, and empty source gate lists; the length bound then forces the corresponding target gate list to be empty. The probability equality concerns the specified index-zero wires and all normalized input vectors.