Adjacent Gaussian gates are exactly special-unitary parity blocks, with common phases separated

For every natural number $n$, every injection $w:\{0,1\}\hookrightarrow\{0,\ldots,n-1\}$ satisfying $w(1)=w(0)+1$, and every complex matrix $U$ indexed by $n$-bit strings, the following two equivalences hold. To specify them, let $c_{2k}=Z^{\otimes k}\otimes X\otimes I^{\otimes(n-k-1)}$ and $c_{2k+1}=Z^{\otimes k}\otimes Y\otimes I^{\otimes(n-k-1)}$, where $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$, and $Z=\operatorname{diag}(1,-1)$, with Boolean indices ordered false, true. For a real $2n\times2n$ matrix $h$, set $G(h)=\exp\!\left(i\left[i\sum_{r\ne s}h_{rs}c_rc_s\right]\right)$. The conditions on $h$ below are $h_{rs}=-h_{sr}$ and $h_{rs}=0$ whenever either $\lfloor r/2\rfloor$ or $\lfloor s/2\rfloor$ lies outside $\{w(0),w(1)\}$. For two Boolean-indexed matrices $A,B$, define $K(A,B)$ on two-bit strings by setting its $(x,y)$ entry to zero when $x_0\mathbin{\mathrm{xor}}x_1\ne y_0\mathbin{\mathrm{xor}}y_1$, to $A_{x_0,y_0}$ when both parities are false, and to $B_{x_0,y_0}$ when both are true. Its embedding $E_w(K)$ has $(x,y)$ entry $K(x\circ w,y\circ w)$ if $x$ and $y$ agree on every wire outside the image of $w$, and zero otherwise. First, there exists an $h$ satisfying the stated conditions with $U=G(h)$ if and only if there exist unitary $2\times2$ matrices $A,B$, each of determinant $1$, with $U=E_w(K(A,B))$. Second, there exist unitary $2\times2$ matrices $A,B$ with $\det A=\det B$ and $U=E_w(K(A,B))$ if and only if there exist $z\in\mathbb C$ with $|z|=1$ and an $h$ satisfying the stated conditions with $U=zG(h)$. Such an injection $w$ cannot exist when $n<2$.