Nearest-neighbour matchgates admit strong classical simulation

Let $F$ assign to every finite Boolean string $s$ a natural width $w(s)$ and a finite ordered list of applications, each containing unitary $2\times2$ complex matrices $A_g,B_g$ and two distinct ordered wire indices. Let $I$ assign a unit-norm two-component complex vector $\phi_{s,i}$ to each wire $i<w(s)$. Assume that every application satisfies $\det A_g=\det B_g$, and that the absolute difference between its two wire indices is at most $1$. Assume also the following effectiveness conditions. Write $u(k)$ for $k$ true bits and $T$ for the encoding that concatenates, for each constituent string $x$, the prefix $u(|x|)$, a false bit, and $x$. The structural string $T([u(w(s)),u(L),T([u(i_1),u(j_1)]),\ldots,T([u(i_L),u(j_L)])])$, where $L$ is the application count, must be computable exactly in polynomial time. For each of the two complex data lists—first, the entries $00,01,10,11$ of $A_g$ followed by those of $B_g$, application by application; second, the false and true components of $\phi_{s,i}$, wire by wire—there must be a polynomial-time string function which, on $T([s,u(j),u(m),[b]])$, approximates the indicated entry’s real component when $b$ is false or imaginary component when $b$ is true, to absolute error at most $2^{-m}$, for every valid index $j$ and natural precision $m$. Here an output string is interpreted as a signed binary integer divided by $2^m$, with its first bit indicating a negative sign when true and its remaining bits giving the binary magnitude; missing bits contribute zero. Define the input vector by $\Psi_s(x)=\prod_{i<w(s)}\phi_{s,i}(x_i)$. An application acts on its ordered pair of wires by the matrix whose $x,y$ entry is zero if $x_0\mathbin{\mathrm{xor}}x_1\ne y_0\mathbin{\mathrm{xor}}y_1$, and otherwise is $(B_g)_{x_0y_0}$ for odd parity and $(A_g)_{x_0y_0}$ for even parity; embedding it on all wires gives zero between strings that differ outside the selected pair. Let $U_s$ be the product of these embedded matrices in reverse list order, so the first listed application acts first. Then there exists a polynomial-time computable string function $f$ such that, for every $s$, natural $m$, wire $k<w(s)$, and Boolean $b$, the signed dyadic value of $f(T([s,u(m),u(k),[b]]))$ approximates

to absolute error at most $2^{-m}$. Moreover, for every such $s,k,b$,

where $Z_k$ is diagonal with entry $1$ on strings whose $k$-th bit is false and $-1$ on strings whose $k$-th bit is true, and the inner product conjugates its first argument. Empty application lists give the identity. Width zero is allowed, with an empty application list and an empty product defining $\Psi_s=1$; all conclusions requiring a wire index are then vacuous.