Nearest-neighbour PQP matchgate languages lie in deterministic P

For every language $L$ of finite Boolean strings, suppose there exists a family of widths $w(s)$, finite gate lists, and output wires $o(n)<w(\mathrm{true}^{\,n})$, with $n\le w(\mathrm{true}^{\,n})$ for every $n\in\mathbb N$, together with a polynomial $p$ with natural-number coefficients satisfying $p(n)\ge2$ for every $n$. Every gate has unitary Boolean-indexed complex $2\times2$ matrices $A_g,B_g$ and an injective ordered wire pair $v_g$; at each unary string require $\det A_g=\det B_g$ and $|v_g(0)-v_g(1)|\le1$. Its local matrix uses $A_g$ on even-parity rows and columns, $B_g$ on odd-parity rows and columns, indexing each block by the first bit, and zero between different parities, while leaving all other wires unchanged. Put $u(j)=\mathrm{true}^{\,j}$ and $T([a_1,\ldots,a_k])=\big\Vert_j(u(|a_j|)\,\mathrm{false}\,a_j)$. Assume polynomial-time computability of $s\mapsto T([u(w(s)),u(\text{gate count})]\mathbin{+\!\!+}[T([u(v_g(0)),u(v_g(1))]):g\text{ in order}])$ and $x\mapsto u(o(|x|))$, and polynomial-time approximation of the complex gate data as follows: for every valid entry index $j$, precision $m\in\mathbb N$, and Boolean $b$, input $T([s,u(j),u(m),[b]])$ yields a number within $2^{-m}$ of the real part if $b=\mathrm{false}$, or imaginary part if $b=\mathrm{true}$. Entries are listed gate by gate, $A_g$ before $B_g$, each in order $(0,0),(0,1),(1,0),(1,1)$; an output string denotes its binary tail divided by $2^m$, negated when its first bit is true, with missing bits zero. Let $U_n$ be the reverse-order product of the embedded gate list at $\mathrm{true}^{\,n}$, the empty product being the identity, and let $a_x$ be $x$ padded with false to that width. Suppose that, for every $x$, the quantity $\sum_{z:\,z_{o(|x|)}=\mathrm{true}}|(U_{|x|})_{z,a_x}|^2$ is at least $\frac12+2^{-p(|x|)}$ when $x\in L$, and at most $\frac12-2^{-p(|x|)}$ when $x\notin L$. Then a polynomial-time Boolean-string function exists which returns exactly [true] on $L$ and exactly [false] outside $L$. Polynomial time throughout means computability by a TM2 Turing machine with identity input and output encodings, and every input condition includes the empty string.