QSP signal matrix and ordered phase sequence

Let $x\in [-1,1]$, put $s(x)=\sqrt{1-x^2}$ using the nonnegative real square root, and define the Hermitian-reflection signal matrix $W_R(x)=\left(\begin{array}{cc}x& s(x)\\ s(x)& -x\end{array}\right)\mathrm{.}$ For each real $\varphi$, define $Z(\varphi )=\mathrm{diag}(e^{i\varphi },e^{-i\varphi })$. For an integer $d\ge 0$ and real phase list $\mathrm{\Phi }=({\varphi }_0,\dots ,{\varphi }_d)$, the length-$d$ QSP sequence is the ordered matrix product $U_{\mathrm{\Phi }}(x)=Z({\varphi }_0){\prod }_{j=1}^d(W_R(x)Z({\varphi }_j)),$ where the product is written left to right in increasing $j$ and matrix multiplication acts right to left. For $d=0$ this is $Z({\varphi }_0)$. This definition fixes the sign of the off-diagonal signal entry, both endpoint phases, and the product order.

The phases are in matrix order: when the factors are applied as chronological circuit layers, the rightmost factor acts first, in accordance with Denotational semantics of a finite layered circuit. In the ordered signal basis, if both coordinates are nonzero, applying $Z(\varphi )$ multiplies the ratio of the second coordinate to the first by $e^{-2i\varphi }$; if either coordinate is zero, that relative phase is undefined, while the displayed matrix action remains defined. This is the relative phase convention fixed by Relative Phase.