Bounded-error nearest-neighbour matchgate languages lie in deterministic P

For every language $L$ of finite Boolean strings, suppose there exists a family assigning widths $w(s)$ and finite gate lists to all strings $s$, with output wires $o(n)<w(\mathrm{true}^{\,n})$ and $n\le w(\mathrm{true}^{\,n})$ for every $n\in\mathbb N$. Each gate consists of unitary Boolean-indexed complex $2\times2$ matrices $A_g,B_g$ and an injective ordered pair of wires $v_g$; at unary strings, require $\det A_g=\det B_g$ and $|v_g(0)-v_g(1)|\le1$. Its local matrix uses $A_g$ on the even-parity block and $B_g$ on the odd-parity block, each indexed by the first bit, is zero between blocks, and leaves other wires unchanged. Require the following effectiveness conditions: putting $u(j)=\mathrm{true}^{\,j}$ and $T([a_1,\ldots,a_k])=\big\Vert_j(u(|a_j|)\,\mathrm{false}\,a_j)$, the functions $s\mapsto T([u(w(s)),u(\text{gate count})]\mathbin{+\!\!+}[T([u(v_g(0)),u(v_g(1))]):g\text{ in order}])$ and $x\mapsto u(o(|x|))$ are polynomial-time computable, and there is a polynomial-time function returning approximations within $2^{-m}$ to each real or imaginary gate-matrix entry at input $T([s,u(j),u(m),[b]])$. Here $j$ indexes the list of entries of $A_g$ followed by $B_g$, in gate order and entry order $(0,0),(0,1),(1,0),(1,1)$; $b=\mathrm{false}$ selects the real part and true the imaginary part; and the output is read as 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 embedded gates at $\mathrm{true}^{\,n}$, with the empty product the identity, and $a_x$ the input $x$ padded with false to that width. Suppose, for every $x$, the sum $\sum_{z:\,z_{o(|x|)}=\mathrm{true}}|(U_{|x|})_{z,a_x}|^2$ is at least $2/3$ when $x\in L$ and at most $1/3$ when $x\notin L$. Then there exists a polynomial-time function on Boolean strings returning exactly [true] for members of $L$ and exactly [false] for nonmembers. Every polynomial-time requirement here means computability by a TM2 Turing machine with identity input and output encodings; all conditions include the empty input.